Evaluation and simulation of thermodynamic data estimationsestak/yyx/DataEvaluation.pdf ·...

Evaluation and simulation of thermodynamic data estimation

G. Moiseev, J. Leitner, J. Sestak, V. Zhukovsky EMPIRICAL DEPENDENCES OF THE STANDARD ENTHALPY OF FORMATION FOR RELATED INORGANIC COMPOUNDS ENHANCING GLASS FORMERS Thermochimica Acta 280/281 (1996) 511 521 G. K. Moiseev and J. Sestak SOME CALCULATIONS METHODS FOR ESTIMATION OF THERMODYNAMICAL AND THERMOCHEMICAL PROPERTIES OF INORGANIC COMPOUNDS Prog. Crystal Growth and Charact. Vol. 30, pp. 23-81,1995 J. Sestak, J. Leitner, H. Yokokawa , B. Stepanek THERMODYNAMICS AND PHASE EQUILIBRIA DATA IN THE S-GA-SB SYSTEM AUXILIARY TO THE GROWTH OF DOPED GASB SINGLE CRYSTALS Thermochimica Acta 245 (1994) 189-206 J. Šesták, V. Šestákova, Ž. Živkovič, D.Živkovič ESTIMATION OF ACTIVITY DATA FOR THE GA-SB, GA-S AND SB-S SYSTEMS REGARDING THE DOPED GASB SEMICONDUCTOR SINGLE CRYSTALS Pure Appl Chem 67 (1995) 1885 B. Stepánek, J. Sesták, J.J. Mares and V. Sestáková THERMAL CONDITIONS OF GROWTH AND THE NECKING EVOLUTION OF Si, GaSb AND GaAs: Glide phenomenon in the gas bowl Journal of Thermal Analysis and Calorimetry, Vol. 72 (2003) J. Sestak, D. Sedmidubsky and G. Moiseev SOME THERMODYNAMIC ASPECTS OF OXIDE HIGH Tc SUPERCONDUCORS: Data evaluation Journal of Thermal Analysis, Vol. 48 (1997) 1105-1122

therm0chimica acta

E L S E V I E R Thermochimica Acta 280/281 (1996) 511 521

Empirical dependences of the standard enthalpy of formation for related inorganic compounds enhancing glass formers 1

G. M o i s e e v a'*, J. Le i tne r b, J. Sestfik c, V. Z h u k o v s k y a

a lnstitute of Metallurgy, Ural Division of Russian Academy ~f Sciences, Amundsen str. 101, 620016 Ekaterinbur g, Russia

b Department of Solid State Engineerin 9, Institute of Chemical Technology, Technickd 5, 166 28 Praha 6, Czech Republic

¢ Division of Solid-State Physics, Institute of Physics, Academy of Sciences c~f the Czech Republic', Cukrovarnickd 162 O0 Praha 6, Czech Republic

d Department of Analytical Chemistry, Ural State University, Lenin str. 51,620291 Ekaterinburq, Russia

Abstract

For the double oxides and other related double AxByC z compounds in the system AC BC, linear correlations have been observed between standard enthalpies of formation of the double c o m p o u n d s AxByC z from the component compounds AC and BC (AH°cc in kJ (g-atom)- 1 and values of the sum of the molar fraction enthalpies of the component compounds AC and BC (A/tf ° = XAcAH°(BC), in kJ (g-atom) 1 In general, the dependence of H ° =f (A/4f °) has

• f , c c

a minimum, its branches being described with the help of linear equations (the average deviation from the known AH°cc values was less than + 4.7% for 121 double compounds in 34 systems). A conclusion has been drawn that these regularities (linear approximation rule LAR) are characteristic for different types of inorganic double compound•

Keywords: Double compounds; Double oxides; Enthalpy; Pseudobinary systems; System AC BC

1. Introduction

Several methods of calculation have been proposed for the determination of s tandard enthalpies of complex compounds [1 8]. These methods are usually based on certain regularities assuming relative and/or similar substances•

* Corresponding author. Dedicated to Professor Hiroshi Suga.

0040-6031/96/$15.00 ~ 1996 - Elsevier Science B.V. All rights reserved SSDI 0 0 4 0 - 6 0 3 1 ( 9 5 ) 0 2 6 6 2 - 2

512 G. Moiseev et al./Thermochimica Acta 280/281 (1996) 511 521

In this report we present some empirical dependences for standard enthalpies of formation of double oxides and other double compounds from the component compounds (AH°.c~, in kJ (g-atom)- ~ as well as some relationships between the AH°c~ values and values of the standard enthalpies of formation of the component compounds (A Hf °, in kJ (g-atom) ~. The information has been collected from the analysis of 34 pseudobinary systems, i.e. oxide-oxide (26), compound-H20 (5), halide-halide (1) and halide NH 3 (2).

2. Method of investigation: oxide-oxide system as an example

For double oxides AxByO z in the system AO BO we have investigated the dependence

A H O (axByOz) = f(A/~r 0) (1)

where AHf°¢c(AxByOz) is a standard enthalpy of formation of the double oxide AxByO z from the component oxides AO and BO and A/7 ° represents the sum of molar fraction enthalpies of the component oxides AO and BO according to the following relation- ship.

A/4 ° = XAoAH°(AO)+ XBO A H ° (BO) (2)

AH°(AO) and AH°(BO) are the standard enthalpies of formation of the component oxides from the elements and XAO and Xuo are the molar fractions of the component oxides in the double oxide AxByO.~ with a given composition. Throughout the paper all enthalpy values are expressed in kJ (g-atom)- 1 of the relevant oxide at a temperature of 298.15 K. AH°.cc (AxByO z) is given in relation to the standard enthalpy of formation of double oxide AxByO z from the elements (AH°el (AxByOz)) by the following equation

anoxc = aHoe, _ A H o (3)

For determination of the shape of the dependence given in Eq. (I) we constructed o (A~SyOz) vs. A/4 ° =f(AH°(AO), AH°(BO)) for each system under graphs of AHfx c

investigation. It is worth noting that IAH°(AO)[ > IAH°(BO)I in all cases. The characteristic groups of dependences Eq. (l) are shown in Fig. I. As it can be seen from this figure, te dependence Eq.(1) exhibits a minimum and its branches can be approximated by linear equations of the form

A n ° , ~ = A + S A/~o (4)

For calculation of the constants A and B linear regression analysis was used with constraints of the following type

AH~,~=0 if XAO=I or XRo=l (5)

3. Results and discussion

Calculated results are given in Table 1 for all 34 systems. All enthalpy values for the simple and double compounds necessary for the calculations were compiled from Refs.

G. Moiseev et al./Thermochimica Acta 280/281 (I996) 511 521

AI~r(AC ) AH~(BC) AH~(AC) AH~(BC)

_ l I ' , ~1' 1 ' ~1~ 1 y " . \ i~

[ II \ . . . . / I I t / " I I I L 1 ¢ I I ~ I [

[ I 1 I A H ~ cc I I I ~ / I aH~,cc

513

IP I II b I

I

I~ ,k/'" I

,Y i '

AJt~(AC) AH~(BC)

, \ ' II I b x ~ ill C

a~? *nL'

AH~(AC) ,',H~(BC)

It ~ I I

f ~ aH~x ~

AH~(AC) AH~(BC)

I

I : t [ I ~,

a~ an~.cc

Fig. 1. Characteristic plots for the individual groups of compounds (groups a g in Table 1: the left-sided example: a (BzO3-Na20), b (V2Os-NasO) and c (UO 3 LizO) types; right-sided examples: d (SiO 2 PbO), e (SrO V205) and f(CaO-B203) types; symmetric example: g (TiO 2 MgO) type. The solid and dashed lines represent known and estimated data respectively, points designated "1" show existing and estimated minimum values. The cases a and d can be determined only after adding further data,

[5 and 9-17] . The average deviation between experimental values of AHf°~c and the values calculated according to the Eq. (4) (with constants A and B presented in Table 1) was less then _+ 4.7% for all the 121 double compounds accountered in the 34 systems

in question. In the "left-sided" systems (Fig. 1, groups b and c) the minimum of A H°,c~ is shifted to

the higher concentrat ions of BC in the complex compounds; in the "right-sided" systems (Fig. 1, groups e and f) it is shifted to the higher concentrat ions of AC in the complex compounds . In the "symmetric" systems (Fig. 1, g roup g) the minimum of AH°.¢¢ is observed for equimolar composi t ions (XAc = XRC). Because of a lack of complete information about the AH°,cc values for groups a and d it is not possible to

514 G. Moiseev et al./Thermochimica Acta 280/281 ( 1996 ) 511 521

Table 1 Input da t a and the cons tan t s of Eq. (4) for var ious pseudob ina ry systems

System AC BC and AH°/ (kJ AHj°cc/(kJ(g-atom) 1 AH°cc = A + B A H ° / ( k J g-a tom) 1 %

double c o m p o u n d s g-a tom) Exp. Calc. A B

Group "a"

S i O 2 - R b 2 0 [9]

1. SiO 2 303.30 0 0 2. Rb2SiO 3 - 2 0 8 . 2 0 - 4 6 . 5 0 - 4 8 . 4 0

3. Rb2Si20 s 239.90 - 3 4 . 1 0 - 3 2 . 5 0

4. Rb2S i40 9 - 2 6 5 . 2 0 22.10 19.50

5. R b 2 0 - 113.00 0

B203 N a 2 0 [9]

1. B203 254.60 0 0 2. N a B O 2 - 196.35 - 3 3 . 4 0 - 3 7 . 5 0

3. N a 2 B 4 0 7 - 2 1 5 . 8 0 - 2 5 . 5 0 - 2 5 . 0 0

4. Na2B6Olo - 2 2 5 . 5 0 - 2 0 . 6 0 - 1 8 . 8 0 5. N a 2 0 - 138.10 0

T iO 2 N a 2 0 [9]

1. TiO 2 314.07 0 0 2. Na2TiO 3 - 2 2 6 . 0 9 - 3 9 . 1 3 - 3 9 . 1 3

3. Na2Ti205 255.10 26.80 26.05

4. N a 2 T i 3 0 7 --270.10 --20.70 19.60 5. N a 2 0 138.10 0

SiO 2 K 2 0 [9]

1. SiO 2 - 3 0 3 . 3 0 0 0 2. K2Si40 9 - 2 6 6 . 7 0 - 2 1 . 0 0 - 19.8

3. K2S i20 s - 2 4 2 . 2 0 36.00 - 3 2 . 9

4. K2SiO s 211.90 46.10 49.4

5. K 2 0 120.50 0

KCI AICl 3 [9]

1. KCl 218.40 0 0 2. K C I ' A I C 3 - 197.20 8.73 10.50 3 .3KCI 'AICI~ 207.80 8.71 - 5 . 3 0 4. 3KCI '2AICI 3 20t .40 8.44 - 8 . 4 4

5. A1C13 - 176.00 0

Group "b"

V 2 0 s N a 2 0 [9]

1. V 2 0 ~ - 2 2 1 . 5 0 0 0 2. N a V O 3 - 179.80 - 3 2 . 7 0 - 3 0 . 9 0 3. N a 4 V 2 0 v 165.90 - 4 1 . 5 0 - 4 1 . 2 0 4. N a 6 V 2 0 s - 159.00 - 4 5 . 2 0 - 4 6 . 3 0

5. N a 2 0 --138.10 0

BaCl 2 HzO [9]

1. BaCI 2 286.20 0 0 2. B a C I 2 . H 2 0 - 180.70 - 2 . 7 0 - 2 . 6

- 1 5 5 . 5 5 5 0.513 0

(points 1 4) - 4 . 9 +4 .7

+11 .6

164.050 - 0.644 0

(points 1 4i - 12.3 +2.O +8 .9

- 139. 700 0.644 0

(points I 4) 0 +2 .7

+5 .6

- 163.848 0.540 0

(points 1 4) +5 .9 +8 .6

- 7 . 4

108.582 - 0 . 4 9 7 0 (points 1 4) - 2 0 . 3

+ 20.9

0

163.854 0.740 0

(points 1 4) +5 .5 +0 .7 - 2 . 4

-- 7.937 - 0.028 0 (points 1 3) - 1.9

G. Moiseev et al./Thermochimica Acta 280/281 (1996) 511-521 515

Table 1. (Cont inued)

System AC BC and double c o m p o u n d s

A/4°/ (kJ AHf°cc/(kJ (g-a tom)-1 AHf°,¢ = A + BA/~°/ (kJ g-a tom) i % g-a tom) 1

Exp. Calc. A B

3. BaC12.2H20 - 158.90 - 3 . 3 0 3.50 4. H 2 0 - 9 2 . 2 8 0

KAI(SO4)2-H20 [9]

1. KAl(SO4) 2 - 2 0 5 . 8 5 0 0

2. K A I ( S O j 2 . 3 H / O -- 122.93 - 2 . 5 4 - 2 . 6 6

3. KaI(SO4) 2 . 1 2 H 2 0 - 103.80 - 3.37 - 3.30 4. H 2 0 92.28 0

B e S O ~ - H 2 0 [9]

1. BeSO 4 200.87 0 0 2. B e S O 4 - 2 H 2 0 - 130.40 3.86 - 3 . 6 6

3. B e S O 4 - 4 H 2 0 116.40 - 4 . 1 8 - 4 . 3 9 4. H2O 92.28 0

Z r O 2 - B a O [5, 9, 11, 13]

1. Z r e O 2 - 366.85 0 0

2. B a Z r O 3 321.80 - 2 5 . 0 9 - 2 5 . 0 9

3. B a 3 Z r 2 0 7 - 3 1 2 . 8 0 31.15 30.97 4. Ba2ZrO 4 306.75 - 3 1 . 8 7 - 3 3 . 4 7 5. BaO 276.75 0 -

C a ( N O 3 ) 2 - H 2 0 [9]

1. Ca(NO3) 2 - 104.30 0 0

2. C a ( N O 3 ) 2 . 2 H 2 0 - 9 6 . 3 0 - 2 . 0 5 - 2 . 0 1 3. C a ( N O 3 ) 2 - 3 H 2 0 - 9 5 . 3 0 - 2 . 3 4 - 2 . 2 6

4. Ca (NO3)2 .4HzO - 9 4 . 7 0 - 2 . 4 1 2.41 5. H 2 0 - 9 2 . 2 8 0

Group "c"

SiO 2 BaO [9]"

1. SiO z - 303.30 0 0 2. BaSiO 3 - 289.90 32.05 33.60

3. BaS i20 5 294.40 - 2 1 . 8 7 - 2 2 . 3 0

4. Ba2SiO 4 - 2 8 5 . 5 0 38.71 - 3 5 . 4 0 5. BazSi30 s - 2 9 2 . 6 0 - 2 6 . 7 9 26.80

6. (BasSisO~8) - 2 8 6 . 9 0 - 4 1 . 0 8 - 4 t . 0 8 7. BaO - 276.60 0 0

P z O s - N a 2 0 [9]

1. P205 - 2 1 3 . 1 4 0 0

2. N a 3 P O 4 - 156.86 - 6 8 . 7 6 --68.90 3. N a 4 P 2 0 7 -- 163.09 66.74 - 66.72 4. N a s P 3 0 ~ 0 -- 166.24 --62.83 --75.04 5. N a : O - 138.1 0

SiO 2 N a z O [9]

1. SiO z 303.30 0 0

--6.1

- 10.428 0.052 0

(points 1-3) + 5.2 - 4 . 9

- 204.296 - 0.557 0

(points 1 4) 0

+0 .6

- 5 . 0

- 2 6 . 2 2 8 - 0 . 2 5 2 0

(points 1 4) + 1.9 - 3 . 6

0

- 7 5 9 . 6 8 0 - 2 . 5 0 5 0

(points 1 3, 5, 6) 4.7 - 1.8

1125.836 4.067 + 8.6

(points 4, 6, 7) 0 0 0

- 2 8 4 . 1 8 0 - 1.333 0

(points 1 4) + 1.0 0 19.4

- 141.277 0.466 0

--6.604 - 0 . 0 3 2 0

(points 1 3) - 4 . 6

2.8

516 G. Moiseev et al./Thermochimica Acta 280/281 (1996) 511-521

Table 1. (Continued)

System AC-BC and double compounds

A/4°/(kJ AH°¢¢/(kJ(g-atom) ' AH~¢¢ = A + BA/4°/(kJ g-atom) ' % g-atom) i

Exp. Calc. A B

2. Na4SiO, , - 193.10 -40.90 -40.90 3. NazSiO a -220.20 -38.50 -38.50 4. NazSizO 5 -248.30 -26.00 -25.90 5. Na20 -138.10 0 0

UO3-Li20 [14]

1. UO3 b -305.13 0 0 2. Li2U3Olo -278.68 -11.87 -10.65 3. LizUO,, -252.22 -21.29 -21.29 4. Li4UO s -234.55 -23.33 -28.40 5. Li20 - 199.31 0 -

AI20 a Na20 [9]c

1. AI20 3 - 333.07 0 0 2. NaAlllOxv -317.05 -5 .70 -3 .92 3. NaAlsO s -300.50 -9.05 -8 .07 4. NaAIO 2 -235.60 -23.83 -23.83 5. NasA10 4 - 160.90 - 10.81 - 5.6 6. Na20 -138.10 0 0

U O 2 ( N O 3 ) 2 H z O [9]

1. UO2(NO3) 2 - 122.70 0 0 2. UO2(NO3)z'HzO - 107.50 -2 .06 -2 .54 3. UO2(NOa)2'2HzO -02.30 -3 .47 -3.41 4. UO2(NO3)2'3HzO - 100.00 -3.71 -3 .80 5. UOz(NO3)z'6HzO -96.60 -3 .60 -3 .60 6. H20 -92.28 0 0

TiOz-SrO [5]

1. TiO z -314.07 0 0 2, SrTiO a -305.04 -27.64 -26.89 3. Sr2TiO 4 - 302.02 - 23.03 - 21.45 4. Sr3Ti20 v -303.23 -25.78 -25.78 5. Sr4Ti3Olo -303.76 -26.69 -27.66 6. (Sr6TisO16) -304.22 -29.32 -29.32 7. SrO - 296.00 0 0

ZrO2-Li20 [16] e

1. Z r O 2 -- 366.85 0 0 2. Li2ZrO 3 --283.08 -7.38 --7.30 3. Li6Zr20 7 -266.33 -7.48 -8.03 4. LisZrO 6 -232.82 -4 .49 -4 .02 5. (Li22ZraOzv) -269.85 -8 .45 -8.45 6. Li20 - 199.31 0 0

Group "d"

Si02-PbO [9]

1. SiO 2 - 303.30 0

(points 1, 3, 4) 102.753

(points 2, 5)

- 122.784 (points 1 4)

-81.418 (points 1 4)

33.752 (points 4-6)

65.452 (points 4, 5)

- 20.570 (points 1-4)

76.751 (points 5, 6)

-934.762 (points 1, 2, 6)

-21.45 1055.530

(points 3 7)

-31.955 (points I, 2, 5)

23.877 (points 3-6)

4.075

0.744

- 0.402

- 0.244

0.244

0.474

-0.168

0.832

-2.976

3.566

- 0.087

0.120

0.037

0 0 + 1.4 0

0 + 10.3 0 -28.4

0 +31.2 +10.8 0 +48.2 0

0 -23.3 + 1.6 -2 .4 0 0

0 +2.7 -6.1 0 -3 .6 0 0

0 + 1.2 - 7.4 + 10.9 0 0

G. Moiseev et al./Thermochimica Acta 280/281 (1996) 511-521

T a b l e 1. (Con t inued )

517

Sys tem A C - B C a n d A/4f°/(kJ AHf°cc / (kJ(g-a tom) - ' AH~,¢c = A + B A / 4 ° / ( k J g - a t o m ) 1 %

d o u b l e c o m p o u n d s g - a t o m ) 1 Exp. Calc. A B

2. P b S i O 3 - 2 0 6 . 2 0 - 3 . 5 3 - 3 . 6 3

3. P b 2 S i O 4 - 173.80 - 2 . 4 2 - 2 . 4 2

4. P b 4 S i O 6 - 147.90 - 1.69 - 1.45 5. P b O - 109.10 0 0

Z r O 2 S r O [5]

1. Z r O 2 - 366.85 0 -

2. S r Z r O 3 - 3 3 1 . 4 3 - 14.95 15.64

3. S r 4 Z r 3 O l o - 3 2 6 . 4 0 - 13.91 - 13.42

4. S r 3 Z r 2 0 v 324.34 - 13.27 - 12.510

5. S r 2 Z r O 4 319.59 11.43 - 10.42

6. S r O - 296.00 0 0

C s 2 0 T e O a [15]

1. C s 2 0 - 1 1 5 . 7 7 0

2. C s 2 T e O 3 111.40 - 54.14 - 59.97

3. C s z T % O s - 1 1 0 . 1 1 - 4 0 . 1 6 - 3 9 . 9 3

4. C s 2 T e 4 0 9 - 109.08 30.55 - 24.00

5. T e O 2 - 107.53 0 0

Group "e"

CaO V 2 0 5 [9]

1. C a O - 3 1 7 . 5 5 0

2. C a V 2 0 6 - 2 6 9 . 5 3 - 15.95 - 16.52

3. C a V z O 7 - 2 8 5 . 5 6 - 2 3 . 9 0 - 2 2 . 0 3

4. C a 3 V 2 0 s - 2 9 3 . 5 4 - 2 4 . 7 8 - 2 4 . 7 8

5. V 2 0 s - 2 2 1 . 5 1 0 0

S r O - V 2 0 ~ [5]

1. S r O - 296.00 0 2. S r V 2 0 6 - 2 5 8 . 2 0 - 2 4 . 4 6 - 2 4 . 4 6

3. S r 2 V 2 0 7 - 2 7 1 . 2 0 - 3 5 . 6 4 - 3 2 . 6 2

4. S r 3 V 2 0 s - 2 7 7 . 3 8 36.16 - 3 6 . 6 8

5. V 2 0 5 - 2 2 1 . 5 1 0 0

N H 3 ( g ) H g I 2 [9]

1. HN3(g) 73.52 0

2. H g I 2' 1 .333NH 3 - 57.05 - 4 7 . 0 5 - 54.60

3. H g I z ' 2 N H 3 - 6 0 . 9 5 - 7 1 . 2 0 - 6 4 . 3 2 4. H g l z . 6 N H 3 - -68 .05 - -84 .08 - -82 .46

5. H g I 2 - -35 .13 0 0

NH3(I ) H g l 2 [9]

1. NH3(1) - -66 .00 0 - 2. Hgl2" 1 .333NH 3 52.76 - 6 3 . 5 2 - 5 5 . 7 0

(points 2 -5 )

130.588 0.441

(poin ts 2 -6 )

1667.670 15.508

(poin ts 2 -5 )

76.195 0 .344

(poin ts 2 -5 )

145.446 0.657

(poin ts 2 5)

87.505 2.491

(points 2 -5 )

110.961 3.159

(poin ts 2 5)

- 2 . 7

0

+ 14.2

0

- -4 .6

+ 3 . 5

+ 5 . 7

+ 8 . 9

0

- 1 0 . 8

+ 0 . 6 + 2 1 . 5

0

- 3 . 6

+ 7 . 7

0

0

0

+ 8 . 5

1.4

0

- 1 6 . 1

+ 9 . 7

+ 1.9

0

+ 12.4


T a b l e 1. (Con t inued)

Sys tem A C BC a n d

d o u b l e c o m p o u n d s

A/4f°/(kJ AHf°cc / (kJ(g-a tom) ' AH~,~ = A + BA/4r°/(kJ g - a t o m ) t % g - a t o m ) 1

Exp. Calc. A B

3. H g I 2 " 2 N H 3 55.94 65.73 65.73

4. H g I 2 - 6 N H s - 6 1 . 6 0 - 7 7 . 4 0 - 8 3 . 6 0

5. H g I : 35.13 0 0

Group "f "

C a O SiO 2 [9]

1. C a O - 3 1 7 . 5 4 0 0

2. C a S i O 3 310.42 - 16.70 16.40

3. C a 2 S i O 4 - 312.80 - 18.90 - 16.40

4. C a 3 S i O 5 314.00 12.70 12.20

5. C a 3 S i 2 0 7 - 3 1 1 . 8 4 - 19.70 - 19.70

6. S iO 2 - 303.30 0 0

C a O T i O 2 [9 ] f

1. C a O - 3 1 7 . 5 4 0 0

2. C a T i O 3 315.81 16.66 16.58

3. C a 3 T i 2 0 v - 3 1 6 . 1 5 - 13.40 14.80

4. C a 4 T i 3 O l o - 3 1 6 . 0 5 - 17.51 - 18.90

5. (Ca9TisO25) 315.90 17.50 17.50

6. T i O z - 3 1 4 . 0 7 0 0

M o O 3 K 2 0 [10] g

1. M o O 3 - 186.27 0 0

2. K 2 M o O 4 - 153.39 - 5 6 . 8 2 - 6 1 . 0 0

3. K z M o 2 0 7 164.37 40.64 40.64

4. K 2 M o 3 O l o 169.83 - -30 .50 - 30.50

5. K 2 M o 4 0 1 3 -- 173.12 - 2 5 . 4 1 - 2 4 . 4 0

6. K z M o s O 2 5 - 179.38 - 14.34 - 12.80

7. K z O -- 121.50 0 0

S r O W O 3 I-5] h

1. S r O - 296.00 0 0

2. S r W O 4 253.36 41.76 - 4 1 . 5 6

3. S r 2 W O s 267.60 43.93 49.94

4. S r 3 W O 6 274.68 - 4 0 . 5 6 - 3 7 . 4 8

5. (SrgWsO24) - 2 6 5 . 6 0 53.16 53.16

6. W O 3 - 2 1 0 . 7 2 0 0

A120 3 C a O [9]

1. A I 2 0 3 - 3 3 3 . 0 7 0

2. C a A I 2 0 4 - 3 2 5 . 3 1 - 3 . 6 9 - 3 . 5 5 3. C a A I 4 0 7 - 327.90 - 5.00 4.74

4. C a 3 A I 2 0 6 - 3 2 1 . 4 3 - 1.57 - 1.76

5. Ca16A12019 - -318 .48 - -0 .53 --0.41

6. Ca12A114033 - 3 2 3 . 3 0 - -2 .57 - -2 .62 7. C a O - 3 1 7 . 5 5 0 0

0

- 8 . 0

0

- 1095.800 3.451 0

(points 1, 3 5) + 1.8

+ 13.2

698.584 1.303 + 3 . 9

(poin ts 2, 5, 6) 0

0

- -3388 .470 - 10.671 0

(points 1, 3 5) + 0 . 5

+ 9 . 5

3003.402 9.563 - 7.4

(poin ts 2, 5, 6) 0

0

- 345.540 1.855 0

(poin ts 1 6) - 7 . 3

0 208.158 1.728 0

(poin ts 6, 7) + 4 . 0

+ 10.7

0

520.920 - 1.760 0

(poin ts 1, 3 5) + 0 . 5

13.6 205.412 0.975 + 7.6

(poin ts 2, 5, 6) 0

0

146.094 0.460

(poin ts 2--7) + 4 . 0

+ 5 . 2

- 1 2 . 1

+ 2 2 . 6 - 2 . 0

0

G. Moiseev et al./Thermochimica Acta 280/281 (1996) 511-521

Table 1. (Continued)

519

Sys temAC B C a n d k/4°/(kJ AH~,~,/(kJ(g-atom) ~ AHOcc=A+BAHO/(kJg_atom) 1 % double compounds g-atom) i

Exp. Calc. A B

C a O - B 2 0 3 [9]

1. CaO 317.55 0 2. CaB20 4 - 286.05 - 17.59 - 15.95 3. CaB40 7 275.53 14.97 - 10.62 4. CazB20 5 -296.57 -21 .27 21.27 5. Ca3B20 6 - 301.80 22.82 - 23.92 6. B20 3 -254.55 0 0

BaO WO 3 [5]i

1. BaO -276.80 0 0 2. BaWO 4 -243.76 -48 .17 44.47 3. Ba2WO 5 -254 .80 53.60 -59 .32 4. Ba3WO 6 -260.28 -52.11 55.51 5. (BasW2010 --257.90 -63 .50 -63 .50 6. W O 3 -210.71 0 0

T i O 2 - M g O [9]

Group "g"

1. 7~O 2 314.07 0 0 2. MgTiO 3 -307.46 5.78 -5 .25 3. MgTizO 5 309.67 - 2 . 9 4 -3 .49 4. Mg2TiO 4 - 305.25 - 2.80 3.49 5. MgO - 300.85 0 0

128.870 (points 2 6)

0.506 +9.3 +29.0

0 - 4 . 8

0

- 929.989 - 3.360 0 (points 1, 4, 5) + 7.7

- 1 0 . 7

283.595 1.346 6.5 (points 2, 3, 5, 6) 0

0

249.450 - 0.794 0 (points 1 3) +9.2

18.8 238.950 0.794 - 24.6 (points 2, 4, 5) 0

The interpretation was made taking into account the phase Ba8Si5018 [17]. b A Hf °(UO3) was taken as the arithmetic mean A Hr ° value for known phase modifications, In the same way we determined AHf ° values for other simple compounds with different modification stable under standard conditions. c With the supposition that we analysed the all existing double oxides in the system AIzO 3 Na20, the AH°c~ data for NasAlO4 are not completely correct. In this situation the investigating system is symmetric-type. If phase Na6AI40 9 exists (A /~o = 220 kJ (g-atom) ~ AH~c c = - 32.5 kJ (g-atom) 1 the system is "left-sided". a The interpretation was made taking into account the possible existence of the phase Sr6TisO 16 (A H°xc is minimum). ~The same as d with the phase LizzZrsO27. fThe same as d with the phase CagTisOzs. gThe same as d with the phase Ki2MoTO27. hThe same as d with the phase Sr9WsO24. ~The same as a with the phase BasW2011.

c l e a r l y c l a s s i f y t h e s e s y s t e m s a c c o r d i n g t o t h i s s c h e m e . F o r t h e s a m e r e a s o n s w e c a n n o t

d e t e r m i n e t h e m i n i m u m v a l u e o f A H ° c c s u f f i c i e n t l y a c c u r a t e l y fo r t h e m a j o r i t y o f t h e

s y s t e m s in g r o u p s b, c, e a n d f.

B y a n a l y z i n g o f t h e o b s e r v e d r e g u l a r i t i e s it f o l l o w s t h a t a c o m m o n l i n e a r e q u a t i o n

f o r t h e b r a n c h e s o f t h e d e p e n d e n c e E q . (4) c a n b e w r i t t e n fo r a l l a s s u m i n g o n l y l i m i t e d


Table 2 Values of s t anda rd en tha lpy of fo rmat ion from c o m p o n e n t from c o m p o n e n t oxides for some complex oxides es t imated with the help of the equa t ions presented in the text

Double oxides accord ing AHf°~c/(kJ mol l) Double oxides accord ing AH~eo/(kJ tool ')

to Ref. [17] to Ref. [17]

N a 6 S i 2 0 7 - 737.6 Ca2B6011 - 239.0 N a 2 S i 4 0 9 - 230.6 Na2B8OI 3 - 345.5 Na6SisO19 567.1 Na4B,oO17 665.5

K2Si307 - 2 9 6 . 3 NaB9014 - 179.6 Ba3SiO 5 - 237.9 C a s A I 6 0 , 4 - 66.8

Ba3Si5 ° 13 - 526.1 Ca4AI6013 - - 70.0 Ba2Si, 203 , - 33.3 Sr 6, V, sO6, - 2968.3

Ba8Si5018 - 1273.5 CasTi4013 - 3 6 1 . 5 Pb3Si207 34.8 NaaTisO14 331.8

Pb , 1Si302 v - 63.6 Na2Ti6Ol 3 - 975.2

information about the double compounds available: for the "left-sided"

AH°,c¢(i) A nf°¢c = A H O ( ~ - - ~-/so (i) [AH°(AC) - A/S ° ] (6)

and for the "right-sided"

AH°o¢(i) AH o AH°~ = A/So(~_--~O(BC)[ f -AH°(BC)] (7)

AH~c(i) and A/S°(i) are relevant enthalpy values for a reference double compound i for which reliable enthalpy data are available.

If we take the observed regularities as acceptable (we can name them as Linear Approximation Rule-LAR), we can point out some possibilities of its application. For example it is possible to revise the known enthalpy data of double compounds. In particular this revision was performed for some complex compounds in Table I. It is also possible to estimate unknown AHf°,c~ values of some double compounds, if the reveantA/S ° is located in the range of know A Hf°,c~ data; such examples are presented in Table 2.

4. Conclusion

On basis of the analysis the 34 inorganic pseudobinary systems the empirical dependences for the standard enthalpies of formation of the double compounds from the component compounds (Linear Approximation Rule) have been proposed. The LAR can be used for revision and correction of known AH°cc values and for estimation of unknown values, for related double compounds. With the help of LAR the standard enthalpies of formation from component oxides have been calculated for the complementary 20 double oxides. It seems reasonably to assume that the observed regularities

G. Moiseev et al./Thermochimica Acta 280/281 (1996) 511 521 521

are co r r ec t and sufficiently re l iab le for re la ted d o u b l e c o m p o u n d s in o t h e r i n o r g a n i c

systems.

Acknowledgements

This w o r k was ca r r i ed o u t in the f r a m e w o r k of p ro jec t A2010532 s p o n s o r e d by the

A c a d e m y of Sc iences of the C z e c h Republ ic .

References

[I] L. Pauling, The Nature of the Chemical Bond, 3rd edn., Cornell University Press, 1954. [2] D.E. Wilcox and L.A. Bromley, Ind. Eng. Chem., 61 (1963) 32. [3] P. Peix, J. Solid State Chem., 31 (1980) 95. I-4] S. Aronson, J. Nuclear Mater., 107 (1982) 343. 1-5] H. Yokokawa, N. Sakai, T. Kawada and M. Dokiya, J. Solid State Chem, 94 (1991) 106. 1-6] A.G. Moratchewski and I.B. Sladkov, Thermodynamic Calculations in Metallurgy, 2nd edn., Metallur-

gia, Moscow 1993 (in Russian). I-7] L.A. Reznitskz, Inorg. Mater. 26 (1990) 1359 (in Russian). 1-8] J. gestfik, G. Moiseev and D. Tzagareishvili, Jpn. J. Appl. Phys., 33 (1994) 97. [9] H. Yokokawa, J. Nat. Chem. Lab. Ind., 83 (1988) 27 (in Japanese).

1-10] S. Crouch-Baker, P.K. Davies and P.G. Dickens, J. Chem. Thermodyn., 16 (1984) 273. [11] S. Dash, Z. Singh, R. Prasad and D.D. Sood, J. Chem. Thermodyn., 22 (1990) 557. [12] S. Dash, Z. Singh, R. Prasad and D.D. Sood, J. Chem. Thermodyn., 26 (1994) 737. [13] S. Dash, Z. Singh, R. Prasad and D.D. Sood, J. Chem. Thermodyn., 26 (1994) 745. [14] E.H.P. Cordfunke, W. Ouweltjes and G. Prins, J. Chem. Thermodyn., 17 (1985) 19. [15] E.H.P. Cordfunke, W. Ouweltjes and G. Prins, J. Chem. Thermodyn., 20 (1988) 569. [16] G.P. Wyers, E.H.P. Cordfunke and W. Ouweltjes, J. Chem. Thermodyn., 21 (1989) 1095. [17] F.Ja. Galachov (Ed.), Diagrams of Refractory Oxides, Nauka, Moscow 1985 1988 (in Russian).

Pergamon Prog. Crystal Growth and Charact. Vol. 30, pp. 23-81,1995

Copyright ~c~ 1995 Elsevier Science Ltd Printed in Great Britain. All nghts reserved

0960-8974/95 $29.00

0960-8974(95)00011-9

SOME CALCULATIONS METHODS FOR ESTIMATION OF THERMODYNAMICAL AND THERMOCHEMICAL

PROPERTIES OF INORGANIC COMPOUNDS

G. K. Mo iseev* and J. Sestak l -

*Institute of Metallurgy, Ural Division of the Russian Academy of Sciences, 620219, Ekaterinburg, Russia

1"Institute of Physics, Czech Academy of Sciences, 18040 Prague, Czech and Slovak Federal Republic

CONTENTS

]hNTRODUCTION ................................................ 25

I. STA~0ARD ~THAf21ES OF FORmaTION ...................... ...... 26

I.q. Standard enthalpies of formation of condensed compounds... 26

I.q.I. Method of electronegativities ........................... 26

1.1.2. Empirical ratioB for calculation of~H~98 of Hisham and

Benson halogens ......................................... 28

o 1.1.3. Quasiadditive method for estimation of~H298 of ionic

compounds ............................................... 29

1.1.4. Cell model of ~kiedemA R ................................ 29

O 1.1.5. Empirical equations for calculation of~H298 for crystal

hydrates, ~mmoniates, alcoholates ....................... 37

1.1.6. Increment methods ....................................... 38

1.1.6.1. Calculation of~H~98 based on the use of effective

charges on atoms in molecules and ions ................ 38

1.1.6.2. ~e Van M.s method ..................................... 39

1.1.6.3. B.K. Kasenov's method ................................. 40

1.1.6.4. Variant od Ducros M. and Sannier H. increment method.. 41

23

24 G.K. Mo iseevandJ . Se~ak

o 1.q.7. Calculation of~H298 of complex compounds, which can be

presented by the sum of n-simple crystal compounds ...... 42

q.q-7.q.D.S.Tsagareishvili's method ........................... 42

1.d.7.2. Method of "average contributions" of ~oiseev G.K ...... 42

1.q.8. Comparison methods of estimating~H~8 .................. 49

1.2. Standard enthalpies of gaseous compounds .................. 50

1.2.d. Method based on atomization enthalpies of similar

substances .............................................. 50

q.2.2. Method based on the use of chemical bonds energy ........ 52

II. INC~T OF ~THALPY OF COMPOUNDS ~I THE RANGE O-298.qSK..52

2.d.Increment of enthalpy of condensed compounds ............... 52

2.d.q."Triangle" method ........................................ 52

2.1.2. method of the Institute of Metallu_rgy, Georgian Acad.

of Sci .................................................. 52

2.d.3. Empirical methods ....................................... 53

2.2. Increment of enthalpy of gaseous compounds at 298 K ....... 53

2.2.1. General approach ........................................ 53

n o H o 2.2.2. Estimation of ~298- 0 for ideal gases ................... 54

III.ENTROPY ~ STANDAaND CO~ITICVS ............................. 55

~.C. Condensed compounds ....................................... 55

3.1.1. Increments method of Latimer ............................ 55

3.1.2. Increment methois of Kum~k V.~. and Tsagareishviii D.S..56

o ~.1.3. Some empirical formulae for calculation of $298 ......... 56

3.2. Gaseous compounds ......................................... 62

3.2.1. Some empirical dependencies ............................. 62

IV. H~T CAPACITY .............................................. 63

4.1. Condensed compounds ....................................... 63

4.1.1. Heat capacity at standard conditions .................... 63

$.1.2. Calculation of temperature dependency of heat capacity..64

4.2. Heat capacity in the liquid state ......................... 70

EstimationofThermodynamicalandThermochemicalP~pe~ies 25

4.3. Heat capacity of gases .................................... 71

v, T~ERATURE, ENTROPY AND ~THALPY OF N~ELTING ................ 74

5.1. Melting temperature ....................................... 7d

5.2. Entropy and enthalpy of melting ........................... 76

CONCLUSION .................................................. 77

REFERenCES .................................................. 78

INTRODUCTION

Development of thermodynamic simulation (T S) /I, 2/ and its

application for studies of different processes in systems with

participation of tens and hundreds of condensed and gaseous

compounds leads to the necessity of intensive and qualified

estimation of thermodynamical and thermochemioal properties of

new compounds; creation of special database (DB) for program

packages of complete thermodynamic analysis (CTA) with the use

of computer.

Till recently researchers could use only experimental methods

for obtaining this information. But parallelly created and

developed empiric, semiempiric, quasi- and strictly thermodynamic

calculation methods make it possible today to get fast "thermody-

namic" data. In many cases calculation information : -a) can not

be received in the experiment; -b) are more accurate, than those

received in the experiment; -c) are much cheaper than those obta-

ined in the experiment.

The purpose of this work is short description of some of our

calculation methods for estimation of the main thermodynamic and

thermochemic properties of condensed and gaseous compounds. Survey

of all published methods was not our aim.

Calculation methods are oriented to obtaining of those proper-

ties, which, with the use of sub-program "TERMOS" of CTA package

26 G.K. Moiseev and J. Sestak "ASTRA" /I, 3/ allow to find temperature dependencies of reduced

Gibbs energy for each individual compound.

o 0 0 0

For condensed substances these are A~29~ ~2ge~ ~298- ~o' 0 0 0 ~0 Tmee~a~me~ , ep(cr)= }(T) an~ Cp(e) ; ~or gaseous a~gB, ~;~B' Hzg~- o '

Cp= ~(T) . AS a r u l e , c a l c u l a t i o n dependencies are oriented

to obtaining values in the technical system of units. In this

paper we didn't describe calculation methods and algorithms for

estimation ¢*= f T ) ,'3,. Z. ST~OARD E~THALPIES OF ~OR~TION

1.1. Standard enthalpies of fox-mat;on of condensed compounds

O

To estimate AN29 8 it ia possible to use Berkenheim's method.

It is based on Mendeleev's half-sum ~ule, according to which nume-

rical value of physico-chemical property of a compound is equal to

arithmetical mean from the values for the neighbouring compounds

in the period or row of the System. If comparison goes along the

row, then for account of differencies in valency comparison should

be made per g-atom of these compounds.

Compared compounds may have common cations or anions. For

example: For CdBr 2 standard enthalpy can be calculated as

0 0 I0 A~298= O.5A~298(~_n,~r2) @ 0.5&H298 (a~r2~ or as

O

I.q.I. Method of electron eRativities /g/

Enthalpies of formation of ionic and metallic compounds can be

estimated from the equation O

= - 23.0GGZ (&A- , kca /mole ~ 2 e 8

where ~ and ~ - va lues of e l e c t r o n e g a t i v i t i e s , ~ - number of

valent bonds in the compound, equal to

~ , ~ - number o f ca t i ons or an ions; ~ ' B A - va lency o f ca t i on

or anion.

1 Li

2 Na

3 K

4 Cs

5 Be

6 l~g

7 Ca

8 Ba

9 Y

10 La

11 Ce

12 Th

13 U

14 Ti

15 Zr

16 Hf

17 V

18 Nb

Estimation of Thermodynamical and Thermochemical Properties

Table 1. Electronegativities of elements

27

e!e- ~,(]3) ment

0.95

0.9

0.8

}[ ele- ~,(B) N ~!e- ~,(B) N ele-16,(B) !ment nent ment

l 19 Ta 1.3(II$ 3~ Pt 2.1 55 S 2.5

20 Cr 1.4(11) 381Cu 1.8(i) 56 Sb 1.85

21 1.6(lID 39! 2.O(i~ 57 Bi 1 . 8

0.75 22 IMo 1.6(IV~ 40 Ag 1.8 !

1.5 23 W 1.6(IV)41Au 2.25

1.2 2a Nn 1.4(IIi 42 Zn 1.5

1.0 25 1.5(II~ 43 Cd 1.5

0.9 26 Re 1.8(V) I ~ A! ft.5 |

1.2 27 2.2(VID45 In 1.5

28 Fe 1.7(II I 46 T1 q.5(1) 1 . 1 5

1.fl 29 1.8(II~7 1.9(II~

58 Sn 1.7

59 Pb 1.6

1.4(Iv)30 Co 1.7 48 Sn 1.7

fl .$(IV) }1 Ni 1.8 ~9 Pb 1.6

1.6

1.5

1.4

32 Ru 2.0 50 F 3.5

33 KU 2.O5 51Cl 3.O

Pd 2.0 521I 2.55

1.5(III~5 Os 2.0 53 O 2.9

1.65 36 Ir 2.1 54 Te 2.1

Electronegativity values are given in Table I.

Remarks. There is some uncertainty when using (1) for estimation

of standard enthalpy of metallides formation, as the concept of

valency, for ex., of anion, is difficult to apply here. Thus, when

measuring ~ one should base on elements with stable valency. Fo~

ex., for metallide ~gAg the number of bonds on ~g is 2; for Ag,

to (2), number of bonds is I. Then &~98 from the according

first option =-16.6 kcal/mole; from the second -8.4 kcal/mole.

O Experimentally obtained value is ~H298 =-9.2 kcal/mole /4/.

28 G. K Moiseev and J. Sestak

Thus, i t i s e x p e d i e n t to a s sume t h a t g roup ~lgag has t h e a v e r a g e O m ntunber o f bonds 1 .5 and to f i n d f rom ( q ) & ~ 1 2 9 ~ = - 1 2 . ~ k c a l / m o l e .

i f t h e e l e m e n t has v a r i a b l e v a l e n c y , i t i s n e e e s s a ~ j to

coordinate valency o£ "cation" and "anion" in metallide. ~or ex.,

for MgTI magnesium has stable valency 2[ 21 - valencies d and 3.

O ~xperimentai value ~ H298~-12.0 kcai/moie /~/. AYter calculation

&Te(.O c at ~q.5 and B=2, [email protected] kcal/mole; at 8("3 =1.9 and

~=2, &H~8=-22.6 kcal/mole. Assuming the existance of hypothetic

TI( , I ) withS1~,,)= [~T¢(,,,) + 6Ted, ) ] 0 . 5 - - q . 7 , we g e t o

a H298=-11 • 53 kcal/mole.

Accuracy of evaluation of standard enthalpies of formation

for ionic compounds is higher than for metallides. It is better to

use this method for simple substances.

For complex compounds there is the following possible vray for

o estimating aH298. For related substances (q) can be written as:

~.gg

~or these substances we find first average value A, average values

of ~ cation and ~ anion, number of valent bonds Z from (3), and

o then estimate ~H298.

O ~ For example, when estimating&H298(YBa2Cu307) /5/ was found

A-11.13 (for Y203, BaO, CuO); Z-l@ (by oxygen: 7x2--14); ~ . ' I . ~ 9 4 .

(average for Y}+, Ba 2+, Cu 2+) and ~ -}.44 (average for oxygen

in above-mentioned oxides). After calculation we got &H~98=-687.7

kcal/mole. Experimentally received value -6~9.7 kcal/mole /6/.

o 1.1.2. Empirical ratios for calculation of aH298 of Hisham

and Benson halogens

In /7/ was found the ratio 0 0 0 0

g.r , (,,)

where a and b - numerical coefficients, n - formal valency of

metal (~), including NH~.

Estimation of Thermodynamical and Thermochemical Properties 29

In Table 2 are given recommended equations for calculations.

Discrepancies o£ calculated and experimentally obtained values

usually are not more than experiment error. In paper /7/ it is also

pointed out that for hydrated compositions there is correlation

° . ,

where A and k-constants, ~ - number of ~ter molecules in the

compound. In /7/ are given also , probably, the most complete and

o systematized data about ~H298 for different halogens.

o 1.1.3. ~uasiadditive method for estimation of AS298 of ionic

c omoound s

In /8/ are suggested equations based on the use of Born-Saber

thermo chemical cycle:

to r L ~ - rl~3~_- m.L LQ z~H298= ~. K" ' kca!/mole, k6)

v,;here ~K and h a - charges of cation and anion, n - number of atoms

in ~o~ecule of com~ound~ J , , J~ , k a~d Lo_ - p a t t e r e r s , f o ~ a

from the follo~&ng ratios: 2

: ]K~,~ - r % (~ .~ + % ) . (170-138 .8 r~ • 1 7 . 4 r K ), (7)

t~= 5.g rL~. ry,, (8) t~ 5@&& r&.

Calculated parameters ~L , L~ are given in Tables 3-8.

1.1.4. Cell model of Miedem A.R.

Cell model for estimation of enthalpies of formation of alloys

and other binary compounds containing metals, has been explained in

a number of papers /8-10/ and tested with good results.

According to the model, enthalpy of phase AI_xB x from metals

A and B, can be calculated from tabulated values of electronegati-

vity ~/, atomic volumes V and electronic densities on boundaries

of Wigner-Seitz cells ( ZWS ):

30 G. K. Moiseev and J. Sestak o

Table 2. l::quations for calculation of AH298 for halogens, kcal/mole

• O Equation in which~H ~s equal toAH2~ 8

Groun IA of metallic halo~ens~ includin~ ammonium halocens

(~C:) = 0.~7 H(~r) + 0.18~ H(~L~) - 3.~

(MCI) = 0.702 H(~i) + 0.298 H(~I) - 8.8

(~_Br) - 0 . 839 H(~_I) + O.161 H(NLF) - 6.5

<~r) = 0.~59 H(~Z) + 0.5~ ~(,~:Cl) - 1.7

( ~ I ) - 1 . 192 b_'('~LBr) - 0 . 1 9 2 H(N0~) , C.7

(me~-) . 1.195 ~(:~ci) - o.195 ~(~) + ~.~

(~ t I ) - I.~#'~= H t ~ C l ) - 0 .~25 H ( ~ ) + 12.5

Group IIA o£ metallic halo.~ens

(~C12) - 0 . 856 H(~t l2) + 0 . I ~ H ( ~ 2 ) - 4 1 . 2

(MC12) = 0 . 8 6 2 H(~LBr2) , 0 . 1 3 8 H ( ~ 2) - 9 . ~

(MBr 2) -- C .993 H ( ~ I 2) + 0 .007 H ( ~ 2) - 37.O

(NLBr 2) -- 0 . 9 5 2 H ( ~ I 2) + 0 .0~8 H(~C12) - 35 .0

( ~ I 2) = 1 .007 HQ~Br2) - 0 . 0 0 7 H ( ~ 2) + 37 .3

( ~ I 2) = 1 .050 H(~Br2 ) - 0 . 0 5 0 H(~C12) + 36 .8

(M I2 ) - 1 .168 H(~C12) - 0 . 1 6 8 H(NL~2) + z#8.1

(~Br2) = 1.160 H(NIC12) - 0 . 1 6 0 H ( ~ 2) - 10.8

Sub~rouos cf metallic ha!o~ens

(~CI n) -- 0.683 H(MIn) + 0.517 H(~P n) - 3.71 n

(MCI n) = 0-831 H(NaBrn) + 0.169 H(~n) - 1.52 n

(L~r n) ~ 562 H(MCIn) + ~ ~ Wf ~ =~ = ~- ; ~.4- 8 ~Mln) - ~.~pn

~tBrn) 0.822 H(~In) + 0.I~8 H(~:Fa) - 2.~a n

(~,!! n) = 1.217 H(MBrn) - 0.217 H(~CFn) - 3.21 n

(~In) = 1.a64 H(MCIn) - 0.@6@ H(~Fn) + 5.34 n

(~LBr n) = 1.20~ H(I,IC!n) - C.203 H(~n) + q.&~ n

Halogens of rare-earth metals, excluding ~uI~ and GdZ~

(~Sr~) . 0 .933~ t ( ~ C l 3) + 0.O66? H(ZI~) + a 9 . 7

(~ Im) = C.56L6 H( '~CI=) + 0 . 0 6 6 7 HE~L~=) + 159. 9 J w w

([Cl=) = 1.0715 H('EBr=) - 0.0715 H(~=) - 53.3 j


-4/3

where P and Q - model constants and

_-

g

where C A

(9)

( 1 0 )

(11)

- atomic concentration and C s - surface concentration of A

atoms of A kind.

31

Value Q/P for evenly group or ~amily of binary compounds is

constant and can be found from the main known experimental values

O

AH298"

In Table 9 are given the main model parameters, necessary for

or~/- potential of electrons (elect- the calculation, chemical

ronegativity parameter); ~w$- electron density on the boundary of

Wigner-Seitz atomic cells and V m- mole volume. Units ~W$ are such

that density = 1 for lithium; one unit of density described from

0 In paper /8/ are given comparison results of AH298 for inter-

metallides (total 51), received in experiments and calculations.

0 The same method is used for estimation of &H298 for metal

1

hydrides /9/. For calculations it ~s assumed that _ ~ =5.2,

~/3 =1.5; VH=I.7 cm3; ~H(1/2H2+Hmetal)=lOO kJXg atom H) and

R/P=}.9. We've made calculations and comparison with experiment

for 35 binary hydrides.

Equation for calculation of enthalpies of formation of ternary

hydrides from intermetallides ABn, was suggested:

where AN n - intermetallide; x, y and F - parameters,

, (42)

given for

some metals in Table 10.

32

Ion

G. K. Moiseev and J. Sestak

Table 3. Parameters for anions of halogens, O 2-, S 2-

ia Ja at n k

+1 +2

~- 7.8

Cl- 10.7

Br- 11.5

I- 12.7

0 2- 16.5

S 2- 21.7

-9~. 5 -11o.7

+~

-126.9

-7.6

16.9

5o.~

172

3o9

16.5

51.5

98.9

153

362

40.6

86.1

147.1

133

415

+4 +3

-I~3.1 -159.?

6a.7 88.8

12o.7 155.3

195.9 2LF$. 4

113 94

AL68 521

*6

-~75.5

112.9

189.9

292.9

74

574

In paper /10/ were offered algorithm and base of initial data

for calculation of AH~98 by Miedem's method. This algorithm was

realized as a program "NIX" for computer in the Institute of

Metallurgy, Ural Div of Russian Acad. of Sciences.

In paper /qd/ v~th the use of NLiedem's method were calculated o

~H298 for alloys Sc, Y, lanthanides, Th, U and Pu with a number of

easy-to-smelt metals (A1, Ga, In, T1, Sn, Pb, Sb, Bi). In Table 9

are given parameters for calculation of AH~98_ , calculated by

authors of /11/. Values of R/P, B 2 for easy-to-smelt metals are

given below:

Metals A1 Ga In T1 Sn Pb Sb Bi

R/P,B 2 0.O 0.45 0.25 0.25 O.10 0.O 0.30 0.30

Table 4. Parameters for anions Se 2-, Te 2-, H-

Ion nk-+d +2 +3

ia Ja ia Ja ia Ja

Se 2-

Te 2-

K-

23.07

8.95

3~1-3.6

47.7

2a.15

2a.

7.86

45o.9

a62.7

15.8

22.9~

L

a78.8


Table 5. Parameters for single- and double-charged oxygen-containing anions

Ion nk=+fl +2

ia Ja ia Ja

*CH-

IO=

sro~ J

C!0~

czol J

nol J

no~

vol

CO~-

SO~- WO~-

SeO 2-

TiO~-

Fe202-

SeO~-

1~oO~-

ZrO~-

Hf0~-

A1202-

SiO 2- J

CrO 2-

9.13

1o.13

'10.76

'12.'17

10.2a

9.78

9.85

22. =~ J~

23.77

22.05

23.37

22.27

20.83

22.51

-4'1.

- -37-3

7.6

19.2

-5 .8

-37.8

-9.6

141.9

118.5

54.1

179.7

31.6

6.3

95.o

8.83

11.54

9.11

10.76

20.5~

22.07

19.87

21.03

22.17

18.98

17.69

20.9

29.32

19.50

20.95

18.22

18.72

22.01

-89. q

-5~.4

4 6 . 6

-43.4

-'167.0

145.5

147.9

21.8

85.8

223.2

-16.3

-25.1

203.1

~3.6

-16.A~

2O. 2

- 209.9

-18.8

149.1

* at nk=+ 3 ia--9.68 and Ja---~%~.fl.

34 G. K. Moiseev and J. Sestak

Table 6. Parameters for cations at n~ =- l

J Ion

~S. 2 Y

78.9 La

96.3 Ce

107.0 Pr

1o2.9 Nd

195. ~ c~

551.~ Ko

22~ .0 Er

528.0 AI

3~6.2 Ga

5~x~.6 In

~9.3

371 .c TI

~95.3

5~. 3 ~o

(39.77) (17o3.o) w

~06 ~n

9~.0 ~e

! 8~8.7

1727, o Co

(1742.3) Ge

197.2 Sn

679.1 Pb

(32.94) (1149.o)Pt

3039.4 Pt

3005.6

4161.4 U

t1232,1) As

4 7 3 . 7 Sn

1072.8 Re

511.e ~

1811 . I *Sc

Ion n k ~ •

Li +1 6.a 5

Ha +1 7-~5

K +d 8.92

Rb ÷1 9 - 36

Cs + I 10.10

Cu +~ 10.31

+2 19.8C

Ag ÷1 14.41

Be +2 15.03

Mg +2 16.29

Ca +2 17.57

Sr +2 17.97

Ba +~ 18.8z~

Zn +2 19.57

Cd .2 2C. 58

Si +4

Ti +2 17.5

+3 28.92

+~ a~ .0

Zr +4 ~1.47

E£ +~ (~1.83)

Hg +1 10.35

+2 23-78

V +3

Nb +5 57.21

Ta +5 (56.93)

Sb +3 32.12

Bi +3 [33-75)

Cr +2 28.71

+3 30.87

Ni +2 19.25

Th +a 4&. C6

- calculated by us.

n k

6

÷~

÷~

÷~

÷~

÷3

÷~

+~

÷3

÷3

÷I

+3

+I

+3

+4

+6

+2

. 2

*3

~2

÷4

÷~

. 2

÷2

+3

+4

ea

÷3

+2

+3

+1

÷3

i k J- .<

8 T ~

<2s.75)

<30.~7)

(22.99)

( I t . 26)

(30.52)

( ~ o . ~ )

(29.C5)

27.50

3o. 78

9.55

34 .73

10.28

(38 .75)

(46.12)

( 6 9 . 8 )

Lea2.8)

< ~2e. 2)

<gp .e )

,9~r.s)

-?~2.a)

~953.e)

683.8

1067.9

162.9

4105.8

179.8

(1~52.8)

(2094.8)

< 4387.7)

19,5a

19.05

31.42

19,18

483.8

a97.2

1125.3

508.4

(42.05)

(~-1. ~8)

20.81

(25.42)

(58.06)

(50.90)

42.04

(31.25)

(20.83)

(35.25)

8.37

(2~)

~1556.1)

(1828.~)

551 .o

(754.7)

(1'G2. I)

(241o.4)

1748.2

( 1 1 3 1 . 6 )

(552.2)

(1323.9)

112.O

(882) •

Ion

I

Li

Na

K

Cu

Ag

Ng

Ca

Sr

Ba

Zn

Cd

Hg

A1

Ga

In

T1

Pb

Sb


Table 7. Parameters for cations at n .= -2

nk ik Jk Ion n k i~

2 ~ 4 > 6 7

+1 6.z~ 2 V +3 30.18

+1 8.00 61.7 Fe +2 16.69

+1 9.78 112.8 +3 27.53

+1 7.38 76.4 B +3 23.95

+2 17.69 393.@ V,n +2 16.01

+1 8.81 128.3 +~ 40.72

+2 14.39 178.3 As +3 27.61

+2 16.54 [email protected] Be +2 11.71

+2 17.09 266.4 Bi ÷3 28.69

+2 18.08 310.I Ce +3 27.37

+2 16.52 308.4 Co +2 16.51

+ 2 17.66 568.5 Ge ÷4 39.96

+2 19.35 463.7 La +3 26.9

+3 23.79 581.6 No ÷~ 40.13

+3 27.66 805.1 Ni +2 16.70

+3 27.93 841-7 Re ÷4 ~2.18

+1 7.39 75.7 Sn +2 15.94

+3 29-65 974.6 +4 39.19

+ 2 17 • 26 364.6 W +4 39.58

+3 27.67 858.6

Jk

899.8

Fj8.O

837.8

599.5

28~ jeJ

1665.2

863.8

90.3

915.5

712.1

3~.7

1619.9

695.6

1620.6

~o.6

1767.2

3o~. 5

1575.o

1592.5

Table 8. Parameters for cations in complex oxygen compounds at na=-2 of multiatomic anion

35

Ion i k Ion i k Ion i k Ion i k

Li +

Na +

K +

Cu 2+

Fe 2+

6.5

8.02

9.62

16.5

16.10

Ag +

Nn 2÷

Be 2+

~g2+

Ni 2+

7.70

15.6O

11.80

14.30

15.88

Ca 2÷

Sr 2+

Ba 2+

Co 2+

16.52

17.3o

18.2O

15.82

Zn 2÷

Cd 2+

T1 +

pb 2+

15.72

16.82

s.9o

16.60

36 G K Moiseev and J Sestak

Table 9. Parameters for calculation of AH£9 ~ for alloys b y / 8 / a n d / l l/

Element

Sc

Ti

V

Cr

~n

~e

Co

Ni

Y

Zr

Nb

Me

Tc

Ru

Rh

Pd

La

Hf

Ta

W

Re

0s

ir

~t

Th

U

Pu

2 ~,

5.25 1.27

~3.}) ~1.~2) ~.65 1 .z~7

z~. 25 1 .6~

.85

4.~5

~.9~

i.10

5.20

~.20

}.~0

@.0

a.65

5.}

~.55

5.6

5.05

}-55

~.o5 I

~. 80

5.5

5.55

~.55

5.85

~. 50

~.05

I~.8o I(;o?) I

~ . 7 3 5 .7

1 .61 3 . 8

1 . 7 7 3 . 7

1 . 7 5 3 .5

4 . 7 5 3 . 5

1.21 7 . }

( 1 . 2 2 ) (7.36) 1.~9 5.8

1.62 4,9

• 77 4.4

1.81 #.2

1.87 z,,1

1.76 4.4

1.65 ~-3

1.09 8.0 47,98)

1.43 5.6

1.63 ] ~.9

1.81 ~-.5

1.90 ~.~

1.89 Z~.2

1.83 ~.2

1.78 ~.~

1.28 7.5

1.56 5.6

(in brackets)

2/5c Element ~)/ , ~ 4/3 2/.5 2 V cza- ,o~!t '~ "~ cm

G.q Li 2.85 O.98 5.5

u . 8 Na 2.70 0.82 8.3

~,1 ~ 2.25 0.65 12.8

Rb 2.10 0.60 16.6

Cs 1.95 0.55 16.8

cu 4.55 1.47 5.7

Ag ~.45 1.39 ~.8

Au 5 . 1 5 1.5'7 " . 8

Ca 2.55 0.94 8.8

Sr 2.%O 0.8~ 10.2

Be 2.~2 0.81 11 .~

Be 4.2 1.6 2.9

~g 3.~5 1.17 5.8

Zn z~.10 ~.52 ~-~

Cd z~.05 1.2~ 5.5

Hg 4.20 1.24 5 • 8

A1 z~. 20 1.59 a. 6

Ga ~.10 1.31 5.2

In 5-90 1.17 6.5

T1 5.90 1.12 6.6

Sn 4.15 1.2a 6.~

Pb 4.10 1.15 6.9

Sb ~. L~O I . 26 6.6

Bi 4.15 1.16 7.2

Si ~.70 1.50 ~.2

Ge ~.55 1.57 ~.6

1 . ~ 5,2

By 1111

Ce 3'.02 1.07 ~.5z~ Tb 3.15 1.2 7.20

Pr 3.03 d.O8 7.57 Dy 3.23 1.22 7.12

Nd 3.O~ 1.41 7.51 Ho 3.20 1.2~ 7.07

Sm 3.10 1.10 7.36 Er 3.24 1.26 8.99

Eu 3.16 0.9 9.42 Tm 3,2~ 1.27 6.90

Gd }.lB 1.19 7.35 Yb 3.20 0.95 8.50

Lu 3.~o 1.30 6.81


Table 10. Parameters, necessary for calculation of AH;9 s of hydrides, formed

from intermetallides AB~ by (12)

~etal A

Ti, Hf, Zr,V

Sc, Nb, Ta,

T,a, Y, rare-earth

Th,U,Pu

AB n ABnHx+y

AB 5 ~5H5 ~B 3 ~B3E 4

AB2 AB2H3.5

AB ABH 2

AB 5 ABhH 6

AB 3 AB3~

AB 2 AB2H 4

AB ABH2- 5

x

2

2

2

1.5

2.5

2.5

2.5

2

parameters

3

2

1.5

0.5

3.5

2.5

1.5

0.5

f

0 .1

0 . 2

0 . #

0 . 6

0 .1

0 . 2

0 . 4

0 . 6

37

o 1.1.5..Empirical equations for calculation of AH29 8 for

cr~stal h~drates, ammoniates, alcoholates /12/

Enthalpies of hydratation of crystal salts H°hyd(~aXb) are

estimated as: O O O

and can be calculated by two-parameter equation

where m and C - different constants for each salt. Value m is close

to -3 kcal/mole; value C depends on the nature of salt and for

one- and two-valent metallic salts is in the range -I-8 kcal/mole.

Enthalpy of formation of solid crystal hydrates, ammoniates

and alcoholates can be expressed by equation

where Y - H20 , NH 3 or CH3COH. A andeS-- different constants for

each salt.

For hydrates values o~1 and

A=+155oi,,-226.0 (16)

3 8 G.K. MoiseevandJ. Sestak

~or n=~7, average deviation received for ~H ° f o r each

series is less than ~ kcal/mole at max deviation ~ 3 kcal/mole.

For 42 compounds enthalpy of dilution of salt MaXb can be

correlated to the added number of water moles by equation:

a~o = Z ~Cno ) ano + 5%+ c

H o where a, b and c - constants; ~ dil(no )- integral enthalpy of

dilution. This equation is valid at no=456.

-O Authors used experimental data about AH298 from paper /4~/.

t~7)

I.fl.6. Increment methods

o I.d.6.1. Calculation of aH29 $ of compounds, based on i he use of

effective charges on atoms in molecules and ions /da-16/

method of estimation is based on the equation: ,~ o =. 0 ~' %~ Ec.b..~H0 ÷ ~H~ , ~ /mo~e (48)

o o where EC.L.- ener~j of crystal lattice, H C and H~- entha!pies of

formation of cation and anion.

Ec. Here A- Madelung constant; qc- cat ion charge, qy and qx- e f f ec t i ve

charges on atoms of anion YXnm. Values q can be found during equali-

zation of potentials, given as dependency

q =Aq 2 + Bq + C + M, (20)

where A, B, C - atomic parameters (in thous.cm -1) (Table 41);

m-- a " M - correction. Values q for YX n - nlon are found after solution

~y- rl;~ = O> ] £,.,. - r ' , . q . , ~ , - _ ~ . ]- ( 2 4 )

In (49)~values RCL Y and RCI~X - distances between ions in lattices

CT.y and CLX. For ex., for BaCrO 4 RBaCr=4.25 ~ and ~aO=~.44 ~.

Values AH~ and AH~ are taken from the reference literature.

O In /15/ are given data on AH298 for salts with 23 cations

(alkali and alkali-earth metals, Sc 2÷, Ti 2÷, V 2+, Cr 2+, ~n 2+, Fe 2+,


Table l l. Atomic parameters

89

iParame- ters

A

B

C

0

27.8

147.5

126.9

28.0

~63.6

151.6

C1

Io9 .7

136.7

Br

7.6

79.0

123.¢

w

3 . 7

93.o

63.0

~o

5.5

71 .¢

59.0

~n

7.88

92.~9

~ .96

C AI V Cr H B N

13.21

81 .o7

109. ~

2d9.2

1 o 9 . 7

12.22

80.28

55.84

A

B

C

27.8a

102.o

79.23

82.3o

5~.Ia

28.0

119.2

Io5.7

29.55

~. 3

123.7

Co 2+, Ni 2+, Cu 2+, Zn 2+, Cd 2+, Ra 2+, N_~) and anions of metavanadates,

o oxalates, chromites and aluminates, and AH298 o£ these salts.

o Besides, ~H298 were calculated for 52 sulphates, carbonates, sul-

phites, rhennates, chromates, titanates, molibdates and tungstates

of different metals.

In /16/ are given H° 298 ~ ~ _ $-' of salts with anions: ~oF~-, WF~-, WCI

- 3- W2CI~- ' W3Cl~, ~o~, w%, WCl, w~, ~ocl 2-, wcl 2-, ~ocl~-, ~o2cl 9 , w 4-,

and also enthalpies:~of anions solvation (x~H s) at infinite dilution.

In paper /I~/ are givem initial data for calculations and stan-

dard enthalpies of ammonium halogenides; perchlorates of alkali

metals, nitrates of alkali and alkali-earth metals, nitrites, hydro-

fluorides, permanganates, rhodanides of alkali metals, alumohydrides

and azides of alkali and alkali-earth metals; borhydrides of alkali

metals.

q.I.6.2. Le Van ~.'s method /47/

It is based on the use of empirical equation 2

where n a and n c- number of anions and cations in molecule; A and C -

constants (Tables 12 and 13).


Table 12. Values of C, kcal

cation

Ag +

AZ3 +

Ba 2+

Be 2+

Bi 3÷

Ca 2+

Cd 2+

Ce 4+

Co 2+

Or3 +

2

-22

-219

-211

-156

-112

-205

-92

-296

-82

-174

cation

Cs +

Cu +

Cu 2+

Fe 2+

Fe 3+

Hg +

Hg 2+

in 3+

K +

Lq 3+

-106

-20

-51

-89

-101

-29

-~0

-155

-107

-290

cation

.5

Li +

Mg 2+

~n 2+

Na +

Ni 2÷

Pb 2+

pd 2+

Ra 2+

Rb +

C

6

-108

-dT?

-125

-107

-78

-78

-49

-213

-106

cation

7

Sb 3+

Sn 2+

Sn ~+

Sr 2+

Th 4+

Ti 2+

TI +

U4+

UO 2+

Zn 2+

8

-94

-97

-13o

-206

-- j~j

-13o

-~9

-229

- 309

-105

O To calculate ~ H298 for hydrated compounds in /17/ the following

equation is offered:

o = mH~98(x) -71n, kcal/mole AH298(x nil20) , (23)

where x - water-free compound, n - number of water molecules in

crystallohydrate.

1.1.6.3. B.K. Kasenov's. method /18,19/

It is based on the dependency, o±'fered for arsenates of alkali and

alkali-earth metals /18/r

where the first term is standard enthalpy of cation in water solu-

tionl K - conversion coefficient, n - metal valency, AHi(298) -

standard enthalpy of anion.

~O In /18,19/ are given~H298 for arsenates of alkali, alkali-

earth and rare-earth metals.


Table 13. Values of A, kcal

anion A anion A

~2of -357 ~o~ - ~ so~- £ s 0 £ - '110 ~nO~ -9AL F,_SO ~

J

~Zsog -18o ~07~ +~ ~so Z

~o~ _~ ~o~- -~o ~o~-

~,; ,~o ~of -~35 ~o~- cJ_o- +12 ~ ' Z -274 s4o ~-

coe=- ~-85 u3P2o ~ -5or voO-

doo} -Ira s~c, a -15? zn% z-

i%of -25~ ~io}- -19o

anion A anion A

-75

-106

-q 36

-173

-75

-154

-217

- 26~

-227

-80

-19o

-315

-182

-233

-3

-202

HC00-

CH3COO-

CH3CH2CC0-

C20 ~-

C3H20 ~-

C~H40 ~-

-68

-78

-82-

-131

-136

-l@fl

41

1.1.6.4. Variant of Ducros M. and Sannier H. increment method

/20/

For calculation of standard enthalpies of formation and free Gibbs

energies were offered equations, kcal/mole:

- - (r~A+%)(&-x~) +n~.¥A + %.% + r~,, w~ 2

o I t i i W ~

wZ where ~- number of bonds~ n A

(25)

(26)

and n B- correspondingly number of

42 G.K. Moiseev and J. Sestak

anions and cations in molecule. Parameters XA, XB, Y~, YB' WA' WB

a,~d also X I, ~, Yi, ~' wi' % are given i~ Tables I~, 15 (~o~

Eq.(25)) and in Tables 16,17 (for Eq.(26)).

Taking into account, that O o ~L c,

then Eq.(25) and (26) can also be used for calculation of standard

entropy of compounds by equation 0 - 0 o r'L 0

after calculation of values of standard free Gibbs energy and r%

°

o q.1.7. Calculation of~H298 of complex compounds, which can

be presented b,y the s_um n - simple crystal comDounds

q.fl.7.1. D.S. Tsa~areishvili's method /21~

It is found empirically that if electronegativity of central

cation by Poling /22/~C~1.9 , then for complex oxide ~ompounds

it is possible to find standard enthalpy of formation from "simple"

oxides by equation

~H~98(from oxides) =-7.0 m, kcal/mole, (29)

where m - number of oxygen ions in the compound.

Then standard enthalpy of ~omplex compound from elements is

o Ho Z~H298 = ~ 298i + H~98(fr°m i-oxides) =TA O H298~7.0 m, kcal/mole (30)

where H ° 298i- standard enthalpy of simple oxides.

.O Calculated feom (~0) values of ~H298 for complex oxides in the

system Y-Ba-Cu-O, including phases of high-temperature superconduc-

tors, agree with those received experimentally /21,2~/.

1.1.7.2. ~ethod of "average cont~i.butions" of ~oiseev O.K.

O Initially it was offered for estimation of AH298 of binary nitrides.

Calculation is done by equation


where n M, and n M,,- number of atoms of metal (I) and metal (If)in

complex nitride; ~,~ ,and ~ M " - enthalpy per I atom M' and ~'' in If

simple nitrides MaN b and Mc'Nd; ~and ~- enthalpy per I atom of

nitrogen in these nitrides; ~/and - atomic fractions of nitrogen

contained in M'aN b and M'c '~d' when~/ +~I~=I; n N- number of atoms of

nitrogen, not bonded into simple nitrides.

Method is based on Eq.(30), to be more exact, on its qst part.

aH2~je=~m L. zg@L. + A,~ (52)

where n i- number of moles of simple i-th nitride in the complex one;

A - standard enthalpy of formation of complex nitride from simple

ones. Analysis shows that

r t / ~ , ~ + n.M, ~ g~,~ ~n. L- H ° L .F c w "~

( ~ )

Physical meaning of Eq.(34) is as follows. If a complex nitride is

a stoichiometric sum of simple nitrides, then the last item is

zero. Then A is the average energy, additionally needed for bonding

two simple nitrides into molecule. If a complex nitride is not a

stoichiometric sum of simple nitrides, then excessive energy per

each atom of"superfluous" nitrogen (a) is equal to average energy

of this atom's bond. Example for Li~rN. Let's consider this

compound as 0.66~ Li3N + 1.0 ZrN + 0.333 N.

Then for calculation from (31) initial data will be the following: !

nLi=2 nzr Li3 (zr )11 S. ~#=~H~8(ZrN)/1 ; total amount of nitrogen atoms in T.i3N is

0.667xi = 0.667; in Zr~: lxd = 1. Their sum is 1.667. Then

,,~t! 1 ~/=0.667/1.667 = 0.~ and Z = /1.667=O.6 nN=0.553. o

For simple nitrides values of ~H298 are taken from reference-

books, for example /2~/.

~ethod was tested on different classes of complex compounds.

Some results of comparison v~Lth the experiment are given in Table 18

4 4

Oa

0

r...q

._=

;>,

X

"8

0

:::S "a >.

.d @

6s

+ r*%

• ~ C'

+ Od

~q

4- O4

~ O Pq

4" K", D'--

Cq

&

G. K. M o i s e e v a n d J. S e s t a k

kO O4 .t- LO O4 0"~ 0 ~'~

-t- u~ oO ~ ~- ~- 4- u'-', ..1- O,j ~ ~ ~*'~ q3 C--- ~ 4 - -1- O",

u~ Ix ~, LP~ ~ ~ t rk CD L0 r '~ u'~ L0 gO t •

~t" 04 + [',,- C--- + hG k.0 4- ~,O "m" ~:- 0 GO 04

O ~ ~ L0 -1- ~% , • ~ " " ~ dS" ,-t- • . O d ' , ~ . C ' ~ ~.0 ~ l C ~ , ,o ,

E-~ + L0 L~ U-'-, -.~" + K", ~-', E~-

L0 ,4 " E ~ ~1 to, l'o, L0 re-, 4" 00 G + ~ 0 d", ~ ~ ~h <D LP~ 0 0 ~ cO 4- U~ ~ crX ~ - ~ ~ ~ G" E"- O cO

,,-, ¢- ,.,-~, ~ 4 . - , g °" -.# 4 .# ~ - . - r

Od Om Od ~ 0 O", ~ " o,.I co T---

t,o., I 0 ,1

u " \ ~-- OJ O", ~ u'., cO cO OJ ~..~..~..~..~..~, u"., OJ 0", 0", '~" 0 ~t_.~ ~0 ~o, 4 - CO 0", u",. 0 K",

O~ I ~-

~ ~ ~ ® ~ o ~ d", LP'...q" i LO cO r~h ~ 6J 0 ~'- sO ~'~ 0 0 • ~ ~ ,, • • D"-- • ~ • . d ~ o", ,-- ~ g - d , J . . J

~ ,.,.-., . ~ ,--r d ,A ,.o

o,,, d.., o.~ ,..0 Lr', kO t,,o, co E",.,. ~--- ~:- co u-,, cO ,.O

~ d "!. " j ~ d , g ~' ,-4 , - c o

- - - - i kO OD L"~ u-~ CM oO ~ ~0 E~- LF~ OJ CO ~ ~ # ~

co ~. u ~ oo o~o~ ~ ao ~ ~o ~- ~ ru • ~ • • • • • • (~ " O~ •

,-- ,-,", o', ~ , 4 ~ d c-- ,"- c..- ,4 o ~ - ~ ~l_ I ~l_ ~'- d ' , ~'- ~- ~-

~ I ̧ ~ 1 ~ 1 ~ 1 ~

Est imat ion of The rmodynam ica l and T h e r m o c h e m i c a l Proper t ies

L"-- 4- cO CO ~+ to, + <r L-~ um 04 C-- K'~ <0 ~- 04 04

~, ,o- , • : ~ ~ , o U--- 4. <

+ 4- ,O ~ 4 - 4 4- cO 14-

0-4 o ' , ~ • d M ~ ~ ,~o , - .

l, + 4- ÷

~o d d 4- ~ 0 4 ~ d

o~ O um ~- o', co 4- • ,4 ,.4] O 4- ~ 4- • c--

4- 4- + LO oo ~ - .q.- u", 0", 4 - c'd.

kO to, 04 .,-4 ~-. ~ E".- .~ 4 " um tO cO 04 .,1 03 Om (',J kid E4 04 0", ~'-

d~< jdd d~A

+ 04

LiD ~ + 4- 0", O0 t,O, "C" 0 04

cO~ 6gJ

4-

t o t',(", E~ Um U'X

4~d2-

<Oi .,4 L0 <0 <0

~ 6 4 j ~ 4 d g ~ d d ~ b ~

OJ O O + ..1- u'~ O,'~ 4- t'~ E,- 04 4- 0-, D-- u~

4,- ~o L'~ ~c'~ cO 4- 4- 4- £x.. K-~ cO um 0", 4- O-~ E~- CO "~ t*m ~ O m + ,..D oJ ~- m, ~ to, <J- ÷ cO kO E'-- 4-

r- um u~ 04 4- 0 cO am, v"

L.m LD Um Q3 .[~ 4- CO E'~ O4 CO to, + 04 re', L~- .4- I'o, U*~ E'~ L'--- L0

L~ C"- ~" 0 4 0 " , ÷ Um kO -,1- (~ +

• o o ~ o', ~ o t, ~ ,4 ~ ! S ~°~r4 = 4 A g ~ , - , ~ - , - ,- ,--,--

o' , 1

45

46 G K Moiseevand J Sestak

Table 15. Values of parameters X A, YA and WA in Equation (25)

"2 ~ 4 # ~" ? #

x 9.154 10.581 7.129 8.782 8.~62 8.@5 16.137

z 86.189 413.038 1.006 110.999 6.122 -d4.0c9 -91.7c,~

W 61.58~ 58.205 ~174.046 _>00.685 3.686 ~7.@63 -105.72~

CZ- ClO 2 ClO~ ClO~ CN- C~S- Co 2-

x 8.459 8.95 8.136 7.731 9.778 8.885 8.115

Y "10.996 -13,752 -15.256 ~21.337 -33.085 -3~.665 89.@35

W 27. 532 67, @77 6~.2~ 70.6~7 -la. 2a9 2}.083 86.698

,- .-

x 7.96~ a.8~7 8.8o5 8.979 9.292 10.889 9.491

Y "IL~D.094 110,30@ 32.862 73.306 129.887 -7- 161 "150.a9a I

w 87.181 161.62 121.5&61 63.601 7a. 701 -76.786 71. 867

2- HSO: I- I0~ m,~ me,, '~ L::~ ~ HS-

x 9.81~ 8.#37 10.127 8.6}5 8.397 8.631 9.058

Y [email protected] 267.#94 3}7.873 -36.#23 167.658 -1.525 30.511

w @0.778 z~6. 593 -7@.@3 75.536 84.9#5 -22.115 63.8

~oof ~3- ~? '~ i ~o? ~o? 0 2-

X 8.085 15.576 8.755 I0.9al 8.722 8.322 11.026

Y 169.~8 -59.7#7 -88.13 -5.513 -3,5~ 12.70 1C. ,9

W 86.701 -29.906 }7.797 -15.70( 51.057 66.668 @7.658 j

2- ~,}- PO? PO~- oo11- OH- one- 020 @

X 7.87} 10.202 11.66 7.95~ 12,10@ 8.055 7.82}

Y -10.98 }6.383 -37"155 I 125.027 31.75a "165.25 ~ 191.699

W 67.907 63.185 }}.977 [ 88.}~- -108.519 155.0~ 92.8#1

P2C7

X 7.35 8,C85 8.122 7.~95 ~,~2 8.52 7.53

Y 370.992 152.815 3.628 88.1&9 ~2a. 292 55.#19 55.27

W 103.332 z~6. 546 -~2.29 73.3C5 9}.093 85.598 91.@13

SiO~- SiO~- TeOg- TiO~ UO~- VO~ WO 2-

X 8.808 8.215 9.2#1 8.@86 9.~06 8.984 8.}8

Y 204.625 205.431 107.577 218.256 363. 558 200.125 197.~17

W 90.108 86.652 59.6 70.08 -10,627 1j.~6 89.9@9

Zr0~-

8.@13

222. 609

97.@O5


0

.=.

.¢

r~

E

©

_= > ~g

[ -

48

X I

y,

W'

G K Moiseev

Table 17. Values of parameters

c co? 1.z2o 12.984 15.475

76.971 426.985

-13.853 -9.1oi

C #- C!-

and J Sestak

X'A, Y'A and W' a ix, Equation (26)

X' 18.195 12.604

Y' -224.717 10.971

W' 296.396 72.721

X' 12.77 12.078 I}.7 17.873

Y' 140.579 168.108 75.996 -52.547

W' -I71.716 -100.605 -172.251 417.786

HPO~- Z- I~ Mo0~- - - #

X' 12.632 12.273 12.12 12.643

Y' 269.542 -19.395 50. 392 210. 231

W' -78.384 193.493 -57.53 -95.655

X' 12.97 12.718 15.377 14.566

Y' 0.09 15.309 ?7.892 53.606

W' -~>a~.135 -68.261 -7.626 -129.991

PO~- P20~- ReO x S 2-

r

12.376 13.086 12.386 12.533

174.083 146.663 -0.a69 -19.212

-114.52 12a. SL3 110.962 -15.206

czo clo Z c s- 11.779 11.74 13.301 12.073

-35.671 -36.6#5 -z4J4".011 -56.762

34.C88 83.945 92.087 172.839

F- H-

12.647 12.264 12.211 12.704

277.501 499.242 I43.052 -36.213

-1~ .489 -18o.0al -2.119 188.81

X'

y,

W'

SeO~- SeO~- SiOU- X' 12.25 12.198 13.339

Y' 90.145 104.365 251.439

W' -87.2~6 - - j ~ . ~ j ~ 9~a -107.758

VO~ WO~- ZnO~- d

X' 12.501 12.994 12.206

Y' 177.154 2#6.699 250.751

w' -22.201 -177.735 227.751

13.386 13.764

141.634 148.164

96.265 94.973

17.083 12.629

-95.759 -99.925

~7.53 55.966

C20~- p3-

12.463 12.097

155.794 -109.143

-62.476 278.608

so~- so~-

12.195 42.113

111.725 175.254

-81.206 -157.726

14.104 13.136 12.792

318.7198 246.6~- 5 314.509

-la9.995 133.6~6 -34.57

Estimation of Thermodynarnical and Thermochemical Properties

Table 18. Comparison of calculated (by (31)) and experimental (by (25)) values

of AH~98 for complex compounds

O Compound -~298' kca!/mole A, % Remarks

KScF 4

K3A12C19

KAIC14

7 CaAl2Si208

K2ZrSi207

by /25/ ( 60.o)

683.6

286.0

727.14

q012.~5

878.84

calculatior

663.9

722.3

~8.9

808.4

1104.8

8 9 1 - 7

Average ~+=

+5.66

+16.5

+11.2

+9.2

+1.5

+9.2

49

Data of /25/ are

erroneous, as condi-

oo pound

is not fulfilled.

and show that when it is used there is a definite tendency to over-

state the results.

q.4.8. C omoarison methods of. estimatin~ AH298o

These methods are well-known, so we'll give only a short

characteristics.

I. Method of Karapetyants ~.H. /26/

~ethod is based on comparison of values o£ the given property

An two analogous rows of similar compounds at the same conditions

o = AH~98(1) + AH298(II) A- B, (35) O

where AH298(II) and H~98(I) - changes of enthalpy at formation

from the elements of similar compounds of two rows (I) and (II);

A and B - constants for the given group of compounds.

O ~H298= AZ + B, (36)

where Z - any other thermodynamic property.

2. Method of Kapustinsky A.F. /27/ demands the knowledge of

H o values of A 298 for several similar compounds O

50 G K MoiseevandJ Sestak

w h e r e Z - v a l e n c y o f c a t i o n o r a n i o n i n t h e c o m D o u n d , ~ - a t o m i c

number in the system of cation or anion; A and B - coefficients for

the given group of compounds.

Yore detailed data about other, more particular methods of

calculation of standard enthalpy of compounds is given in /4, 25,

28 , 29 , 3 0 / .

q.2. STANDARD ~TH_A/ZPIES OF (~ASEOUS CO&fPObqKDS

q.2.q. Method based on atomization enthaloies of similar

substances, /3q/

Usually we find linear dependency between enthalpy o£ atomization

of similar substances and some correlating parameter.

Atomiza%ion enthalpy is thermal effect of ~eactiom of compound

decomposition into similar simple substances in the gas phase:

Atomization enthalpy is connected with H~98~ of the compound

a%t (KA )=aN2 (K)* o

Then

Atomization enthalpies of elements (standard enthalpies of

formation of simple substances in the state of one-atom ideal gas)

are given in Table 19.

As correlating parameter are used: atomic number of element

(cation or anion in the compound); electronegativity of anion;

covalent radius of anion. Correlating parameters for halogens are

given in Table 20.

O The order of AH298 calculation is the following. Compounds,

H o similar to the studied one, should be selected, for which A 298

are known. From Eq.(39) &HA~ (of compounds) are found ~dth regards

to A HA~ (of elements) from Table g9.


Table 19, Atomization enthalpies of simple substances, k J/mole

Element HAT ~lement

AI ~2~ Fe

As 300 Ga

B 5~0 Ge

~a qSO H

Be 326 HS

Bi 209 Hg

Br q~2 I

C 7q 5 In

Ca q77 Ir

Cd q 22 ~g

CI ~21 Mn

Co 428 ~o

Cr ,~O0 N

Cu 339 Nb

79 Hi

HAT Element HAT Element HAT

~7

28o

218

62~

6q

~07

2~o

67O

147

28O

66O

~7o

73o

Np

o 0

Os

P

Pb

Pd

Po

Pt

Pu

Re

Rh

Ru

S

Sb

500

25O

79O

33O

195

372

13O

565

550

775

555

655

275

266

Sc

Se

Si

Sn

Ta

Tc

Te

Th

Ti

U

V

W

Zn

Zr

277

2~2

~oo

782

66O

19o

59c

~70

530

535

85o

610

BI

Then it is shown in the ~raphic form in coordinates "~H "AT

(of compound) - correlating parameter". When !inesr ~ependenc V is

obvious, by the method of least-squares equation of dependency

"~HAT-parameter" is received. Then from the kno~m parameter of the

studied compound v~e find _~ts ~HAT. &H298kofo comnound)~ is calculated

from Eq.(40) ~th regards to HAT(Of elements).

Error for halogenides at the use of parameter-atomic number

~15%; electronegativity or radius +5% /31/. Thuogh parameter -

atomic number - is more universal: not only anion, but also cation

can be varied in similar compounds.


Table 20. Correlating parameters of halogens

Halogen

F

CI

Br

I

Atomic number

9

-~?

35

53

Electronegati- vit,y1 eV

4.0

3.3

3.0

2.66

Covalent radius I nm

0.06~

0.099

0.11~

o.153

1.2.2. Method based on the use of chemical bonds ener~ /92/

It is desirable that compared compounds should have the same stoi-

chiometry, similar molecular configuration and similar atomic pro-

oerties.

For ex., to find ~H~98(BC12F)_ we assume that

Then

° ° ( ee2)+ ° MF#). ,',14zsg =- 2/5 ,-~Izo 8 "~2~

ii. Increment of entha!py of compounds in the range 0-298.q~K

2.1. Increment o$ enthalpy of condensed compounds

2.1.1. "Triangle" method

O O The easiest way to estimate the value H298-H 0 is to use the equ

ation

O 0

received at the assumption that in the range 0-298.15 K heat capa-

city changes linearly from the zero value to Cp298.

2.q.2. Method of the Institute of ~etallur~, Georgian

Acade~4y of Sci. /33/

Offered for calculation of binary inorganic compounds:

O

t'l~ejS- ~° a : 20~'. ~298 e~ <- ~ 2 9 8 / 2 . ~ . ~ . ) , c a l / g atom, (42)


for complex oxygen compounds:

o ° = 24g o o gZ~e.p~_-£,2~,/47~ , cal/g atom. (43)

2.1.3. Empirical methods

They allow to receive desired values from the average values of

o o H298-H 0 for molecules with the same number oZ atoms and the same

class of compounds.

For condensed oxides:

o o C~) ~8-Ho =326.7 + 721.67 n, cal/mo!e, ~ =2~8~

chlorides

0 . ~°z~8-~o =-600 + 1350 n, cal/mole, rb =2&6 (~5)

Graphic methods can be used for a number of similar compounds,

increment of enthalpy for one of which is not known. In coordina-

,,{~ o H o. -o tes ~298- 0)-~298 (or N of not common atom in molecule of the

similar compound)" dependency is built; its analytical form is

o found and by parameter $298 or ~ increment of enthalpy is found.

2.2. Increment of enthalp,y of gaseou s comoounds at 298 K

2.2.1. General apDroach

In the range 298.15 K-OK with the decrease of temperature

real gaseous compounds can condense in to liquid at T v and then

crystallize at Tm, that is why estimation of substances gaseous at

298 K can be rather difficult. In the scheme is shown the change

of C from 0 to 298 K for the most common case. P

In general increment of enthalpy is the sum of three snm~ands. o o

Hz98-FIo --- A.~e~' J- &[qE4- ~@~ (46)

where every s~mmand in (46) exactly and approximately can be calcu-

lated from the following equations with regards to the scheme:

Tm

~ = I [cr(~O(')]~r -~ o.~ % [e r (~>(r .O], (~7) o L

Tm z98 "%: 5 [o~(~){o]ar-- o ~ @~- r,,3{rceO)-~ ] + [er~) 2983 ] • (~9)

Tv

54 G. KMoiseevandJ. Sestak

Then: Tm Tv {SZ

~ o- ]cr@OC)ar Cr[a)(r)aT + ~ C ~ ) ( r ) / r o T~ Tv

Thus, for exact estimation of increment of enthalpy it is

necessary to know temperatures of evaporation-condensation (Tv) ,

of melting-crystallization (Tm) , temperature dependencies of heat

capacities of all phases. Approximate calculation, based on the

assumption about linear dependency Cp(T) in every aggregate state of

substance (gas at T~298 K) needs the knowledge of values of heat

capacities for gas, liquid and crystal at Tm, T v and 298 K.

o o 2.2.2. Estimation of H298-H 0 for ideal Rases

Though, usually in thermodynamic calculations gaseous substa-

nces are considered ideal gases, that is final size of gas molecu-

les and intermolecular interaction are not considered. It is assu-

med that up to 0, K gaseous state of substance is preserved. This

H ° H ° allows to estimate 298- 0 with the use of temperature dependency

Cp(T) of gas, obtained at T~ 298 K or to apply quantum-mechanical

calculations

Cp = Cprog r + Cosci I + Cturn , (51)

where constituents of progressive, oscillatory and turning degrees

of freedom of gas molecule are in different dependencies from the

temperature, but at T÷O they also tend to zero /~/.

For estimation of enthalpy increment it is possible to use the

"triangle" method (Eq.(4q)); empirical dependencies of average valu-

o o es H298-H 0 from the number of atoms in molecules of similar compounds.

For ex., for gaseous oxides and chlorides

o o H298-H 0 = 1410 + 420% cal/mole, (52)

o o H298-H 0 = 700 + 735n, cal/mole, (53)

where n - number of atoms in molecule of compound.

EstimationofThermodynamicalandThermochemicalPrope~ies 55

III. ENTROPY IN STANDAZO CONDITIONS

3.1. Condensed compounds

3.1.1. Increments method of Latimer /~>/

For ionic compounds entropy is calculated as an additive value

by values, empirically found for cation and anion constituents of

the compound. In Tables 21, 22 are given data for calculations.

Example of calculation for A12(SO@)3:

o $298= 2.8 + 3'13.7 = 57 cal/(K mole)

o

Table 21. Contributions of cations to Sz9 s

Element

Ag

AI

As

Au

B

Ba

Be

Bi

C

Ca

Cd

Ce

Co

Cr

Cs

Cu

Dy

Er

aS, cal/K

12.8

8.0

11 .a5

15.3

4 . 9

13.1

15.6

5.2

1 9 . 3

12.9

13.8

10.6

10.2

13.6

10.8

14./-1

14.5

Element

Eu

Fe

Ga

Gd

Ge

Hf

Hg

Ho

In

Ir

K

r,a

Li

Lu

Mg

~n

~o

Na

aS, cal/K

14.1

10.4

11.2

%.3

11.3

14-.8

15.4

1~.5

13.o

15.2

9.2

13.8

3.5

1 4 . 8

7 . 6

lo.}

12.3

7.5

Element

Nb

Nd

Ni

Os

Pb

Pd

Pr

Bt

Ra

Rb

Re

Rh

Ru

S

Sb

Sc

Se

Si

AS, cal/K

12.2

13.9

lO.5

15.1

15.5

12.7

13.8

15.2

15.8

11.9

1 5 . 0

12.5

12.5

(8 .5)

13.2

9-7

<11.6)

8.1

Element

Sm

Sn

Sr

Ta

Tb

Te

Th

Ti

TI

U

V

W

Y

Yb

Zn

Zr

Z~S, cal/K

14.1

13.1

12.0

14,9

14.3

C13.q)

15.9

9.8

15.4

16.0

10.1

15.0

12.0

14.7

lo.9

12.1

56

Anion aS, cal/K, at cation

4 ' I I q 2

F- 5.3

ci- 9.7

Br- 1~.0

l- qa.6

OH- 5.0

0 2- 4.3

S 2- 8.0

G. K. Moiseev and J. Sestak

Table 22. Contributions of anions to S~98

Anion mS, cal/K, at cation

H

H.6

8.1

40.?

13.o

1.0

3.65

cha£~e 2 3

H

H.H 4.0

8.q 6.9

1 0 . 9 9 . 9

43.6 12.5

4.5 3.0

o.5 o.5

5.0 5.~5

6 7 SO~- 22

Se 2- 10.1

Te 2- 10.9

CO~- 15.2 J

SiO~- 14.5

PO~ 24

charge 2 3

8 _ 9

47.2 13. c,

7.1 8 .0

10 .2 10 .3

11 .4 8 .0

1o.5 7.o

17.o 12 .0

4 ~b

dO

7.8

8.8

3.1.2. Increment methods of Kumok V.N. /~6/, and Tsasareishvili

o.s. /37/

o and C of cations In Tables 23, 24 are given increments $298 P298

and anions, corresponaingly, from /36/.

O is calculated by the scheme: a~C rio+mS A $295

Author of /36/ did the work on correlation of increments values by

method of the least squares for a lot of data. This allows to con-

sider the received data more accurate, than those given in Tables

2d and 22.

D.S. Tsagareishvili /37/ also suggests increments of anions

O and cations for calculating $298 of crystal inorganic compounds.

O 3.1. 3 . Some empirical formulae for calculation of $298

On the base of statistical analysis of the known data were

O found average values of $298 for different compounds with the same

number of atoms in molecule. The following equation was offered /38/:

where A - number of atoms in molecule. Average deviation of values

i s ~ 35%.

EstimationofThermodynamicalandThermochemicalP~pedies 57 ~rom / '39/ entropy linearlj depe~ds on !ogari~hm of molecular

weight 0

9

where M - molecular weight, A and B - constants for similar compounds.

From the values of entropy of two substances from the group of com-

pounds , it is possible to find unknow~ entropies for ether com-

pounds of this group.

In /29/ in given the equation

~e_Q~L~98j/~ x ~ c~ns'~-~ 4,@~, , J/(K mole) (56)

where n - number of atoms in the compound. Equation is valid for

haloid salts and su!phides.

O To estimate $298 for a number of compounds it is possible to use

additive sum of their components' entropies. In particular, for

solid ordered alloys /4/, silicates /39/, intermetallides, phosphi-

des, tellurides, selenides, binary oxides /29/.

~ethods of comparison calculation, offered in /26/ are considered

the most accurate. In the rows of similar substances the values of

entropy are approximately connected by linear ratios: D o

o

where indexes I and II are related to rows of similar compounds;

A, B, a, b - empirical constants for these rows.

In /4d/ are offered equations for calculation of entropies of

metal chlorides as functions of their molecular masses (~) (cal/K

mole):

Monochlorides

Dichlorides

o

S~s~45o M-9.1 ,

~richlorides ~ -- Z~ g ~ M - 22.4 ~ (59)

Tetrachlorides $~z~8- 8i.9 ~M- 449.0.

In /25/ are described more complicated calculation methods, offered


Table 23. Contributions of cations to S298 and

Cation z~S <bloo~)

I 2

H20 4'1.2

~3 65.a

NO + 80. S

N~ 65.8

Li + 14.5

~a + 32.6

K + ~7.2

Rb 56.0

Cs 67.6

Pr + 78.6

Cu + 42.7

Cu 2+ 36.5

Ag + 58.7

Ce 2+. 4 8 . 8

Ce 3+ 48.0

C e 4'+ 42.5

Pr 5+ 55.~

~d ~÷ 4 8 . 2

Pm ~+ 56.8

Sm 2+ 51 - 5

Sm 3+ 50.0

Eu 2+ 68.0

Eu 3+ 49.3

~3+ 53.7

Tb 5+ 59.0

Tb 2+ 65.1

~C P

a c . 8

7C .4

56.2

20.7

26.8

28.0

30.8

3'1 . I

29 .5

25.5

25.0

28.6

27.6

31 .~

28.2

31.5

28.3

3'1.4

35.7

~#+.4

29.1

33.3

27.8

33.o

24.3

Cation & S

4

Au + 29.7

Au 3+ z~O.4

Be 2+ 2.7

Mg 2+ 16.5

Cu 2+ 32. O

Sr 2+ 43.0

Ba 2+ 5 ~ - 6

Ra 2+ 61

Zn 2+ 3~. 2

Cd 2+ 46.4

Hg 2+ 124.7

Hg 2+ 6~. 3

B ~+ 4. ?

UO 2+ 77- 3

NP 4+ 55 - 2

Pu 2+ 53.5

ima 3+ 50.5

Pu ~+ 57. I

Ti 2+ 23.3

Ti 3+ 23.7

Ti ~+ 29. I

Zr 2÷ 35-O

Zr 3+ 3"I .0

Zr 4+ 23.6

Hf ~+ 32.7

S± ~-+ 16 .9

&Cp

6

q2.6

22.2

27.3

29.3

28.4

29.6

25.5

28. O

52.6

27.7

6.1

64.6

32.7

40.7

28.4

35.1

2'1.3

23 .3

25.5

24.7

25.O

22.9

2O. 2

12.1

C,,~, J/(K mole)

Cation AS AC P

b g i

AI ~+

Ga 2+

Ga ~+

Ca÷

in +

in 2÷

in 3+

T1 +

TI 3+

Sc ~

y3+

y2÷

La 2+

La3 ÷

Bi 3÷

Cr 2÷

Cr 3÷

Cr ~+

~o 2÷

~o ~

w

Tc ~-+

~n 2+

~n 3+

~n ~

Re 4+

Fe 2+

}I . e

29.8

(39.~)

53.3

47.0

38.5

68.7

4 8 . 7

20.5

50.8

32.2

42.0

40.4

52.8

36.4

25.8

25.6

19.3

28.8

42.9

4 6 . 5

34.7

28.4

39.3

42.3

9

17.6

<22.75) 2fl .6

(2~.9)

23.7

26.5

25.7

3o.9

21 .2

24.0

22.5

29.5

29.3

29.0

21.0

29. I

21 .8

23.5

2'1 .&

21.6

30.5

27.9

25.0

21.2

28.7

q

Tb z~+

Dy 2+

Dy3 +

Ho 2+

Hc 3+

Tm 5÷

yb 2+

k-c3 +

Lu3 +

Th 2+

Th 3+

Th 4+

U 2+

U 3+

U 5+

US+


Table 23. Cont.

2 ,~ ,, q. ~ 6

67.1 - Ge 2÷ 36.7 25.8

55 - 2 84.0 Ge a~+ 26 .~ 27.0

58.9 31.0 Sn 2+ 51.9 27.8

4`6.2 26.1 Sn 4+ 37.3 25.8

G0.1 29.6 Pb 2+ 62.6 29.3

58.1 29.1 pb ~ aT. 3 2~ .9

=L. ~ 33- 3 V 2+ ~= ~. 2 2"I,. 6

51.I 29.0 V "~+ 27.1 25.0

.~-= .,. ~ - 32.5 V z~+ 2 5 . 9 23.9

36.5 26.7 Nb ~+ 27.6 2~.0

47.9 26.d Nb z~+ 3q.7 27.5

~3. ~ 29.7 ~b ~+ L 1 . & 2 5 . 7

L-1.5 2 8 . 2 Z'TcO t 5 7 - 7 5~.,2

53.0 ~C.O NbO 3+ 3"1.L LLT-6

5J .~ 3,~.! Ta ~+ a< .O 27.7

=c.d ~0.~ Ta ~+ ~3.0 26 3 jj

=~ As ~+ ~ ~7.8 ~F.8 ~9.~ 26.7

58.8 ~4.2 Sb 3+ 50.2 ~O.j~

7 8

Pe 3+ 27.6 26.2

Co 2+ 37.6 3q.3

Co 3+ 12.5 12.4

Ni 2+ 28.6 26.7

Ni 3* 28. C -

Pd 2÷ ==.5 2C 6

Pt a~ 28.2 24`. 2

~e~N + "151. n "t 3.5.1

59

by V.A. Kireev. For ex., it is possible to use the equation:

o = X ' ~ - '~L* ''~'" (60) L = l

0 where ~- number of similar atoms in the compound i S i - standard

entropies of atoms in the compound; AS - change of entropy in the

reaction of compounds formation from elements (found from data for

similar compounds).

For ex., let's calculate S~98(ZrC12) from data for T~C12.

For TiC12


Table 24. Contributions o f anions to S ~ and C , .I/(K mole)

Anisn

K-

D-

T-

C!

cicZ i

c i c l

ClO 2

BrCl

2

zc 2

At

0 H -

0 2 -

q C 2

$ 2 -

d

2- s5 SC~-

SC~-

2

SS

99.2

~C .C

22.9

58.~

6s. 5

q26.5

72.~

2~. 7

~ .7

7~.9

~0.9

62.~

27.4

~.}

162.5

75.1

7~. 5

85.7

C Anion

d£. %

2C. }

2&. C

5a.s

79 .a

77.5

92.q

26.5

77.9

2~. 9

2 6 . 4

~? .~

33.7

~7.9

24.6

~2 .2

~o9.8

6 7 . 0

73.9

SeO~- .1

SeO~-

Te 2 -

TeO~-

p~

2- P2

04

HpO 2-

o£q P20~ -

As ~

As03

AsO~.

HAsO 2-

As20 ~-

Sb-~-

BiF-

" S

75

a ? . o

bc .8

9 2 . 2

6~ .5

~2.3

88.8

76.9

21 .5

26 .0

59 ,~

68.6

76.5

92.5

I06

127.5

!77.3

3~.o

?q .~

87.5

9~.~

qO6.&~

q~9.

~6.7

58.9

~.g

36.0

~C Anion Ix S J.

72..5 HCO 2 ~. £

~ 6 . 6 CHTCC7 " ~8.,3

2 L . c ~ ~n~,,~ t 28. a .,1£_

- S z 2 ~ . -

~5 .5 SiO~- 6 2 . ~

5o.0 s~ ~c~- 9 L.

~9.~ Si~- 9 c . 3

Z @ o . J . o

c~.-9 ~ GeO6- oN. P

2~. " s~og- 7~. 9

6 6 . 0 B ~ ~-r"

73.9 go~ ~5.c

77.6 BO 2 }6.7

9"I .2 BF~ 1q6.9

~o~.9 s~f s~.6

q20 .0 A1H~- 9 8 . 6 o

2~.9 A.1H Z 6zl-. ~ }

69. "I Ale 2 33.5

- AI~ 6- 141.3

96.8 GaO 2 ~-9.2

- GaH~ 82.5

25 .% Bel~ 74.7

23.@ Bell- "109.7

1~.'1 CuCI 2- 178.4

5&~.~ AgT~ 2~7.2

U

3"q .y

qq2.a-

23.=

75. ~

~9. T

q 5 - 5

68.2

165 .c

qq m - /

52.c

63.~

o~

68.0

qqS.0

LL?.~

.1~r: ~/.9

6 8 . 9

22.?

96 .q

123.0

qO~ ,6


Table 24. Cont.

SC~Y- "1 ~.',. 2

S ~2- ~.6 2~3

S2C2- 1~-.3 .~

S ~0~- 4 ~,.-. = r_ -

Se 2- =~. 2

ZrO~- 6T.~

H~o~- ~. P

vat 7~.3 P

vo~ ~c.o

V20 < q68.0

crc< ~5..

c~- - P

CrO 2- 9~. 6

~r2u 7 91.2

~ioC 2- 65.9

~Io0~- 97.2

6C.6 0¢~- 5&.Y 59.~

- SCN- 66.5 ~'~ - p

~5C.~ co~- ~.~ ~=.~ P

n~.S C2C i- 7~.~ 79.6

T~.= Mc2C 8" 175.2 163.6

f - WC 9C ~ 89. ?

7c.9 w2c ~- ~85.2 181.~

8707 ~ MnC2 56.1 -

158.9 NinO~ ~24.6 9~.~

7a.9 Mnc~- - 86.8

79.~ MnO~- - 97.5

8&.9 ReO~ 1~5.6 96 . a

86.4 ReC!~- 9~c 4 156.6

q66 = ReBr~- :~ .. .. .c~.7 ,,C ,a

Fe02 5 = 6 59.7

89.8 ~e(~0~-2?9.5 232.~ I

v 8 f

uc~ ~oo.~ J

UC i- qO~.~

u~o~- q99.~

~zo 3 ~9.

TiO~- ~r ,'~o

Ti20 ~- ~0~.6 D-

COCI~ ~c~

coCC~)£-26c.~

PtCI~- 86.0

P~Cl~- 239.G O

PtBr~- 29~.9

62.~

97.9

q7 q Zz

~q -9

~5 -9

~2fi.~

<22.8

223.9

la9 .a

T h e n f r o m (60)

(62)

In /42/ is offered equation for calculation

pies for tungstates of rare-earth elements

O --0 /(~)

of standard entro-

C63)

61

w h e r e S z ~ g X SZ~I~ >/Sz~g

t h a t i s , sum o f s t a n d a r d e n t r o p i e s o f o x i d e s o f r a r e - e a r t h m e t a l s

a n d t u n g s t e n t r i o x i d e ; x , y - s t o i c h i o m e t r i c a l c o e f f i c i e n t s ; N A -

62 G.K. MoiseevandJ. Sestak i +

mole fraction of rare-earth metal oxide, equal to x/kx y).

In /42/ there is data on standard entropy for 3q tungstate.

3.2. Gaseous comoounds

3.2.1. Some emoirical dependencies

In /4/ are recommended the following empirical equations for

calculation of standard entropy of gases (Table 25), depending on

their molecular mass (m) and mumber of atoms in molecule (n).

In /29/ is given empirical equation for two-atomic gases:

$29g 2 2 0 . 8 + O . t 8 ~ - 4 0 0 ~ M ~ ~ - ~ (65)

where ~ - m o l e c u l a r w e i g h t between 20-300~

for many-atomic gases:

In /32/ it is recommended to calculate standard entropy of gases

by equation

9g8

where M - molecular mass, a and b - coeffioients, given for ~ases

of different composition in Table 26.

For estimation of average standard entropy of gases depending on

the number of atoms in compound, in /A3/ the following equation

is offered:

$298 3 5 5 + 8 . 0 A , Ka~/Kmo£, (68)

where A = I - 8 . A t Am8 app rox ima te average s t a n d a r d e n t r o p y i s

96.73 cal/(K mole).

Table 25. Empiric equ~ions ~rcalcul~ion of S~8 ofgases, cal/(K mole)

o S ° n S298 n ..... 298

I

2

P

26.5 + 79 ig m, t 1.6

24.2 + 16.3 !g m, +1.4

9 ÷ 26.7 ig m, + q.8

45

5

1.8 + 35 Ig m, + 1.6

-~I.5 ÷ ~9.5 ig m, + 2.7


Table 26. Coet'ficicnts in Equation (67)

63

Kind of a b ±&,°/o compol~nd

AB

AB~

AB~ P

AB 4

/

AB.

o.o67

o.~

o.q9

0 . 2 a

o.3o

o. 31

5.15

~.97

~.79

4.55

~.2~

4.d4

3.0

2.7

2.8

2.5

~.6

IV. HEAT CAPACITY

4.1. Condensed compounds

4.1.1. Heat capacity at standard conditions

Usually increment methods are used, that is heat capacity of a

compound is a sum of cation and anion constituents. Increments

found in /36/ are given in Tables 23 and 24.

Increments of anions and cations of standard heat capacity,

found in /4/, are given in Tables 27 and 28.

B.K.Kassenov /~/ offered a scheme of anions increments

(P~98) for calculation of standard thermodynamical constants of

salts ( Fz~ & a z~@ ) %Z~@ ~nJ ~ 2~ ) with the use of standard

reference data on increments of cations, considered in water solu-

tion at infinite dilution (F~98[Mn+J)

o ~o - n* FL

where K - conversion factor.

On the base of similarity factor,

O

expression for calculation of standard heat capacity of salts was

offered o L


Table 2 7 Anion constituents o f C~8 , cal/K

[Anion 91 F C! Br I 0 S ., Se ~::~ OK

. : c . , . . ~ > . . , 6.2 c.p ....

~ n i c n SO~ XO~ P CO 5 S i C!~D 4 MoO 4 WO~ CO L

~a 1 8 . - ~ 1 5 . 4 5 . 6 1 4 . 0 5 . 9 2 1 . 7 2 1 . 6 9= ~ ~ . 6

Data on increments of standard entropy of cations[~n~" ~is taken

from the reference literature. Values of increments of anions

9Z~g~m-jF~] for salts of alkali metals are given in Table 29 toge-

ther with values K andS.

Estimation of heat capacity at standard conditions is not

enough for estimation of its temperature dependency at constant

pressure.

4.1.2. Calculation of temperatur e dependency of heat capacity

Usually it is shown as

Cff = b.I#% r - c IO T -z (72)

Table 28. Cation constituents o f heat capacity, cal/K

Metal ~ 2~etal ~ Metal ~K ~ e t a ! ~w,, M e t a l ~ K

A~ 6.15 Cr 5-5 in 5.8 Ni 6.6 Sr 6.1

Al 4.7 Cs 6.. ~ Ir 5.7 P 3.4 Ta 5-5

As 6.0 Cu 6.0 K 6.2 Ob 8.4 Th 8.1

Ba 6.5 ~e 6.2 La 6.1 Or 5.g Ti = ~ jo~

Be 2.3 Ga 5.0 Li @.7 Rb 6.3 TI 6.6

Bi 6.a Gd 5.6 Mg 4.7 Sb 5.7 U 6.4

Ca 5.9 Ge 4.8 ~n 5.6 Se 5.1 V 5-3

Cd 5-5 H£ 6.1 ~a 6.2 Si - Y 6.C

Ce 5.6 Hg 6.0 Nb 5.5 Sm 6.0 Zn 5.2

Co 6.7 Ho ~ ~-5 Nd 5.8 Sn 5.6 Zr . c.7

Sc 5.12


Table 29. Values of" increments of" anions, K and N for calculation C~,,

65

Anions

¢icl

CIC Z

*SO 2-

*SC2-

SeC~-

J

H0~ 2 a _ _

~CC~-

C 20~ -

cH3co 2

Z

Re0~

i~o0 2-

*CrO~-

CI

Br

!-

60, ~+P.6 J - -

/9.9+o.o

~" ~+7.9 OC°

sc.O+_.~.~

9.6÷C.~

32.8+2.3

_-I. 8+C.

62.L+2.2

52..~+~.¢

-88 ~+-~a - j ,#.~

z~-. 6_+z~. 7 --iF'

25.9

47.o

-77.~

78.7

94.6+ 0 . 8

q2a.2+~.O

20.5+ ~. &

~8 . 5

27.~+~.~

-6.7tC.4

2q .5_+5.2

27. ~_L.9

34.~t5.2

K

o. 548

o.57

c.595

O. 7C7

r~r~ C. F ,/'q

~ .OP~

O. 8L

0.~57

0.~

q .OCt

q. 009

o . S q q

C. 8~-8

4.558

0.578

0.556

0 . 6 ~ 2

0. 769

0.573

o. 746

O. ~'~2

0 . 6 0 9

o. 594

o.559

fl

C. 67 •

c.7~a

O. 878

u.o78

~.6Ed %

C. $7

0 .76q

0 . 5 8 2

0.757

o. G6~

q .c9c

0.63. ~

I .o79

0.69 Z~

Q ~

C <'Pro

C. 7nZL

$.~67

0.6~

0 r, P . CAI-O

0.625

o.552

o.552

- for compounds v, ith K, Rb, Cs.

66 G.K. Moiseev and J, Sestak

and also for calculation of reduced Gibbs energies in sub-program

TER~OS of package "ASTRA-~" /I/.

Nany methods have been offered for calculation of temperature

dependency of heat capacity.

One of them is method of Landia i.A. /45/, in detail descri-

bed also in /25/, based on the use of standard entropy and melting

temperature. Calculation is done in the following way.

a ) atomic heat capacity at constant volume (cal/K atom) is

found for oxygen-free compounds:

where S ~ = S°//'I1_ ; n - n u m b e r o f atoms in compound; T - temperature,

at which heat capacity is calculated.

If T is more than characteristic temperature 8,

£ -_ ~ s ~ / % ~ , (74)

then calculation is done by equation

c..j: -- a,G - (22oo/[%~C~ + K {r-e }~], (~5)

where K=~/T m , at K~O.35 its value is taken equal to O.5; T m-

melting temperature of the substance.

b) for complex oxygen compounds and complex oxides, in which

number of oxygen atoms is more than number of atoms of other ele-

ment, the following equation is offered

S~+.T nF- " (76)

where n I and n 2 - numbers of atoms in anion and cation of compound.

At high temperatures when ~/qT~ 1.6, calculation is done by

equation:

C A (77)

in which

~£ ~ S~£ (78)

Transition to heat capacity at constant pressure is done by the


e qua ti on

Cp = C,v ~- p - T 3/2 (79)

w h e r e

1.24 r ~ egg'~ZT3/2.1~ 5 P: y-E-~eet <~-vo~: J , (so) p 29e

where Cvo. £ - heat capacity at 298 K.

Mole heat capacity is ~po~" ~'

Temperature dependency is found in the following way:

- Cp i8 c a l c u l a t e d a t th ree tempera tures (Cp.1, Cp2 , Gp ) ; J

- temperature coefficients are calculated by way of putting Cp

into one of formulae and solution of created system of equations.

In method of L.I.Ivanova /46/ it is offered to calculate heat

capacity for various substances and compounds in the range O.3Tph.tr

Tph.tr from equation

v_ Cp = n. (~.2~-5 + "1.987 TpS.£r " ), ~e/K- mo[, (81)

where Tph.tr.- temperature of phase transition, n - number of

atoms in molecule.

To calculate coefficients a, b, c in Eq.(72) a number o£

equations have been offered.

In /29/ are given equations for their estimation, when

values o£ enthalpy at three temperatures or value C and two P298

values of enthalpy at high temperatures are kno~. Then after

solution o£ the system:

C~,: a~- 298. t~6 - c/(2~s.~s)~ (82)

N O _ O

coefficients can be found.

In /4/ the following equations are offered:

©5~ T,.,.,- o.2'~8)~ (85)

68 G. K, Moiseev and J. Sestak

-2 / -~

where T m - melting temperature;K; n - number of atoms in mclecule.

In /¢7/ the following equations are offerea for calculation:

where~=O.228~ andS-Debye temperature. It can be evaluated:

and, thus

K ~ 0.228" 0.~51~. = 0 .0758 q-m_. (90)

Thus, for evaluation calculation of Cp from (88) it is nece-

ssary to know only T m - melting temperature.

If the value C at temperature T is known, then value T Pat

can be found from the Table 30.

To reduce Eq.(88) to Kelly's equation, it is possible to

calculate coefficients by the following dependencies:

a = 5.5,

b = o.125// '#, (9"~)

These coefficients are calculated per g-atom of the substance. If

C is known, from Table 30 it is possible to findS, and P298at

further calculations will be easy.

In some cases for estimation of heat capacity of complex

compounds, like binary oxides, nitrides and other compounds, which

can be presented as a sum of simple compounds, and also for metal-

lides, it is possible to use method of addition of temperature

dependencies of heat capacity of each simple substance (metal),

taken proportionally to mole (atomic) fraction of each of them.

For ex., for Ne~Ne' '

Cp= o7 c r . (92)

T/~

39

58

57

5S

.i..

3~

~z

32

5~

3C

29

26

2F

26

2y

2z~

2Y

22

2.1

2C

29

18

.17

"16


Table 30. Values of functions

C D

2

.1o.5

.1o. ~75

qo.25o

10 .125

qO.O0

~. ~ F

9.s'5

9.625

9.5

9.~75

9.25

9.q25

9 .oOC

8.87~

8.75

8 . 6 2

8 . ~ 9 9

8 . ~7 ~

8 .2a9

8. q2a

7.999

7.67~

7.7~9

'7.623

7.a98

Cp,, =f(T/ z ), calculated from Eq. (88)

T/z c T/s c T/~ P

> ~ > 6 '7

q5 7. 373 30 .=.665 d . ~.~

qa 7.2'#7 2.9 5.632 0.96

13 7 . 1 2 2 2 . 8 5 . 5 9 5 0 . 9 6

i q 6 . 996 2 .7 5 .556 0 . 9 a

q2 6 .£7 2 .6 ~.<14~ 0 . 9 2

1,2 :~.. ,~;, : ~_.° 5 ~-~* . . . . . C. 90

9 = 6 . 6 a 2 . ~ ~ ~na n ~

& ~ ~ . 9 ~ 2 P . 2 ~ . s v 9 0 . 8 ~

- ~ 7 5.2C ~ ~ . 0 o . c , 2 . ~ 0 . 8 2

,- ~c~ o ~ 5."1"1'1 0 .80 7-5 ~ .....

7 .0 6 . ~ 5 6 '1.9 p~. :',_,0 ~,,' O. 78

C. m 6 . 5 6 . 2 9 "1.8 ~ . ~ 8 f 0 .76

6 . 0 E.22q 1.~, ~ . , ~ o . . . . . 0 . 7 a

~ - ~ r~ o . ~ " ~ . ~ '7' O . m

~ . 8 6.845 q.~5 4.278 0 . 6 8

~.6 6 .04~ 4 . 4 a.159 0.66

~ . a ~.q6 "1.35 a.C~q 0.6~

a.2 5.9~5 1 . 3 0 ~. 69.1 C . 6 2

a . O 5 .908 ' 1 .25 =.Ta -4 0 . 6 0

3.8 .,.~= ~68 .1.20 3.579 o.56

3.6 5.825 I.'15 3.~-05 0.56

3.a 5.778 I.'10 j.= 2"19 0.5a

3-2 5.725 1.05 3.021 0.52

cp ~ / ~ Cp

8 9 1o

2.8.13 0.5 0.618

2.726 0.48 0.554

~.6~8 O.a6 O.493

2.5~9 O.L~+ 0.a36

2.~5& O.a2 0.38 3

2 . 3 6 6 O.a-C C . ~ a

2 . 2 7 Z# 0 . 3 9 ~ . ~ ~

2.18C 0. }~ , C. 9FO~.

~rr 7 2. 086 C. 37 O.c.'s ;

-1.99'1 0 . 3 6 0 . 2 4 7

1 . 8 9 2 0 . 3 5 0 . 2 2 8

"1.801 O . ~ 0.2'1C

"t Tn~ O. 0. . ~, ~ 192

643 9 . ~ 2 . . . . . " ~ . ; # 0

q .519 C.3~ C.16

q.a27 0.30 7 . 1 a 6

7= O oO . 132 '1. ~ 6 . ~ 0

1 . 2 a 7 0 . 2 8 0.1"19

• "159 o . 2 7 0."107

q . cT~ 0 . 2 6 0 . 0 9 ~ c

c.99 o.25 o.o85~

o.9"1 o.24 0.0754

0.832 0.20 O.Oa}9

0.757 0.15 0.0186

0.686 o.'1o o.oc55

69

70 G. KMoiseevandJ. Sestak

In /38/ On the base of analysis of known data are offered

dependencies of average heat capacities from the temperature of

number of atoms in molecule of compound (A), cal/(K mole):

4+ C93)

Average error is about ÷ 15%.

When experimental curve Cp=f(T) in the range 0-298 K is ~uown,

then with the use of (~C~/~T)p,298 for T=29& K and also melting

temperature (Tm) it is possible to calculate coefficients in Eq.(72)

with the use of the following formulae /37/ (cal/K mole): o 0.25

¢= o 4 3 z ~ ° ~ g c ~ / ~ T ) p , ze8 - ~ ~ (gs) O

When Tm, C ° p298 and~ (Debye temperature) are known, coefficients

in Eq.(72) can be calculated from equations, offered in paper /48/,

ca~/~ mole:

0 - 0.54 C~Z~S /TrrL, (96)

o

n - number o£ atoms i n m o l e c u l e o f compound; ~ - o h a r a c t e r i - where

Debye temperature, found by the value ~d~with the use of stic

Debye's functions.

4.2. Heat capacit~ in the liquid state

For the purposes of thermodynamic simulation Cp of liquid

usually assumed to be constant.

For its evaluation Kelly /$9/ found from analysis of the

known data that at T m Cp~ 7.25±0.5 cal/(K atom). Substances with

several crystal modifications and low T m (~150~C) are not consi-

dered here. D.S. Tsagareishvili /50/ found that the change of heat

capacity at transition from crystal to liquid state is equal to

(~/4)AS of melting. Then heat capacity of the melt can be calcula-

ted from equation

EstimationofThermodynamicalandThermochemicalPropedies 71

Thus, it is necessary to know Cp(C~)=f(T)~ Tm, ±H m to estimate

CpC ~ ) .

AS pointed out in /31/, for accurate calculation of Cp(1) it

is necessary to know the following characteristics: T andaH of

boiling, T critical and C v- isobaric heat caoaeity of vapor chase.

Then ~

It seems difficult to use Eq.(90). So, for estimation of C (i) it

is possible to use equation:

where C~boiling - heat capacity of vapor at Tboilin6"

Eq.(98) and (99) are valid mainly for molecular liquids.

For the first evaluation of Gp(1) of compound one can use

method of Neimann-Kopp:

Recommended valued of atomic constituents for calculations from

Eq.(98) are given according to data of /3~/ in Table 31.

It should be pointed out that data about methods for calcula-

tion of Cp(1) are very limited. Many researchers think that Cp(!)~

Cp~r) td0-15% at T m. There are even more general evaluations. For

ex., in /29/ it is said that for simple melted substances (elements)

Cp(1)~ ?.#-7-5 cal/(K g-atom); for inorganic compounds = 8-8.q cal/

(K g-atom). Nitrates and sulphates have Cp(1)= 5-7 call(K g-atom);

borates, titanates, chlorates and other salts of oxygen acids have

Cp(1)~7.5-9 cal/(K g-atom); hydroxides and silicates ~ 7 cal/(K g-

atom).

~-3- Heat capacit~ of ~ases

For most gases with simple molecules heat capacities are cal-

culated up to 6000-20000 K by statistical methods and given in the

reference luterature.


TaMe 31. Atomic constituents of Cp(]), J/ (K mole)

Element C Element C Element C O O D

As

B

Ba

Be

Bi

Br

C

Cd

Cl

Cr

Cu

Fe

&2.2

28.2

Z . -~ @

~6.8

,q.-'l .~

11.7

3~ .~-

z,c. _5

~-5.6

29 =

.q-q .0

~}e

H

Hg

!

in

mg

Mc

H

~b

Hi

Cs

P

Pb

~a

q~.C

b':-. ?

29. ~ J

xx © J4 "~

I ,8 .8

25.q

35.2

38.1

25.q

J

"~+-5

~2.7

aa.S

Re

S

Sb

Si

Sn

Ta

Tc

Te

Ti

U

V

W

Zn

Zr

5! .C

~p.p

~S.D

~8.9

~2.7

39.7 21 .~

~p.D

23.0

16.7

53.~

For one-atom gases C s5cal/(K mole); for two-atom - 7 cal/ P298

(K mole). At 298-2300 K for two-atom gases ,frith N~#0 temperature

dependency is described by equation

Cp = g.7+ 463- T, ~ / / K moe. (101)

For gases with ~>q00 Cp- 9 cal/(K mole) in the given temperature

range /4/.

For estimation of standard heat capacity of gases in /31/ a

number of equations are recommended

a) Cp2~8 = ~pmax (6.47. ~04 T~, o.S7i ) , ]/K moe, where - max heat capacity of gaseous substance, Cpma x for linear and non-linear molecules from equations:

(lO2) calculated


where R - gas constant~ m - summarized number of atoms in molecule

of compound.

b) #_n_ C~2~8 = a & M+ b, J/K.mo~, (.1o~-)

where M - molecular mass of compoun@, and coefficients of equation

for different types of compounds are given in Table 32.

In /43/ for calculation of average temperature dependency of

ideal gases the following equations are offered, cal/(K mole):

-6 -i c~,~ (~ 2 - 0 - (.,~,,,-~ ~>o T + (o.o,~6 ~,-oo~')~# ff~ ~,= 2+ 7, ~.1o5)

-:~-~" (o.586A 2.46").4O~.T - 2 A-=7+24 ( '106) 4 0 , ~

where A - number of atoms in molecule.

Since the appearance of Eq.(.103) and (.10&) does not correspond

to conventional form of presentation (72), on the base of the

same initial data as in /43/, is offered a way o£ calculation,

shown in Table 33.

At A~7 coefficients A, B, C, D, E can be calculated for given

temperature ranges from equations:

A = 10.07 + 0.54~,

B = 0.628 + ~.257n,

c = - 3 . o 5 + 5 . 7 8 3 n , ( .107)

D = -0.86 "10 -2 + 0 .386 1 0 - 2 n ,

E = - 0 . 9 10 - 3 + O.q-~ x I 0 - 3 n .

Table 32. Coefficients in Eq. (102)

Coeffi- cients

a

b

AB~

C. Iq/4

TTpe of comoound

o . 1 3 7 o . 1 5 8

3.527 3.655

AB 5 AB 6

O. 182 O. 2O5

3'. 660 3 .677


Table 33. Temperature dependencies of heat capacity of gases, cal/(K mole)

N'~z~ber of atoms in molecule

d

2

interval, K

2

29 8- 3000

5000-6CCC

298-10C0

1CCO- 3007

}C CC- 1C SC

29d-qCCC

fiCO0-30CC

30CC-5000

298-'1000

qOOC-~OCO

}CO0-6CO0

ICOO-SCCC

~COC-600C

29a-Iooo

1 c o c - 3 c o o

3000-6C00

Equation or value

6.978 + 7.~d qO -1' '2

lfi.S~ + 2.O~ 40 -5 T (A ÷ ~T)

d}.@! + 2.9 qO -~ T (B + ET)

"~L.2~ (c)

"12.1 .,- 6 . ~ 1C ~ T

4,-:, ,,. 59 + 6 .4 - 10 -Zi"

I 2 0 . "11

"12.~8 + 1 . 0 7 flO - d T

21 ~ ~ t . 2 4 - =, 10 . 3 T

25.67

I<=.8 ~. + 1 . 5 0 2 !0 - 2 T

2 6 . q 9 + 1 . 7 3 1'0 - 3 T

3! .4

13.2.7 + ~.94- i c -2 T

30./4-6 + 2.214- 10 - 3

37.no

v. TkTWPERATURE, ~TROPY AND ~T~ALPY OF I~T.TI~G

5.1. ~eitin6 te.mperature

Its estimation by calculation is rather difficult.

For molecular inorganic compounds for approximate evaluation

of T m in /3d/ the following formula is offered b4°/3

T, .= @×+ ~ ~ ( l o8 )

where % I'} d'2

where ~- parahor, ~5o[~ density of liquid at Tboiling , g/cm ~, ~Z -

Estimation of Thermodynarnical and Thermochemical Properties 75

m~lar mass, g/mole.

where A ~- surface tension, mJ/m 2, ~ and~ v- densities o£ liquid

and vapor, g/cm 3.

Coefficients a and b depend on the factor of complexity of

intermolecular interaction ~/).

where V* - molar volume of liquid at boiling, that is--4/~ ~ boiling ~m',g Below are given dependencies of a and b on :

~f-faetor a b

< O.O5 133.45 -1.40

O.05-0.1 92.42 0.60

> 0 .1 8z~-.35 1 . 2 ~

Thus, for estimation of T m from (108) it is necessary to

know: ~boig' ' and~ . In /31/ it is pointed out that the

error of estimation is ~15%.

In /51/ it is shown that at investigation of interaction of

melting temperatures o£ many inorganic compounds with melting tem-

peratures of their simple components (elements, oxides, salts),

melting temperature of compound can be presented by additive fun-

ction of melting temperatures of components and their mole fractions,

that is

On the base of the value o£ coefficient K three groups of compounds

have been pointed out.

I)ZK=O.984. Simple compounds of elements with C, Si, B, S, P, Se,

Te, Cr, Ni, Zn, Ge, Sb, Cd, Bi, etc. Average deviation of temperatu-

res, received from (111) from experimentally estimated is ~11%.

Eq.(11W) is not valid for compounds which noticeably sublimate or

dissociate.


2) K=0.80. Compounds which can be described as a sum of oxides

of different elements (anion groups: WO~- TiO~-, TiO~- 3- , , l~D 4 ,

, ~-o0 4 ,

ReO~, etc.). Average deciation of calculated temperatures from

experimentally found is !7-5%- In this group are also included

sulphides, halogenides, hydroxides.

3) K =C.92~. Binary ionic salts of halogenides, sulphates, nit-

rates. Average deviation of calculated T m from the known values

is 211%.

T m can also be estimated when temperature dependency of heat

capacity is known. Assuming according to /4/ that at T m Cp~ 7.25Z0.5

cal/(g atom) K and solving the equation

we get the value of T m-

Besides, T m can be found with the use of different comparison

and confrontation methods.

5.2. Entropy %nd enth~ipF of meltins

For similar substances it is possible to use for calculations

the following dependencies, (cal/K g atom) /5/:

~£=co~s~~ ~ a~m = ~' ~ga~.m T~. (114) It is pointed out in /~/ that for metallides it is possible to

find entropy of melting with the use of equation (cal/K g atom):

& ~A~fa~ ~ g &~Me (115)

for non-ordered alloys and adding the value

where N i- atomic fraction of metal in the alloy, for ordered alloys.

When entropy of element melting in the alloy in unknown, for

binary ordered and non-ordered alloys as approximate values it is

possible to assume values 3.5 and 2.2 cal/K g atom.

EstimationofThermodynamicalandThermochemicalPrope~ies 77

There are practically no empirical regulations for prediction

of& S and ~H of melting of inorganic substances, though one can

use comparison and confrontation methods for rows of close or

similar compounds, and also for compounds with the same structure.

In /51/ an equation is offered for estimation of entropy

change at melting of complex compounds, which can be presented

as a sum of simple ones; or for simple ones, which can be presen-

ted by sepatare elements with known entropies of melting,

&~ ~ Z ~L" ~m (L~ (117)

where n i - number of moles of i-th simple substance (element),

Sm(i) - entropy change at its melting.

For ex., for B4C, Ti2Cr207, Na2SO # ZnSO#, CaAI2Si208:

~Sm=4ASm(B) + ~Sm(C) ,

Sm=2aSm(TiO 2) + ~Sm(Cr203),

Sm=~Sm(Na2S04) + A Sm(ZnS04) ,

Sm=mSm(Ca0) + ASm(A1203) + 2aSm(Si02)

For simple compounds and those which can be presented as a

sum of simple (oxides, nitrides, etc.) average deviation of calcu-

lated values from experimental ones is not more than Z15%.

CONCLUSION

Described methods were used ~or calculations of thermodynamic and

thermochemic properties of about 600 different substances and com-

pounds, and also temperature dependencies of reduced Gibbs energi-

es. Information was input into the database ASTRA-OWN and is

successfully used at thermodynamic simulation (TS) of different

processes by the TS Centre in the Institute of metallurgy, Ural

Division of Russian Academy of Sciences (Ekaterinburg).


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v < m n c i l 3~ t h e i l ' l s L i L u t e o r ).~e,.~a,_.'_ur%7~ U~_±

D i ~ , i s i ~ , n ~ r Russ:_a~i .~cs,<ie~y o ! S c i e n c e s

ELSEVIER Thermochimica Acta 245 (1994) 189-206

thermochimica acta

Thermodynamics and phase equilibria data in the S-Ga-Sb system auxiliary to the growth of doped

GaSb single crystals *

J. Sest%k a,*, J. Leitner b, H. Yokokawa ‘, B. %pBnek a

a Institute of Physics, Division of Solid State Physics, Academy of Sciences of the Czech Republic,

Cukrovarnickti 10, 162 00 Prague 6, Czech Republic

b Institute of Chemical Technology, Faculty of Chemical Technology, Department of Solid State

Engineering, Technicka 5, 16628 Prague 6, Czech Republic

’ National Institute of Materials and Chemical Research, Department of Inorganic Materials,

Tsukuba, Ibaraki 305, Japan

Abstract

An extensive survey of phase equilibria and thermodynamic data (AH, S, C,, AG) on solid and liquid phases is presented for Ga-Sb, S-Ga and S-Sb subsystems which includes tables of the limiting activity coefficients. The construction method for chemical potential diagrams has been applied in the form of log[a(Ga)/a(Sb)] versus log[P(S,)/bar] plots, the usefulness of which is discussed. The ternary phase diagram has been estimated. Using minimization of Gibbs energy the equilibrium composition of coexisting phases in the S-Ga-Sb system has been evaluated regarding the determination of the maximum level of sulphur doping in GaSb single crystals grown by the Czochralski technique without encapsulant. The calculated concentration of dissolved sulphur in GaSb solid was 10’6-10’7 atoms per cm3, which is in good agreement with the experimentally measured values of about lOI atoms per cm3. After exceeding about 1.5 x 10” atoms per cm3 in the melt, the second phase (Ga,S) started to separate spontaneously.

Keywords: Binary system; Phase equilibrium; Semiconductor; Solid solution; Ternary system; Thermodynamics

* Presented at the Czechoslovak-French-Polish Conference on Calorimetry and Experimental Ther-

modynamics: Applications to Contemporary Problems, Prague, Czech Republic, 4-7 September 1993. * Corresponding author.

0040-6031/94/$07.00 0 1994 - Elsevier Science B.V. All rights reserved

SSDIOO40-6031(94)01843-6

190 J. Sestbk et al./Thermochimica Acta 245 (1994) 189-206

1. Introduction

Recently gallium antimonide based crystals have come to represent an important material for the production of various semiconductor devices [ 11. GaSb and related ternaries exhibit interesting optical properties for lasers and detectors in the range up to 2.3 pm wavelength. GaSb is required as a substrate material on which ternary and quaternary alloys can be epitaxially fabricated [2] for optoelectronics or high-speed electronic devices, among which are the low threshold (In,Ga, _-x Sb) Gunn oscillators [3], low noise (Al, Ga, _-x Sb) APDs [4] for the 1.3 pm band, LEDs and LDSs based on Ga, _,Al,As, _-y Sb, [5] for the 1.7 ,um band, superlattices for both the new kind of (AlSb/GaSb) LDs [6] and the high speed (InAs/GaSb) electronic devices [7]. However it seems unlikely that GaSb devices will prove to be of significant interest for hot electron transistors, but the FET performance shows some potential [l].

In the Prague Institute of Physics we have successfully studied [8,9] the deep metastable centres (conventionally called the DX centres [lo]) employing our own Czochralski grown single crystals doped with sulphur. It is, however, necessary to overcome many difficulties in order to optimize the preparation procedure [ 1 l] and to achieve good quality single crystals with tailored doping. This has certainly focused our attention on some thermodynamic aspects of both the growth processes [ 1 l] and phase equilibria [ 121.

2. Crystal growth and previous data treatments

Previous data treatments on the growth of S-doped GaSb have previously been reported [ 13- 151. The behaviour of sulphur during the growth was mainly explained by sulphur evaporation from the melt because it is known that the solubility of sulphur is very low (7.2 ppm) in the melt. Similarly it becomes convenient to have a slight excess of antimony as part of the standard growth conditions to compensate for Sb volatilization and help to preserve the stoichiometry of the grown crystal. For a standard preparation of the GaSb single crystals [9] (grown using the Czochralski method without encapsulant in an atmosphere of hydrogen) the sulphur concentration (calculated from Hall measurements) reached a limited value of 1 x lOI atoms per cm3 in the crystals though the starting amount of sulphur in the melt exceeded about 2 x 102’ atoms per cm3. It seems that after the sulphur solubility in GaSb melt is attained at about 7.2 ppm (0.0043 at.%) [9], sulphur starts to evaporate. The concentrations of sulphur, however, were used in the range 0.0035- 1.112 at.% [9] so that the limiting solubility of sulphur in GaSb(1) was substantially exceeded, and the Ga,S solid is supposedly created. Such a relatively high concentration of sulphur was intentionally taken into account during the mathematical evaluation in order to appreciate the behaviour of sulphur in the case when sulphur cannot further dissolve in the melt. As a result, unwanted higher mechanical stress and tension were found to be created on the solidification interface which consequently disturbed single crystalline growth.

J. Se&k et al.lThermochimica Acta 245 (1994) 189-206 191

Such a treatment was aimed at making a thermodynamic estimation of how the distribution of sulphur between individual phases is carried out and what evaluation method is convenient, For the growth of semiconductor GaSb single crystals doped by sulphur we have evidently confined our activities to an extremely narrow concentration region [ 121, which is not conventional in standard thermodynamic evaluations, Therefore we would like to link our approach to existing thermodynamic data over the whole concentration range. Two approaches will be presented herewith, (i) the qualitative approach based on Yokokawa’s construction of isothermal diagrams of phase stability (chemical potential diagrams) [ 16,171 and (ii) the quantitative approach using Voiika and Leitner’s standard calculation of coexisting phases by minimization of the total Gibbs energy of the system [ 12,181.

We have hence found it useful to present here a survey of existing input data published to date. We do not treat data on the elements separately, because they have been reported in various compendia [ 19-241 with negligible variations, particularly regarding those elements which are dominantly included in the equilibria under question, i.e., Ga(l), Sb(l), Sb,(g), Sb,(g) or S,(g).

3. Thermodynamic data for the Ga-Sb-S binary edges

3.1. The Ga -Sb system

3.1.1. Phase equilibrium data and phase diagrams Original articles dealing with phase equlibrium data can be found in Refs.

[25-33,771 while phase diagrams have been treated in Refs. [34-36,491.

3.1.2. Thermodynamic data on the solid phase Recently experimental data on solid GaSb, particularly C,“,(T), were given in

Ref. [38]; other information is available in standard tables [23,24,49]. Their data are compared in Table 1.

Table 1

Thermodynamic data for solid GaSb

Ref. AHF(298.15 K)/

kJ mol-’

Sz(298.15 K)/

J K-’ mol-’ C$(298.15 K)/

J K-’ mol-’ AG;(985 K)/ kJ mol-’

I491 -41.589 76.065 48.702 - 15.062 [231 -43.932 76.065 48.59 - 18.215 1241 -41.589 76.065 (24.348) a (-3.242) a [51,531 - 14.207 [551 - 16.475 I561 - 15.734 I331 - 17.606

a Error in tabulated data.

192

Table 2

J. Sest2k et al./Thermochimica Acta 245 (1994) 189-206

Thermodynamic data for Ga-Sb liquid

Ref. AGg/(J molk’) =f(x,T) T/K = 985, .xCa = xsb = 0.5

AG:I AH,” i J molF’ J molF’

[481 AC: = xcaxsb( 19665 - 25.1 T) - 1265 4916

[501 AH,” = x,,x,,(531.41 -21599,27x,, + 32175.13~,,~ - 1487 -1049

AS3 J K-’ mol-’

6.275

0.445 - 15788.12~,,~)

AS: = x,,xs,(4.849 - 14.456x,, + 24.521~s~~

- 15.786~s~~)

1511

[521

AHM = x,,xs,4184[1 + 12.35(x

AS$=x,,x,,l.926[1 -31.35(x;;I:.$

AH: = s,,xs,(4962x,, -9715x,,)

AS: = x,,xs,(3.209~,, + 0.456x,,)

Associated solution model

AH,” = xGaxsb( - 1179x,, - 6591x,,)

AS: = x,,xs,(5.297~,, - 1.445x,,)

AGE = xCaxsb( - 3887 - 5.835T)

AG: = x~axsb(~xxGa + WIZXSL, - 4u.~c,xsb)

-1520 - 1046 0.481

- 1045 - 594

- 1427 - 1046

-1445 -971

- 2409 -972

-2197 -1120

0.458

[531

[541

0.387

0.481

[551

1561 [361

[331 t761

1571

wz,/RT = -2221.6/T+ 15.3011 -2.0160ln T

w12/RT = -1657,6/T + 9.6150 - 1.2631 In T

v/RT = 213.90/T + 0.0435

AGE = x,,xs,[( - 4300 - 8.85T) - 1.33T(x,, - xsb)

+ (4500 - 0.79T)(x,, - x,,)‘]

AG: = xo,xs,(Axs, + %I+, + x,,xs,C)

- 3254 - 1075

-2688 -1121

1.459

1.093

2.212

1.591 A = - 13489.4 + 77.77093 - 10.22977 In T

B = -20026.7 + 136.6057 - 18.47543 In T

C = -7454.86 - 1.09227T + O.O1464x,,T

3.1.3. Thermodynamic data on the liquid phase Activity data on liquid phases have been extracted from the e.m.f. [39-42,781 and

vapour pressure measurements [43,44] and also on the basis of the heat of mixing (established by calorimetric measurements [45-47,501). The results of simultaneous optimization of phase equilibria and thermodynamic data were reported in Refs. [33,36,48-57,761. The known data are compiled in Table 2.

3.2. The Ga-S system

3.2.1. Phase equilibrium data and phase diagrams Original reports on phase equilibria were published in Refs. [37,58-621 and the

phase diagram is available from the standard handbook [35].

Tab

le

3 Se

lect

ed

valu

es

of

the

ther

mod

ynam

ic

data

fo

r in

divi

dual

su

bsta

nces

Subs

tanc

e A

HP(

298

K)/

Sz

(298

K

)/

C,“

,/(J

Km

’ m

ol-‘

) =A

+

10m

3BT

+ 10

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3.2.2. Thermodynamic data on solid phases Solid phases are collectively surveyed in Table 3, but the existence of Ga,S and

Ga,S, phases is still doubtful [37,64] despite their data having been presented in the standard tables [23,24,63,85].

3.2.3. Liquid phase thermodynamic data There is only article [60] reporting the activity of sulphur in dilute solution from

both e.m.f. and direct solid-liquid equilibrium measurements, see Table 4.

3.2.4. Thermodynamic data on gaseous substances Recent experimental data on Ga,S gas can be found in Ref. [64] and also in

standard tables [23,24,63].

3.3. The Sb-S system

3.3.1. Phase equilibrium data and phase diagrams Data on phase equilibria were published in the original articles [65-681 and the

phase diagram is presented in the handbook [69].

3.3.2. Thermodynamic data on solid phases The Sb,S, solid phase was recently treated in Ref. [70] and is described in tables

and reviews [ 17,23,24,63]; see the compilation in Table 3.

3.3.3. Liquid phase thermodynamic data Activity data were estimated by vapour pressure measurements [71], from

gas-liquid equilibrium measurements in dilute solutions [ 17-241 and also from the heat of mixing through calorimetric measurements [68]. Limiting activity coefficients are shown in Table 5.

3.3.4. Thermodynamic data of gaseous substances Gaseous substances SbS, Sb2S3, Sb2S4, Sb3S2, Sb4S3 and also Sb2S2, Sb3S4,

Sb4S4, SbsSs have been described in Ref. [ 751 and relevant tables and reviews [ 24,631.

Table 4

Temperature dependence of limiting activity coefficient of sulphur in Ga-S melt (AG for reaction 0.5&(g) = [S],,(at.%))

Ref. AG/(J per g at. S) = RT ln(y’&,,/lOO) In y&a) Method

(at T = 985 K)

[601 -218810 + 126.31 T -6.92 From EMF measurements

1601 -257176 + 78.707 - 17.40 From equilibrium measurements

[hOI -241160 + 51.9T - 18.6 Estimation from AH; of Ga,S(s)

]921 -375800 + 63.8T -33.6 Estimation from AH,? of Ga,S(s)

J. sestcik et al.lThermochimica Acta 245 (1994) 189-206 195

Table 5 Temperature dependence of limiting activity coefficient of sulphur in Sb-S melt (AG for reaction OSS,(g) = [S],,(at.%))

Ref. AG/(J per g at. S) = RT ln(y&,,/lOO) In Y&,, Method (at T = 985 K)

[WQI -69900 + 17.2T -1.86 From equilibrium measurements

[741 -79981 + 7.60T -4.25 From equilibrium measurements

[601 -99110 + 12.7T -5.97 Estimation from AH; of Sb,S,(s)

~921 -86550+ 11.8T -4.54 Estimation from AH; of Sb,S,(s)

Table 6 Selected values of thermodynamic data for dilute solution of sulphur in Ga-Sb melt

AGE(Ga-Sb) = x,,x,,( - 3887 - 5.835 x T) in J mol-’ RT In y&, = -235000 + 116.3T in J per g at. S

RT In y&, = -79981 + 7.6T in J per g at. S

[551 Selected

[741

Selected data for dilute solutions of sulphur in the Ga-Sb melt are shown in

Table 6.

A collective diagram of the binary edges in the Ga-Sb-S system [35,36,69,88,89]

is shown in Fig. 1.

Fig. 1. S-Ga, SSb and Ga-Sb binary phase diagrams.

196 J. sestcik et al.lThermochimica Acta 245 (1994) 189-206

4. Construction of chemical potential diagrams

Methods for calculating thermodynamic equilibria can be classified according to the use of either (i) independent reactions and their equilibrium constants as stoichiometric, or (ii) linear equations in terms of the chemical potentials as nonstoichiometric [ 80,8 11.

Recent developments in evaluating complicated chemical equilibria and also phase diagrams [82] have revealed that the nonstoichiometric approach bears a certain advantage in solving equilibrium problems [83]. However, the chemical potential diagram was initially constructed using the stoichiometric method because of the manually draw diagrams, while for the advanced automatic constructions the combined nonstoichiometric and convex-polygon method was found to be more appropriate. A recently written computer program [84] enables one to treat all elements in an equivalent manner and makes it possible to construct various types of chemical potential diagram in metal-metal-nonmetal systems. For oxides [ 161 it has been shown that a log(a, /u2) versus log P(0,) plot is very useful and the use of log(a,/a,) makes it possible to treat the two metallic elements (1 and 2) equivalently. Such a diagram consists of the stability areas of elements, binary oxides and double oxides; similarly, this is applicable to sulphides, but has not yet been used. Since the local equilibria concerning the particular compound can be represented as a polygon and its neighbours, whole areas of stability can be viewed globally.

The thermodynamic properties employed in this treatment of the S-Ga-Sb system are listed in Table 3 (as originally compiled from the thermodynamic database MALTZ [85] which is mainly based on the NBS tables [86] and other related databooks [ 17,871). The computer program CHD [84] employed can provide any choice of the axis variables. For the present study the log[a(Ga)/a(Sb)] versus log[P(S,)/bar] plot was selected because the sulphur particle pressure is one of the major controlling factors in crystal growth experimentation.

In these plots (see Fig. 2) the stability polygon of GaSb has a certain range of the sulphur potential which corresponds to different states of the S-doped GaSb. Although these diagrams are constructed by using only the thermodynamic data of compounds, stability regions of the respective compounds bear certain information about the third component within the approximation of an infinitely dilute solution. This means that each point of the polygon of GaSb indicates implicitly its dopant level of sulphur. When the thermodynamic effect of doping is to be explicitly considered, the GaSb polygon is slightly modified so that the three-phase GaSb- Ga,S-Sb junction moves to the higher sulphur potential region. In other words, the high sulphur potential side corner of the GaSb polygon will be extended in the direction of Ga,S. In Fig. 2 the stability polygons of GaSb and Ga,S are separated by several orders of the sulphur potential. Because the effect of extension of the stability polygon by the explicit treatment of the solid solution is of the order of log(O.1) or less, it is expected that the solubility limit of sulphur is small and therefore highly S-doped GaSb will decompose into Ga,S and Sb at high sulphur potential regions.

J. Sestdk et al./Thermochimica Acta 24.5 (1994) 189-206 191

Even when the solid and/or liquid solutions are formed they are treated as stoichiometric. This can be illustrated on a schematic plot of T versus log[u(Sb)/ a(Ga)] (Fig. 3) and its comparison with the normal phase diagram for the Ga-Sb system (cf. Fig. 1). For example, at 800 K there are liquid Ga and solid GaSb and Sb. The phase diagram tells us that Ga(1) contains several percent of Sb. However, this effect does not appear explicitly in the chemical potential diagram, which is understandable in view of the standard plot of the Gibbs energy against composi-

-30 -25 -20 -15 -10 -5 0 log [P(S$bar]

Fig. 2. (a)-(b).

198 J. Se&k et al./Thermochimica Acta 245 (1994) 189-206

Ga

-30 -25 -20 -15 -10 -5 0

log [P(Sz)/bar]

IO

-30 -25 -20 -15 -10 -5 0

log [P(S$bar)

Fig. 2. Series of computer print outs of the chemical potential diagrams in the S-Sb-Ga system for

temperatures from (a) S12”C, (b) 612”C, (c) 712°C to (d) 812°C using data listed in Table 3 (for Sb,S,

the first line with AH,? = -205.02 kJ mol-’ [24]).

tion, where chemical potential change between the non-solubility and several percent solubility of Sb is evidently insignificant. Above the eutectic temperature of 862 K, the Sb-rich liquid appears between GaSb(s) and Sb(s) the composition width of which increases rapidly with temperature while that of the chemical potential diagram is rather narrow.

J. Se&k et al./Thermochimica Acta 245 (1994) 189-206 199

(4

1000 --

800 +

7004-- -5

I I

5 0

Log [a(Sb)/a(Ga)]

Sb(l)

aSb

(SI Sbk.)

1000

700

b)

GalLI

-5

iaSI it)

- Sb(l)

;aS (4

Sb(s)

+ 0 --r-

log [a(Sb)/a(Ga)] Fig. 3. Schematic plot of the Ga-Sb chemical potentials as a function of (a) temperature and (b) strictly respecting the stoichiometric assumption.

Figure 3(b) illustrates the phase relations obtained under the stoichiometric assumption which, strictly speaking, is not appropriate here as it does not repro- duce satisfactorily the phase relations associated with eutectics. As a first order approximation, however, the diagram reproduces well the main feature of a proper diagram as the difference between the two figures is of the order of log[a(Sb)/ a(Ga)] d 0.5. The stability field of GaSb(1) gives rise to physically unreliable boundaries between GaSb and both Sb and Ga components. However, the stability region of GaSb(1) indicates well where the 1: 1 composition of a liquid is located in the diagram. As an illustration, estimated forms of the corresponding chemical potential diagrams at 612 and 712°C are given in Fig. 4(a) and 4(b) which show an essential change of the solubility (dashed) areas. This strongly indicates the merit of using chemical potential diagrams even when data are not available for solution phases.

5. Evaluation of equilibrium composition of coexisting phases

The chemical equilibrium in the S-Ga-Sb( -H) system was calculated using a general method based on minimization of the total Gibbs energy of the system for a set of points satisfying the material balance conditions [ 12,181. The calculation program makes use of a modification of the RAND method [ 181 accounting for twenty eight chemical species. The sources of thermodynamic data are shown in

200 J. Se&k et al./Thermochimica Acta 245 (1994) 189-206

Sb

(885 K)

-iO -10

Log [P(Sdbar]

0

-30 -20 -10 0 log [P(Szl/bar]

Fig. 4. Chemical potential diagrams in the S-Sb-Ga system including probable solubility regions

(hatched) and showing the effect of two-fold [24,87] input data on Sb,S, (both values in Table 3)

(broken lines) for temperatures (a) 712°C and (b) 812°C (cf. Fig. 2(b) and 2(c)).

Table 3. The model of a regular solution with a temperature dependent interaction parameter R,,_,,/J = -3887 - 5.883 x T/K [55] was used. Ideal behaviour of sulphur in the melt was supposed according to Henry’s law. The limiting activity coefficient of sulphur depending on the composition of the melt was estimated on the basis of a regular solution model following the equation

RT In YgGa-Sb) = xGaRT In Ysm(Ga) + %bRT In Y&b) - %-SbXGaXSb (1)

J. Se&k et al.lThermochimica Acta 245 (1994) 189-206 201

where xoa and xsb are the mole fractions of components Ga and Sb, respectively; Y&,), Y&), y$oa_sbj are the limiting activity coefficients of sulphur in Ga(l), Sb(1) or GaSb(l), respectively; T is the temperature; and R is the gas constant.

The Ga-Sb-S solid solution was presumed to be pseudobinary solution of components GaSb and S. The value of the limiting activity coefficient of sulphur in this solution was estimated by the Kroger equation [90]

AG:(S) ln ko

lny,“,,,=lny,“,,0+~- S(GaSb)

where AGF(S) is the molar Gibbs energy of sulphur fusion and k&,,,,, is the equilibrium distribution coefficient of sulphur in GaSb, which can be deduced from the condition of a thermodynamic equilibrium between solid and liquid phases. The value y&i) corresponds to the stoichiometric composition of the melt (xoa = xsb = 0.5). For the case of the melting of pure sulphur the temperature dependence of AGz(S) was calculated by the standard equation

G” -G&, = S(i) - 1015.47 + 44.544 x T - 7.033 x T In T (3)

In order to calculate the activity coefficient of sulphur in the solid phase the distribution coefficient of sulphur (k&,,,,, = 0.06) was utilized.

Previously, such calculations of the GaSb equilibrium were conveniently carried out at a temperature of 985 K and atmospheric pressure. The starting substances used were Ga(1) (no = 0.4995 mol), Sb(l) (no = 0.5005 mol), S(1) (no = 10e4 to 5 x 10m2 mol) (and H,(g), no = 1.5 mol). These values corresponded to the usual experimental conditions employed for the growth of S-doped GaSb single crystals. The most important results of the calculation are as follows:

(i) In the case of all initial amounts used for sulphur in the system in question, there exist four phases in the equilibrium: gas phase, melt, GaSb-S solid solution and solid Ga,S. According to the Gibbs phase rule, this system does not have any degree of freedom and therefore the composition of multi-component phases is not dependent on the starting amount of sulphur. The starting amount of sulphur, however, has affected the equilibrium amount of the individual phases and the distribution of sulphur among them. There are substantial differences in stoichiometry for GaSb equilibrium liquid caused by formation of Ga,S solid.

(ii) For determining the sensitivity of the calculated results on the used thermodynamic data used, the equilibrium calculations were carried out using both the highest and the lowest values of the limiting activity coefficient of sulphur in the S-Ga and S-Sb systems, collectively shown in Tables 4 and 5. In addition, equilibrium distribution coefficients in GaSb, kgcGaSbj = 0.1 and 0.01, were applied [9,11]. From the comparison and the indefiniteness of initial thermodynamic data it follows that the calculated results concur but are a rough estimate only.

(iii) The thermodynamical calculation is in good agreement with our previous experiments when growing the S-doped GaSb single crystals [9]. The maximum attainable concentration of sulphur in the single crystalline bowl was about 1017 atoms per cm3. After exceeding this value, the crystals became either polycrystalline or twinned. For this reason it seems that the second Ga,S phase started to separate


spontaneously from the melt, which resulted in impaired single crystalline growth. The so-called constitutional supercooling was not mentioned because it was not likely to occur under our growth conditions and consequent calculations. We can see that the thermodynamically calculated values of sulphur concentration in the GaSb solid ( 1OL6 to 1017 atoms per cm3) correspond satisfactorily with the experimental measurements [9,11].

(iv) For the (Te,S) doubly doped GaSb we have already analysed [ 911 the quaternary system S-Ga-Sb-Te in six binary subsystems Ga-Sb, Ga-Te, S-Ga, Sb-Te, S-Sb and Te-S. It seems that the most important subsystems are Sb-Te, Te-S and the above-mentioned S-Sb. In the case of the S-Sb subsystem and within the region of used concentrations of the elements it became evident that sulphur exists below its melting temperature in the form of Sb2S3, i.e. sulphur can only be bound with great difficulty in the GaSb structure without creating any second phase. However, tellurium can form two solid solutions with antimony from a concentration of about 3 at.% Te. The Te-S pseudobinary cut shows the boundary line of solid solutions as a sulphur concentration of about 15 at.%. For this reason, if the concentration of sulphur exceeded this value, Te-S solid solution appeared and no Te nor free S atoms existed to form a second phase. It is necessary to add that, if the concentration of tellurium increased above the value of 12 at.% without exceeding that of sulphur, the calculated concentrations of dopants were identical to those based upon measurements. However, when the concentration of sulphur exceeded 12 at.% the calculated and measured concentrations exhibited different values independent of the tellurium concentration.

6. Discussion and conclusion

On the basis of above treatments involving chemical potential diagrams we can proceed to estimate the S-Ga-Sb ternary system, see Fig. 5. Because there is no experimental information available we had to make certain estimates on the phase relations including the liquid solutions. The phase relations between the solid compounds were based on the thermodynamic calculations which had been applied in constructing the chemical potential diagrams (see Fig. 2) and then compared with the proposed chemical potential diagrams at 885 and 985 K (see Fig. 4), actually accounting for solubility regions. Since the composition of the liquid phase changes with temperature, the appearance of the phase diagram changes dramati- cally, while the same features appear without large differences from those obtained under the stoichiometric assumptions. We just have to remember that the order of the differences between the two treatments is about 0.5 which is quite a small value compared with the scale of the chemical potential diagram of a ternary system even when the data on solution phases are lacking.

The above tabulated values of heats and entropies of formation of GaSb(s) (298 K) were derived from the temperature dependence of AGF for the reaction Ga(1) + Sb(1) = GaSb(s) published [55] in the form AGF /RT = 5.37 - 7950/T + 0.1 In T. The GaSb(s) temperature dependence of C, as found in tables [23] and

J. Sestrik et al./Thermochimica Acta 245 (1994) 189-206 203

Ga

GaSb

liquid

Sb

liquid

Liquid

Sb

Fig. 5. Hypothetical S-SbbGa ternary phase diagrams proposed on the basis of chemical potential

diagrams (cf. Figs. 4(a) and 4(b)) for temperatures (a) 712°C and (b) 812°C.

the constant B in the standard equation C, = A + BT was fitted in such a way that AC, = -0.831 J K-r at the GaSb melting point (985 K). This modification produces a decrement of a mere 0.8% at 298 K.

Data for gallium sulphides were transferred from tables [23] but were originally published in Ref. [63]. The existence of Ga,S and Ga& is still in doubt, the latter being thought either to decompose at 832°C or to reach a distectic point at 960°C. The origin of Sb2S3 data is again Ref. [23] having been transferred from Ref. [64]. The tables [24] and database [84,85] contain data from Ref. [87] where the value

204 J. Sestcik et al.lThermochimica Acta 245 (1994) 189-206

given for formation enthalpy is quite different, i.e. - 141.796 kJ mol-’ as against -205.016 kJ mol-’ [23,63]. Because the Sb2S3 melting temperature of 823 K is the same in both tables, and our treatment is applied above this temperature range of 885-985 K, the above mentioned AH: difference becomes unimportant (see Fig. 4) which illustrates that this discrepancy does not affect the thermodynamic features of GaSb and related phases. The data for gaseous sulphides were compiled from Ref. [24] having been transferred from Ref. [87].

For the description of nonideal behaviour of the Ga-Sb melts the model of a simple solution containing only two adjustable parameters was applied. According to Ref. [55] the model is capable of reproducing the published experimental data with an accuracy comparable with the other multiparameter equations [51-541, particularly when applied within the 0.4 < xsb d 0.6 concentration region.

The experimentally established values of the limiting activity coefficients of sulphur in the binary systems S-Ga and S-Sb exhibit a rather high dispersity, similarly to the values estimated on the basis of empirical correlations. The presented relations represent only rough estimates, the refinement of which is a matter of further experimental and theoretical studies.

Both approaches presented above, either the construction of chemical potential diagrams [ 161 or the direct calculation of the equilibrium composition of coexisting phases by Gibbs energy minimization [ 181, were found to be valuable for better understanding of phase stabilities in the S-Ga-Sb system.

Acknowledgements

The continuous cooperation with Dr. G.K. Moiseev (Institute of Metallurgy, Ekaterinburg, Russian Federation), Dr. V. Sestakova (Institute of Physics, Prague, Czech Republic) and Prof. 2.D. Zivkovic (Technical Faculty, Bor, Yugoslavia) is much appreciated. The study was carried out under Scientific Project Number 210128, and 104/94/0706. Calculation of thermodynamical data sponsored by the Academy of Sciences and the Grant Agency of the Czech Republic.

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Journal of Thermal Analysis and Calorimetry, Vol. 72 (2003)

THERMAL CONDITIONS OF GROWTH AND THENECKING EVOLUTION OF Si, GaSb AND GaAs

Glide phenomenon in the gas bowl*

B. Štìpánek, J. Šesták**, J. J. Mareš and V. ŠestákováInstitute of Physics, Academy of Sciences of the Czech Republic, Cukrovarnická 10, 162 00 Prague,Czech Republic

Abstract

The configuration of thermal gradient is illustrated for various types of crucible rotation, which isimportant for the creation of dislocations, which decreases along the grown axis of crystal. A newmechanism for dislocation elimination during the growth is proposed to explain this phenomenon,which provides a good agreement with the experimental results. The concentration of etch pits rap-idly decreased from the beginning to the end of the crystals and the dislocation densities in the mid-dle portion of all investigated crystals were found less than 102 cm–2. The shallow vertical tempera-ture gradients and virtually flat solidification interface prevented thermal stress from their buildingup in the crystals. As a result, the dislocation formation had random distribution. Using good neck-ing procedures and choosing an appropriately oriented starting crystal with the shoulder angle<38.94° (assuming growth in <111> direction) it is possible to produce almost dislocation-free crys-tals without resorting to additional doping normally employed to reduce dislocation formation.

Keywords: doping, gallium antimonide, low dislocation density, single crystal growth,thermal gradients

Introduction

The practical importance of semiconductor materials, such as Si, GaAs, GaSb, forelectronics and photonics is, of course, for years beyond any doubt. An introductionof novel epitaxially grown quantum devices, containing low-dimensional subsystems[1] (e. g. quantum wells, quantum dots) as well as further sophistication of classicalsemiconductor devices (IC’s, lasers, detectors) put, however, new heavy demands onsubstrate materials the quality of which is decisive also for the quality of end prod-ucts. Typical requirements of the substrate wafers are extreme chemical purity, latticeperfection, and thermal and mechanical stability. Besides, for special uses (e.g. detec-tors for green house gases) it is necessary to grow also tailored single crystals with thespecial doping [2].

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* Dedicated to the 80th of Professor Vladimír �atava, Ph.D, DSc., emmeritus professor of the Institute ofChemical Technology in Prague, one of those who formed theoretical basis of thermal analysis and ofmaterial science of silicates.

** Author for correspondence: E-mail: [email protected]

Gallium antimonide (GaSb) belongs to the group of III/V compounds. Singlecrystals of GaSb [3] are used as substrate material for the fabrication of long wave-length detectors and lasers (λ≈1.5 µm). High quality GaSb substrates are required forthe growth of the (GaIn)(AsP) epitaxial layers used in optical communications. Qua-ternary systems, Ga1–xInxAs1–ySby or Ga1–xAlxAs1–ySby which are suitable for thesewavelengths, can be lattice-matched to GaSb by differential contraction during thecooling process because the expansion coefficient values of the layer and substrateare very close [4].

Not less important are the studies allied to the system thermodynamics, relatedto the distribution of thermal and concentration gradients [5–7] as well as to the eval-uation of phase equilibrium [7–10]. It is a continuation of our previous papers pub-lished in J. Thermal Anal. [7, 9, 10]. We extend it by our theoretical modelling of theoverall gradient distribution created as a result of local thermal and concentrationconditions, where the type of the externally applied crystal motion and crucible rota-tion become demonstrative, cf. Fig. 1.

J. Therm. Anal. Cal., 72, 2003

2 ŠTÌPÁNEK et al.:GROWTH AND THE NECKING EVOLUTION

Fig. 1 Illustration of the gradients distribution (thick curves with arrows) shown forsimplified arrangement of cylindrical melt container (crucible) heated from sides(shadow) with unmarked crystal growing seed. Upper line shows stationary dis-position while middle line and bottom line portray iso-rotation and coun-ter-rotation, respectively

Experimental

Studies of impurity type and concentration, free carrier concentration, types of conduc-tivity and mainly dislocation density have to be carried out for GaSb single crystals. It isdifficult to anticipate carrier concentration, especially with n-type conductivity, whenpreparing substrates. According to the literature [11–16], the concentration of residual ac-ceptors in undoped GaSb single crystals is 1.0 to 2.7⋅1017 atoms cm–3 p-type and, owing tothe distribution effect of impurities, their concentration is changing in the direction of thecrystal growth. To obtain a substrate with n-type conductivity and a low donor concentra-tion level it is necessary to know to exact correlation between the concentration ofimpuri-ties (deliberately added to the melt) and the consequent carrier concentration.

It is known that the dislocation density decreases in the direction of the crystalgrowth; this decrease is of the order of 102 to 104 cm–2 [17, 18]. Some authors [15, 17, 19]have found that the undoped crystals are relatively poor in quality. Doping of some ele-ments reduces the formation of dislocations as a result of the higher thermal stress whichare always inherent in the Czochralski technique (as manifested by most III–V com-pounds [20]). Using low temperature gradients in the furnace also decreases the possibil-ity of dislocation formation and thus dislocation-free crystals can be grown.

The aim of our investigation was to study creation of dislocation and to find crit-ical source causing multiplication of dislocations. For this study, Te-doped andundoped GaSb single crystals were grown using the Czochralski method withoutencapsulant and under low temperature gradient conditions along a solidification sur-face. The Czochralski apparatus technique without encapsulant was found to be verysuitable for crystal growth (Fig. 2), and the standard distribution of temperature andconcentration are shown in Fig. 3 completing thus the illustrative picture of the rota-tion consequences, cf Fig. 1.


ŠTÌPÁNEK et al.:GROWTH AND THE NECKING EVOLUTION 3

Fig. 2 Schematic diagram of the Czochralski apparatus for GaSb single crystal growth.1 – sliding rod, 2 – top part of the apparatus, 3 – quartz tube, 4 – holder of theseed, 5 – seed, 6 – water cooling, 7 – molybdenum wire coils, 8 – GaSb singlecrystal, 9 – quartz crucible, 10 – graphite cylinder, 11 – melt of GaSb,12 – power supplier, 13 – deuterium lamp

In given crystals, we have found that the etch-pit densities (EPD) decreased withgrowth distance measured relatively to the seed. Such a behaviour had already beenreportted in the literature and various explanations had been proposed:

• according to Benz and Müller [21], dislocation pairs of opposite Burgers’ vectorscan annihilate one another;

• according to Yip and Wilcox [22], dislocations are eliminated by growing out of thecrystal because they propagate normal to a solidification interface that is convex to-wards the liquid. However, none of these mechanisms can explain our results.

• annihilation works only when the dislocation densities are high (over 105 per cm2

[23]);• our solidification interfaces are almost flat and even slightly concave towards the

liquid when growth begins; thus the Yip and Wilcox mechanism does not apply.

To explain our results, we supposed that the dislocations remained in the (111)dense planes of ‘zinc-blende’ structure (schematic positions of <111>plains are shown inFig. 4), even up to temperatures close to the melting point. Inclined with respect to the[111] growth axis, they are gathered on the lateral surfaces during solidification.



Fig. 3 The solid–liquid interface of the growing crystal illustrating the adhering surfacelayer, δ, with the concentration gradient, C, associated with the distribution co-efficient, k. The rising curve shows the temperature profile, T, and the decliningcurve depicts the decrease concentration. The arrows separate the working con-ditions to regions of stable and metastable growth

Fig. 4 Schematic position of <111>plains and their mutual angles

If this were the single operating mechanism, we would get crystals absolutely freefrom dislocations. To take into account the remaining defects, we were led to considerthat even below the CRSS (critical resolved shear stress) threshold (there is obviously nosignificant increase in dislocation density), residual glide can hamper elimination byforcing cross-slip of dislocations about to leave the crystals. Furthermore, direct creationby a Frank–Read mechanism cannot be ruled out, though it is certainly limited.

Let us consider (Fig. 5) an element of crystal of height dz. Let Θ be the angle be-tween the (111) growth plane and the other (111) planes (Θ=70.53°). The disloca-tions in the ring of width dr=dz cotΘ grow out when the interface moves from z toz+dz, unless they cross-slip and turn back to the inside of the crystal. The number ofdislocations remaining between z and z+dz is

N N z z1 1 2=( – ) ( )γ γ d (1)

where N(z) is the number of dislocations at the height z. Supposing the etch-pits to berandomly distributed over the surface of the slice (an assumption we found to be validfor densities over 500 per cm2), γ1 dz simply represents the ratio of the surface of thering to the total surface of the discus of radius R; on the other hand, γ2, is an empiricalparameter depending on the linear density of defects that can cause cross-slip.

γ1 can be easily calculated:

γ ππ

γ1 1

2 2d

dor

2z

R r

R= =

R

cot Θ(2)

Since in our experiments R =0.5 cm, we find γ1=1.4 cm–1.Let us consider that N2 is the number of residual glide-induced dislocations at

the origin; we will assume the equation of N2=γ3 N(z) dz, where γ3 is another empiri-cal parameter depending on the linear density of defects that can cause pinning andmultiplication of existing dislocations throughout the solidification.



Fig. 5 Dislocation elimination on lateral surfaces during growth

A condition of balance of dislocations leads to

d dN N N N z z= + = + +– (– ) ( )1 2 1 2 3γ γ γ (3)

Assuming γ2 and γ3 to be independent of z (γ1 is clearly independent of z since thedislocation distribution is uniform on a slice), the above equation can be easily integrated:

N z N z( ) ( )exp(– )= 0 γ (4)where γ= –γ1+γ2+γ3 and N(0) is the dislocation density in the seed. Measurementsdone before and after crystallization proved that this value did not change duringgrowth. The first experimental points are not exactly at the bottom end of the sam-ples, due to some losses in the polishing procedure.

Linear regression on the experimental data was used to find the ‘best fit’ value ofγ. It can be noticed that in spite of making a lot of simplifying approximations, theempirical equation fits well the experimental data.

Another interesting point is that the value of γ obtained with the best fit proce-dure (cca 1.1 cm–1) is close to the value of γ1 (1.4 cm–1). This indicates that the elimi-nation mechanism is by far the most important one, even though glide phenomenahave also to be taken into account.

For crystals grown in the <111> direction we predicted that dislocations areeliminated during solidification on the lateral surfaces of the crystal owing to theglide phenomenon. We propose that this mechanism explains our results. In order toobtain a true correlation with experimental data, the starting angle between the <111>growth plane and the other <111>planes (70.53°) has to be equal 19.47°. If the start-ing angle between the shoulders of the crystal were below 38.94°, the dislocationswould remain. Existence of such a mechanism was confirmed by studying our pre-pared GaSb crystals. We grew crystals with a different starting angle of crystal shoul-ders or we changed this angle during the growth.



Fig. 6 Dislocation density longitudinal to the growth direction <111>of the GaSb sin-gle crystals. The curves show the dislocation profiles for different crystalsshoulder initial angles. a – for an angle of 14.2°; b – 23.8° and c – first 15 mmat an angle of 46.8° and then 20 mm at an angle of 29°

The dislocation density (EPD) profiles of the crystals with different starting an-gles are shown in Fig. 6.

Crystals with an starting angle <38.94°showed decreasing EPD profiles; but inthe case of the GaSb crystals (Fig. 6, curve (c)), where the starting angle was 46.8°,the dislocation density slowly increased. After reducing the neck angle to 29.0° thedislocation density started to reduce. In the centre of all crystals grown with an start-ing angle <38.94° the dislocation density decreased to a value of <10 cm–2.

The same results were observed for Te-doped GaSb single crystals. If the startingangle was lower than 38.94° the dislocation density decreased so that it was possible toproduce a dislocation free area at the end of crystals. We grew Te-doped crystals withstarting angles of about 24° and the dislocation density was measured using four Te-doped crystals with different concentration of tellurium. For each crystal was the disloca-tion density calculated from wafers taken from the central part of the crystal.

Conclusion

For the low thermal gradient configuration, which seems be most appropriate for therotation set up where the crystal seed rotates fast and crucible slow (under coun-ter-rotation, cf. Fig. 1), the number of dislocations decreases along the crystal. A newmechanism for dislocation elimination during growth is proposed to explain this phe-nomenon. The agreement with the experimental results is very good. The concentra-tion of etch pits rapidly decreased from the beginning to the end of the crystals andthe dislocation densities in the middle portion of all investigated crystals were<102 cm–2. The shallow vertical temperature gradients and virtually flat solidificationinterface prevented thermal stress from building up in the crystals. As a result, thedislocation formation had random distribution. Therefore, Te doping was not foundto influence EPD so there was no evidence of the existence of the so-called ‘harden-ing effect’. Using good necking procedures and choosing a starting crystal shoulderangle <38.94° (assuming growth in <111>direction) it is possible to produce disloca-tion/free crystals without resorting to Te doping to reduce dislocation formation.

* * *

The author express thanks and gratitude to Grant Agency of the Academy of Sciences; particularlythe projects No. A 4010101 and No. A 1010806 are especially acknowledged, as well as one, whichis under the process of application at the Grant Agency of Czech Republic.

References

1 J. J. Mareš, J. Krištofik and P. Hubík, Phys. Rew. Lett., 82 (1999) 4699.2 B. Štìpánek, V. Šestáková and J. Šesták , J. Electr.Eng., 50 (1999) 5.3 V. Šestáková, B. Štìpánek and J. Šesták, J. Crystal Growth, 165 (1996) 159.4 M. Astles, A. Hill, A. J. Williams, P. J. Wright and M. J. A. Young, J. Electron.Mater.,

15 (1984) 41.5 J. P. Czarnecki, N. Koga, V. Šestáková and J. Šesták, J. Thermal Anal., 38 (1992) 575.



6 V. Šestáková, chapter ‘Crucible-free zone melting’ in the book: ‘Modern materials and technolo-gies’ (J. Šesták, Z. Strnad and A. Tøíska, Eds), Academia, Prague 1993, pp. 115–130 (in Czech).

7 J. Šesták and B. Štìpánek (Eds) ‘Thermodynamic applications in material science’ as a specialissue of J. Thermal Anal., Vol. 43, Akademiai Kiadó, Budapest 1995.

8 J. Šesták, J. Leitner, H. Yokakawa and B. Štìpánek, Thermochim. Acta, 245 (1994) 189.9 J. Šesták, B. Štìpánek, H. Yokakawa and V. Šestáková, J. Thermal Anal., 43 (1995) 389.

10 J. Šesták, B. Štìpánek and V. Šestáková, J. Therm. Anal. Cal., 56 (1999) 749.11 D. T. J. Hurle, in P. Hartman (Ed.), Crystal Growth: An Introduction, North Holland,

Amsterdam 1973, pp. 210–247.12 S. Tohno and A. Katsui, J. Electrochem. Soc., 128 (1981) 1614.13 Y. J. van der Meulen, J. Phys. Chem. Solids, 28 (1967) 25.14 B. Cockayne, V. W. Steward, G. T. Brown, W. R. MacEwan, M. L. Young and S. J. Courtney,

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Growth, 78 (1986) 9.16 B. Štìpánek and V. Šestáková, Thermochim. Acta, 209 (1992) 285.17 S. Kondo and S. Miyazawa, J. Crystal Growth, 56 (1982) 39.18 J. P. Garandet, T. Duffar and J. J. Favier, J. Crystal Growth, 96 (1989) 888.19 S. Miyazawa, S. Kondo and M. Naganuma, J. Crystal Growth, 49 (1980) 670.20 A. S. Jordan, A. R. von Neida and R. Caruso, J. Crystal Growth, 76 (1986) 243.21 K. W. Benz and G. Müller, J. Crystal Growth, 46 (1979) 35.22 V. F. S. Yip and W. R. Wilcox, J. Crystal Growth, 36 (1976) 29.23 C. F. Boucher, Jr., O. Ueda, T. Bryskiewicz, J. Lagowski and H. C. Gatos, J. Application Phys.,

61 (1987) 3.



Journal of Thermal Analysis, Vol. 48 (1997) 1105-1122

SOME THF RMODYNAMIC ASPECIS OF HIGH Tc SUPERCONDUCIORS

J. Sestak, D. Sedmidubsky 1 and G. Moiseev a

Division of Solid-State Physics, Institute of Physics of the Academy of Sciences, Cukrovarnick~t 10, 16200 Praha 6 llnstitute of Chemical Technology, Faculty of Chemical Technology, Department of Inorganic Chemistry, Technick~f 5, 167 18 Praha, Czech Republic 2Institute of Metallurgy, Ural's Division of Academy of Sciences, 101 Amudsen str., 620316 Ekaterinburg, Russia

Abstract

Thermochemical and thermodynamical properties of HTSC phases are reviewed for the Y- Ba-Cu--O system and also presented for the newly calculated Bi--Sr-Cu--O system stressing out stoichiometric and phenomenological viewpoints. Simulated data are listed for (/-/~298--/~o), phase transformation temperatures, standard entropies, standard enthalpies of formation, heat capacities in crystalline phase, etc. Pseudobinary phase diagrams are treated showing the effect of oxygen partial pressure particularly illustrated on the (Sr, Bi, Ba)--Cu--O system.

Keywords: data simulation, enthalpy, non-stoichiometry, oxide superconductors, phase diagrams, thermodynamics, Y-Ba--Cu--O and Bi--Sr-Cu--O systems

Introduction

No other field has attracted so much attention and publication activity than that of oxide high temperature superconductors (HTSC). Beside the specific journals ex- clusively devoted to this topic (e.g. Physica C, Supercond. Res. and Tech.) the other journal have also used the opportunity to include HTSC papers sometime publish- ing whole volumes as monothematic monographs, e.g., Thermochimica Acta 1991 [1]. One of the most important and attractive problems are caloric and thermochemical properties where the thermodynamic stability of HTSC associated substances plays a major role. It can be solved by means of thermodynamic investigations including experimental studies as well as theoretical calculations which re- quire a vast amount of data. There are various methods of thermodynamic estimation which provide a feasible and valuable tool of the simulation and extrapolation of thermodynamic data to the range of variables where experimental data are lacking. When using the term thermodynamic "equilibrium" the stability aspect with respect to stable and metastable systems and phases must be considered [2]. There are different wellknown approaches to obtain the temperature, pressure and composition of a multicomponent system at equilibrium in order to study an appropriate

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�9 1997 Akadgmiai Kiad6, Budapest Chichester

1106 ]EST,~K et al.: THERMODYNAMIC ASPECTS

phase diagrams. The obvious method is a purely experimental one, this is -for instance, a direct measurement of phase composition at equilibrium when temperature, pressure and other intensive parameters are defined. Several experiments with different temperatures seem to be a way how to obtain a phase diagram of a system under investigation although such direct experimental studies are laborious and time-consuming moreover bearing a most severe problem how to attain a true equilibrium. Therefore both stable and metastable phase boundaries are simultaneously observed [2] further complicated by oxygen stoichiometry because of a free exchange of this volatile component with environment. The most widely investigated system is that of Y-Ba-Cu-O [3] (YBCO) where three superconductors from the family of phases Y2Ba4Cu6+nO14+8 (with n = 0, 1 and 2) have already been obtained in various forms including bulk crystals. Unfortunately, only a little portion of reports have been directed to study the thermodynamic properties of HTSC phases which is true even for the YBCO system although a self-consistent set of thermodynamic functions was already published by several authors but has to be continued in order to become generally useful. All these aspect will be the main objects of our discussion below stressing out particularly our preceeding investigations.

Our previous studies on the YBa2Cu30 ~ related phases

In the early years 1988-92 we got successfully engaged in studying phase diagrams in the YBCO system [4-12] and succeeded to investigate the pseudobinary cut of YCuO2.5--YBa2Cu306.5--BaCuO2 and the characteristics of YBa2Cu306. 5 phase [4]. An improved experimental determination of oxide phase equilibria was proposed being based on the separation of the melt from an equilibrated sample by soaking melt into the supportiong Y203 pellet and the consequent analysis of its solid residual [5]. This method has been applied to growing single crystals [6]. It matured to several review articles [8-15] and consequent attempt to prepare YBCO doped polycrystalline materials through fast melt solidification and glass recrystal- lization [15-18]. Since 1992 we moved to studying thermodynamics, first simulat- ing the YBCO deposition from vapor/gas [19] followed by thermodynamic and structural considerations on compatibility of various substrates for the YBCO thin film formation [20]. Thermodynamical and thermochemical estimation and calculations methods have been in the center of our interest until now [24-35].

It was shown, for example, that thermodynamic simulation of the YBa2Cu30~ (123~, 5 >__ 8 < 7) formation [21] can be realised on basis of a formal mixing of the end members, i.e. both 123 phases having the oxygen content 6 and 7. To find the enthalpies of mixing, AHmix, the known values of formation enthalpies,//~298, have to be employed from literature, e.g., using less recent series [22]:

8 = 6 6.25 6.47 6.5 6.69 6,93 7

A/-~ZgS = 2630 2671 2672 2677 2689 2713 2772 kJ mol -l

J. Thermal Anal., 48, 1997

~EST/t,K et al.: THERMODYNAMIC ASPECTS 1107

and fitted by a polynomial to match either ideal, regular (all data upper raw 1 or selected data lower raw 2) or subregular solid solution models. The coefficients of the polynomial AI-1~298=a+bS+c82+d83 and the end values of the individual models are as follows:

Coefficient -A/-/~298 Corr. Devia- Model

a b c d 8 = 6 8 = 7 coeff, tion

Ideal -2641 -76.8 2642 2718 0.856 0

1-regular -2634 -124.5 45.7 2680 2733 0.866 +

2-regular -2680 --64.7 -109 2642 2718 0.923 +

Subregular -2631 -220.5 316 -181.5 2631 2717 0.864 -

The best fit provides seemingly the regular model of solid solution with a posi- tive deviation from the linear interpolation of given experimental data. The charac- ter of the cubic dependence allows one to estimate the heat of mixing from the sym- metrical dependence according to:

Regular (I): -57.7 82 with /~nmi x at 13=0.5 equal 14.5 kJ mol -l

Regular (2): 107.7 62 with AHmi s at 13=0.5 equal 26.9 kJ mol -l

On the other hand the change of the oxygen stoichiometry can be viewed as a series of superstructures with characteristic values of 8 following the number of fully occupied oxygen chains, e.g., non (deoxygenated-tetra, 8=6), every two in five (8=6.65), every second chain (8=6.85), and all ( oxygenated-tetra, 5=7) [23].

Thermodynamics, stoichiometry and stability

It was found that the actual formulae of HTSC are sensitive to experimental conditions and can be strongly dependent not only on the overall, but also local, ener- getic state [12]. Possible sources for such stoichiometric changes can be either the action of the external force fields or the effect of the internal make-up. The excess of surface energy of very thin layers enveloping grains results from a very high curvature of extremelly small particles or from pressure contacts between the grains created in well-compacted samples. Similarly a structural misfit of thin HTSC layers deposited on various substrates [20] can produce strain or the strain sites can be created during the formation of off-stoichiometry inclusions with the dimension near lattice parameters. Another source of stoichiometric changes can be located on the boundary of currents passing the crystal lattice or in the vicinity of negative electrodes [34]. In contradiction to structural models we can hypothetically account for the existence of various phases yet unknown by a formal extending the cation ratios to ranges of yet undetermined compounds which is well compatible with our


1108 ~EST.~K et al.: THERMODYNAMIC ASPECTS

Table 1 Hypothetical changes of ratios (Y + Ba):Cu = |

Composition shift Y1Ba2Cu3Oa Y1Ba3Cu2Oa Y2BalCulO 5

supercond, phase "green" phase

Y:Ba:Cu | 1 Y:Ba:Cu | Y:Ba:Cu |

Basic 1 2 3 1 1 3 2 2 formulae

Exchange of 1 1 2 1 1 5 3 2 Yc:~Ba

Multiplication (13 3) - 3 8 5 2.2 of(Y+Ba) (22 3) - 1 6 3 2.4

2 4 7 0.9 1 3 3 1.3 Multiplication of(Cu) 12 4 0.7 2 2 3 1.3

1 2 5 0.6

2 3 5 t 1 4 3 1.7 Unequal change (Yc~Bac~Cu) 4 1 5 1 2 12 1.5

13 5 0.8 1 2 2 1.5

2 1 1 3

1 2 1 3

1 3 1 4

2 2 1 4

(2 1 2) -

(2 1 3) -

2 3 2 2.5

1 4 2 2.5

(I 3 2) -

sources of the stoichiometry

oxygen 05)

stress sites

surface layer

(tetra-ortho)

valence changes

impurities

dopants (M)

changes accounted for YIBa2Cu3Os:

cations ratios (Y:Ba:Cu)

high curvature, interfaces

(Y-Ba) (Cu)

vicinity of charge, supercurrents

(Y-Ba-Cu)

metastability due to freeze-in, inclusions

(Y-Ba)-M

phenomenological thermodynamic calculations. Accordingly, the extent of possible variations of the cation ratios can be formally derived using the schema shown in Table 1, where we can phenomenologically assume the change of cation stoichiometry in three groups and four gradual levels regardless to real structures [24]. For most of the existing and derived compounds their thermodynamic and thermochemical properties were not found to be determinable by conventional estimations and thus our calculations using special empirical and quasi-thermodynamical methods and computer programs [31, 35] became a useful tool.

Simulation of thermodynamic and thermochemical data

We already published reviews and refinements of the known and estimated thermochemical properties such as metastability [25], (H~298-/~o) and the phase transfor-

J. Thermal Anat., 48, 1997

Tab

le 2

The

rmoc

hem

ical

and

the

rmod

ynam

ical

pro

pert

ies

of s

ome

com

poun

ds in

the

Y-B

a-C

u--O

sys

tem

Oxi

de

A/~

z98

/ ~

98 /

/a

~zg8

-/-/~

o / c o

=a

+bx

lO-3

T--c

xlO

ST-Z

/ kJ

mol

-~

J K

-1 m

o1-1

J

mo1

-1

J K

-1 m

o1-1

a b

c

Tph,

r/ M

lphd

%

,oh,

/ ~ C

J

mo1

-1

J K

-I t

ool -

l

t'rl

123-

O6

-25

86

.8•

31

9.8

6

4935

2 30

7.48

76

.28

123-

O7

-27

06

.3+

__

2.4

3

23

.06

51

107

315.

29

54.4

7

123.

5-O

7.5

-27

94

.5+

__

3.5

34

5.15

54

040

305.

12

84.5

6

124-

O8

-28

81

.2+

__

.5.7

3

67

.24

57

750

356.

72

45.7

4

125-

O9

-30

55

.2•

41

1.4

2

6561

0 36

3.27

11

4.67

143-

O8.

5 -3

82

8.3

• 4

54

.15

70

016

312.

96

41

.07

211-

O5

-27

12

.0_

+_

2.6

2

23

.00

35

344

206.

10

35.2

0

YC

uO 2

-103

8.6+

_24.

3 98

.81

1567

7 92

.90

16.3

6

Y2C

u205

-2

214.

8+_5

.1

200.

83

3136

0 20

1.65

41

.85

Y2

BaO

4

-253

3.5_

+0.

7 17

8.81

28

128

164.

65

20.5

3

Y2

Ba2

Os

-313

1.0_

+6.

4 2

45

.16

37

905

227.

88

22.0

5

Y2B

a407

--

4314

.1•

37

7.8

4

5734

6 33

5.28

2

4.6

0

Y4B

a309

-5

666.

6:!:

9.3

42

4.0

0

6604

9 39

3.75

48

.23

BaC

uO 2

-782

.0_-

L21

.4

110.

52

1702

0 91

.45

24.4

2

BaC

u20

/ -8

07.0

_+7.

4 15

1.51

22

707

118.

30

27

.09

Ba2

CuO

3

-14

09

.1 +

_29

176.

87

2673

0 14

5.32

2

8.7

7

BaC

usO

8 -2

65

6.4

• 4

20

.00

65

590

354.

18

113.

42

26.2

13

1373

13

70

362.

12

42

.45

0

1288

11

0900

4

18

.10

22.0

64

1190

23

400

41

6.6

0

42

.47

0

1110

9

20

0

44

5.8

0

22

.06

4

1023

89

00

49

8.2

0

42

.81

0

1170

15

6500

35

7.90

20.9

25

1543

31

200

26

2.0

0

8.05

0 18

43

6450

0 12

5.60

18.1

55

1428

14

30

26

5.1

4

20

.32

0

1673

9

80

0

198.

37

36.7

34

t313

10

200

24

9.2

7

52.4

40

1413

28

100

386.

75

44.9

15

2433

2

45

70

0

510.

18

7.24

5 13

18

2830

0 12

2.46

8.01

0 15

00

4280

0 16

3.76

16.5

94

1123

43

00

176.

00

24

.90

10

73

3100

44

5.05

,-]

>,

o z > N

,..]

,,q

O

11 I0 ~ESTAK et al.: THERMODYNAMIC ASPECTS

mation temperatures [26], standard entropies of formation [27], heat capacities in crystalline phase [28], heats of melting/decomposition [29] and average heat capacities of phases transformation products [30]. The multitude of literature data on properties of complex oxides, their critical analysis and calculation, and also the statistical methods used for the determining reliable data were also described in detail [26-33] and the final thermochemical properties of the complex oxides are given in Table 2.

Standard enthalpies of formation

On the basis of above mentioned treatments [31-35] we analysed available sources and data priority accounting for [33] :

- Repeated or close values reported by different researches using various experimental techniques,

- Results published by representative laboratories, - Most recent papers and - Neglection of extreme values.

Consequently we treated averaged data of 11 characteristic YBCO compounds in order :

- to find some regularities in the published values, - to try the derivation of certain empirical dependences and - to estimate the extrapolated data of related compounds difficult to measure

and/or to exist only hypothetically.

One of the most important characteristics is the standard enthalpy of formation, Af/-/~298(]), which can be presented in the following way:

Af~298(]) = ~'niA/-~298(i ) + A/~298(OX)j (kJ mo1-1) (1)

where m i and A/-/~298(i ) are the number of moles and the standard enthalpy of formation of the i-th simple oxide of thej-th complex, respectively and AH~29s(OX)j is the standard enthalpy of formation of thej-th complex from simple ones. As a result the empirical dependences based on the number, m, of oxygen atoms in the molecule of a compound in question [24] were employed in the three gradual manners. First using a simple relation of Ae/-/~29s(ox)j equal to (am) and to (a+bm) and then a more complex average method of the five simultaneously applied equations for

A/~29a(ox) j = a+bm, = a+bEN, = a+blnY-,N, = a+bY_At and = a+blnY~M,

where EN is the sum of number of elements in a molecule and EM is the molecular mass of a compound. The resulting data of individual treatments are compared in Table 3 [33].


~EST./~K et al.: THERMODYNAMIC ASPECTS I I t 1

Table 3 Standard enthalpies of formation from simple oxides

-~298(ox)lkJ mo1-1

Compound Equation employed: literature -29.274xm 13.3-14.5xm average

data 235.3-51.6xm method (@)

BaCuO 2 73.4u 59.5 - 73.4u

Ba2CuO 3 108T29 89.2 - 108u

Y2BaO4 73u 119.0 44.7 61.2u

Y4Ba309 224.3u 267.7 229.1 184.8~9.3

Y2BaCuO 5 76.4u 148.7 59.2 84.6$2.6

YBa2Cu306 110.7-+54.5 178.4 74.3 126.6u

Y Ba2Cu307 129.7u 14.3 208.2 125.9 134.2u

YBa2Cu40 s 173.6u 237.9 177.5 154.0.7,.5.7

YBalCU3Os.5 241.0u 1.2 252.8 151.7 190.2-T-14.2

M~Y2Cu205 -14.5u 148.7 59.2 -14.5T-5. Ir

YCuO 2 11.3u 59.5 15.7 4.5u

YBa2Cu306.5 - 193.3 100.1 130.8u

YBa~Cu3.507.s - 223.0 151.7 144.6u

YBa2Cu609 - 267.7 229.1 172.7u

Ba3CuO 4 - 119.0 44.7 138.7-+6.4

Ba3CusO 8 - 237.9 177.5 210.9-+5.9

Ba2Cu30 s - 148.7 59.2 147.7u

Y2Ba205 - 148.7 59.2 97.6u

Y2Ba407 - 208.2 125.9 169.1u

_ BaCu202 38.2u 59.5 - _.. 30.1u

y-containing

31.75-0.56 Y_,N

603.72-129.94 lnY_~

32.86-0.25 ZM

718.76-131.32 InZM

40.02-25.45 m

(e) y-not containing (Ba)

-18.166--0.547 EN

255.57-71.283 InZN

-20.25-0.228 7_21,/

303.58--69.17 InZM

-3.4-35 m

m - number of oxygen atoms in molecule, EN - sum of the number of elements in molecule, EM - sum of molecular mass of a compound, M ~ - metastable.

Effect o f the multiplication of oxide layers

The values of formation enthalpy can be consequently used to extract their partial values along the homological series. For example, according to the stoichiometry changes: 1237 --~ 1248 ---> 1259 ~ 1/2 24715 we can write following equations:


1112 ~EST~.K et al.: THERMODYNAMIC ASPECTS

Af/-/~ox(298)(1248) = AfH~ox(298)(1237) + AH(CuO)

Af/-~ox(298)(1259) = Af/-/~ox(298)(1237) + 2AH(CuO)

Af~ox(298)(24715) = 2AeH~ox(298)(1237) + M'/(CuO)

2x1237 1237 24715 124 s -268,4 -134.2 -289.2

-20.8 19.8

1259 -154.0 -173.0

-19.8 1/2 39.6 ~M4(CuO)

(2)

(3)

(4)

The resulting value of the formation enthalpy of adding a single CuO is 19.8 kJ mo1-1. Similarly we can estimate the value for Cu20 assuming

(011)BaCuO 2 = BaO+CuO; (012)BaCu202 = BaO+Cu20;

-73.4 --86,4 ~AH=-13

which is in accordance with the estimate CUO(-155.15) - Cu20(-167.28 ) = -12.13 kJ mo1-1.

Accordingly we can calculate the multiplication of BaO layers assuming:

AfH~ox(298)(2205) = AfH~ox(298)(2104) + AH(BaO) (5)

AfH~ox(298)(2407) = Af/-/~ox(298)(2104) + 3M-/(BaO) (6)

210 220 240 -73 -97.6 -169.1

-24.6 -71.5 -32 ~M-/(BaO)

and

AfH~ox(298)(1438.5) = Ar/-~ox(298)(1236.5) + 2AH(BaO)

143 123 -190.2 -130.8 -29.4 ~AH(BaO)

(7)

Phase diagrams and oxygen partial pressure

Most of the phases in the systems in which the HTSCs have been observed [12- 14], contain copper or oxygen in variable oxidation state. This variation of valency can be achieved either by substitution of some heterovalent cations or by modifica-


~ESTAK et al.: THERMODYNAMIC ASPECTS 1113

tion of oxygen stoichiometry. As a rule, the oxygen content can be varied in some extent by adjusting the state of the surrounding atmosphere. There are two variables which influence the oxygen concentration in a condensed phase [36, 37] - temperature and partial pressure of oxygen in the atmosphere.

The variable oxygen content must be taken into account when a thermodynamic model of a given phase is constructed. At normal conditions, i.e. when temperature and pressure are adjustable parameters, the construction of thermodynamic model means to propose the form of Gibbs free energy as a function of these two parameters and chemical composition. If oxygen is a free component the Gibbs energy is to be recalculated with respect to its activity. For this reason it is convenient to in- troduce another thermodynamic potential (sometimes called hyperfree energy) [37]

Z = G - nd.t o (10)

where G is Gibbs energy and n o and I.to are, respectively the molar quantity and chemical potential of oxygen. For such partially open system the chemical equilibrium is described by analogical relations as for closed system, except that G is re- placed by Z. For instance, the well known equation for equilibrium between a solid phase and high temperature melt transforms to

zq(0 _ Zq(S)- Zq(/ ~rc (l) re(s)- re(l)

(11)

where l and s stand for liquid and solid, respectively, Zq are quasimolar hyperfree energies and Yc is quasimolar fraction of stable component c 1. The term "quasi" means the quantity related to the sum of conservative components.

The thermodynamic models usually contain some free parameters which should be evaluated before they are applicable for the phase diagrams calculations. The equilibrium conditions like (Eq. (11)) can be used to refine the model parameters by fitting them on the experimental data. For the pseudobinary systems (with two conservative components) the usually available experimental data are from DTA measurements which provide the temperatures and compositions of eutectic and peritectic points at various values of Po: For more complicated systems additional complementary methods (e.g. soaking method [5]) have to be applied.

Recently three pseudobinary systems - Sr-Cu--O, Bi-Cu--O and Ba-Cu--O - have been extensively studied [39] by the approach described above. The common component in all of them is copper and it was this particular component which was supposed to be responsible for variable oxygen concentration in the high temperature melt, while the other stable components were considered to bind the amount of oxygen corresponding to their standard valence states (Sr 2+, Ba 2+, Bi3+). Conse- quently the thermodynamic behavior of CuOx melt was of particular interest.


1114 ~EST/~K et al.: THERMODYNAMIC ASPECTS

Particular case o f CUOx-based subsystems

Melt of CuO x

The quasimolar hyperfree energy of the liquid phase CuOx was modelled by the integral of its total differential [37]

T

Z(ql)(T, Po2) = Z(qI)(T*,P@- I(Sq(T)- 1/211oSo2(T))dT- T"

T Po2 - 1/2 RIn(P@~11o(T,P@dT- 1/2 RTI'qo(T, Po2)dln(Po)

T* * PO 2

(12)

For the starting point of integration the eqam!ity of quasimolar hyperfree energies of CuO(s), Cu20(s) ard CuOx(/) at the eutectic point (T*= 1353 K, P~)2=0.531 [36]) was employed. The definite value was calculated using tabular data [38]. The oxygen concentration in the melt was described by empirical relation.

no(T, Po2) = 1 - ( T - Q) 1/2 (a - b(Po~) ~/3 (13)

which fits very well the experimental dependence found by Roberts and Smyth [36] in a broad range of partial pressure and temperature. The temperature dependence of quasimolar entropy was well described by simple relation

Sq(T) = E + F In(T) (14)

while the function So.(T) was taken from thermodynamic tables. All free parameters of the model (Q, a, b, E and F) were evaluated by the regression using the data taken from Ref. [36].

Pseudobinaries M--Cu--O (M= Sr, Bi, Ba)

For intermediary phases Mx..yCuyOs(s) (M=Sr, Bi, Ba) a simple model supposing the oxygen content being independent on both T and P% was used. In such case the hyperfree energy

Zq(T, Po2) = YcuZq(CUO) + (1-Ycu)Zq(MO v) +

+ A + BT + CT In(T) - 1/2 A'tloRT ln(Po2 ) (15)

is constructed as a weighted sum of contributions of respective oxides forming the given phase and the temperature dependence of the Gibbs energy of formation. The last term reflects the difference between the quasimolar fraction of oxygen in the in- terlnediary phase and in the assembly of original oxides. For all considered solid


,~ESTAK et al.: THERMODYNAMIC ASPECTS I 115

phases excepting Sr14Cu24041 the coefficient ATIo which reflects this difference was regarded as zero.

The thermodynamic model for the mixed oxide melt of type Ml_yCuyO,(/) (M=Sr, Bi, Ba) is based on the assumption of mixing the CuO(x ~ liquid phase discussed above and the other part of the melt (MO~ 1~) with fixed oxygen content:

Z(ql)(T, Ycu) = YcuZq(CUO~ l)) + (1 - Ycu)Zq(MO~ l)) +

+ RT[Yculn(u ) + (I - Ycu In(l - u + "QYcu( I - YCu)( I + KYc~) (17)

Here ~ and K are phenomenological interaction parameters of the used quasiregu- lar model being dependent on Po2 as follows

= A + Bln(Po) and K = C + Dln(Po) (18)

The hyperfree energy of MO~ 1) melt consists of two terms: the hyperfree energy of the solid oxide MOv and Gibbs energy of its melting as a function of temperature.

The coefficients in temperature dependencies of Gibbs energy of formation of binary phases (Sr2CuO3, SrCuO2 and Sr14Cu24041 in Sr-Cu-O system, Bi2CuO 3 in Bi--Cu-O system, Ba2CuO3 and BaCuO2 in the system Ba--Cu--O) and those of melting of individual oxides, as well as all other phenomenological parameters were calculated by regression analysis of the DTA-data recorded at two values of Po. in the surrounding atmosphere, namely in pure oxygen (Po~ = 1) and purified air (Po2= 0.21). Their real values are not presented here and the interested reader is refered to original paper [39].

i i i l ~ " ~ i i i J i

1800 ~ ' ~

1700

~ ~ 1600 1526 ",,, Liquid

:~ 150o "~ 1435 ."

. o o . . . . . . . . . . . . . . . . I . . . . rx 1 . . . . . . \ ,. .

~L, 1 '97 1304 ~ C 1:]00 I . . . . . . . . . . . . . . . . ",., " . - , . . . _ / 1286.

. ' 1200 12oo o ' . . . . --sc+%_o -11;,9

3 . . . . . . . . . . . . . . . . . . . . . ~.I d .31 s c + c . o 1o93. 1100 ~n m St4C24+CuO

1 0 0 0 . , �9 , . , I , , I I , I , I , f ,

0.0 O.1 0.2 0.3 0.4 0.5 0.6 0,7 0.8 0.9 1.0

SrO Mole f racLion of Cu CuO x

Fig. 1 Phase diagrams of Sr-Cu-(O) system calculated for Po2= 1 (full lines) and Po =0.01 (dashed lines)

3. Thermal Anal., 48, 1997

1116 ~ESTAK et al.: THERMODYNAMIC ASPECTS

The refined parameters were substituted back to model functions of Zq for the respective phases which were further employed for the numerical calculation of liquidus curves in the phase diagrams. The results are presented on Figs 1, 2 and 3 showing, respectively, the phase diagrams of the systems Sr--Cu--O, Bi--Cu--O and

o9

a5 h ~0 Ca.

E-~

1400

1300

1200

Ii00

1000

S ~ i

Liquid

_ . j ~ . . . . . , -" . . . . . . . . . . . . L2..~5_ _ _

1008 C

o.o o.1 o.e o.a 0.4 0.5 0.6 o.v o.a 0.9 1.o

BlOt. 5 Mole I r a c L i o n of Cu CuO x

Fig. 2 Phase diagrams of Bi--Cu-(O) system calculated for P%= 1 (full lines) and P%=0.05 (dashed lines)

i

1600 ~ ' ! Liquid

i t ' . 1311 1298

1300 1273 . . 1 1 2 ~ j ,,,,v~,

1-12t,~'-- , 2 8 " 2 " - . , ~ , , . . - " 1 2 0 5 1 2 0 0 ~ ~ ., 1 1 7 5

rn I m I J 1100 . . . . . . I . , . I . , , , , , . , . ]

0 . 0 0 . 1 0 . 2 0 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 . 0

BaO M o l e f r a e U o n of C u CuO X

Fig. 3 Phase diagrams of Ba-Cu-(O) system calculated for P%= 1 (full lines) and Po =0.21 (dashed lines)


SESTAK et al.: THERMODYNAMIC ASPECTS 1117

1 6 0 0

1 5 0 0 L i q u i d s Z ~

v 14oo / # q. " , , . / .,.."C,,_o " i X I , 'I - - 2 -

1 3 0 0 . . . . . . ", X _ . . . . . .. , - .

Q . . . . . . .

,_ o

U 0

1 1 0 0 " E/ ( J

130 m I

1 0 0 0 i t i i I i i i

0.0 0.2 0.4 0.6 0.8 1.0

BaO Mole f r u c t i o n of Cu CuO x

Fig. 4 Phase diagram calculated in Ref. [40] using simple regular model (full line) and in Ref. [41] using temperature dependent subregular model (dashed) together with the experimental data from Ref. [42] (dashed-and-dotted line)

Ba-Cu-O for two values of partial oxygen pressure. In Fig. 4 the previously calculated and experimentally determined diagrams for the system Ba--Cu--O are shown for the comparison.

Let us note that although the calculated curves fit very well the experimental data [39], the thermodynamical properties evaluated from the yielded parameters loose their physical meaning out of the interval of temperatures from which the experimental values were available. Thus a better approach would be based on separate refinement of these parameters for solid phases (e.g. from calorimetric data) whose fixed values would be subsequently used for the regression analysis of the model dependencies of the melt. This improvement is a subject of further investigation.

Thermochemical and thermodynamical data of the SrO-Bi203(-CuOx) system and chemical phase diagrams

Currently we have started to calculate thermochemical and thermodynamic properties in the system of SrO-Bi203 [43], see Table 4, trying to extend our estimation to reach preliminary values for the whole SrO-Bi203-CuO x system, see Table 5.

Accordingly it was found useful to employ another representation of the effect of oxygen partial pressure against the phase relations involved which is illustrated in the case of SrO-Bi203 in Fig. 5. A recently written computer program [44] enables one to construct generally various types of chemical potential diagrams in the metal (M1)-metal (M2)-oxygen systems. For oxides [44] and their relation to HTSC [31]


~x

"4

Tab

le 4

The

rmoc

hem

ical

and

the

rmod

ynam

ical

pro

pert

ies

of s

ome

doub

le o

xide

s in

the

SrO

-Bi2

03 s

yste

m

/~29

8 /

~229

8 /

H~29

8"-/~

o / zp

h.tr/

A

nph.

tr/

%=a

+b'lO

-3r-c

'lO~r

-2/

Oxi

des

kJ m

o1-1

J

mo1

-1 K

-1

J m

o1-1

K

kJ

mo1

-1

J m

o1-1

K -l

a b

c

S%B

i209

-4

207.

4 47

1.0

6890

0 12

38

267.

7 39

9.4

68.3

32

.0

Sr3B

i206

-2

443.

4 31

0.2

4506

5 14

83

199.

9 25

4.4

46.8

18

.6

Sr2B

i205

-1

860.

3 26

5.5

3711

0 12

13

132.

5 20

6.0

39.8

14

.3

Srls

Bi2

2051

-1

7928

.9

2607

.0

3763

30

1076

10

99.5

20

72.1

41

1.0

139.

7

Sr6

Bi1

4027

-7

927.

6 13

65.7

19

6020

92

2 51

6.7

1054

.8

221.

5 64

.4

SrB

i407

-1

803.

5 35

1.7

5031

0 80

0 11

3.5

266.

8 58

.7

15.4

Sr2B

i6O

ll

-302

9.6

554.

5 79

450

864

194.

9 42

4.4

90.5

25

.0

SrB

i204

-1

224.

2 20

2.8

2914

0 12

13

101.

5 15

7.6

32.9

9.

9

SrsB

iloO

23

-803

3.1

1175

.3

1696

20

1070

52

9.8

933.

1 18

5.6

62.7

Srs

Bi2

Oll

-5

387.

5 57

8.1

8477

0 13

35

350.

3 49

6.2

82.9

40

.9

SrsB

i6O

l4

-490

2.7

700.

0 10

1975

12

13

366.

6 56

5.9

112.

6 38

.4

Sr6B

i201

1 -4

391.

4 47

1.0

6890

0 12

38

261.

7 39

9.4

68.3

32

.0

Sr6B

i401

5 -5

162.

7 62

0.4

9013

0 12

00

323.

4 50

8.7

93.6

37

.4

Sr24

Bi1

4052

-1

9201

.2

2332

.3

3392

90

1224

12

48.4

19

25.4

34

7.8

143.

5

cp(a

t T>

Tph

.tr)/

J m

o1-1

K-1

534.

7

356.

6

280.

6

2777

.0

1392

.0

346.

8

555.

7

217.

8

1250

.0

670.

2

775.

4

534.

7

685.

8

2594

.0

O>

,-]

>.

t'D

.. o z >

Tab

le 5

The

rmoc

hem

ical

an

d th

erm

odyn

amic

al p

rope

rtie

s of

som

e co

mpo

unds

in

the

Bi-

-Sr-

-Cu-

O s

yste

m

Oxi

des

kJ t

ool -

1 J

tool

-1 K

-t

J m

ol -~

K

kJ

too

l -~

J to

ol -~

K -1

a b

c

Sr2B

i2C

uO ~

-2

09

5.7

30

3.3

44.5

13

72

156.

7 2

47

.8

66.0

20

.3

%(a

t T

> Tp

h.tr)/

J

mol

-~ K

-1

384.

1

,-q

>,

SrsB

i4C

usO

19

-72

43

.0

956.

5 14

2.2

1819

66

1.4

849.

9 17

4.2

95.8

Sr3B

iEC

U20

8 -2

90

6.4

40

1.5

59.5

17

72

254.

1 34

8.5

78.4

3

5.9

Srl

sBi2

2Cul

oO61

-2

03

8.4

30

72.1

4

49

.6

1676

14

38.7

24

69

698.

3 18

2.5

SrgB

i4C

uOl6

-7

09

7.7

84

1 12

0.8

2018

54

7.2

708

150.

6 74

.2

Srv

Bi2

Cu2

0~

-53

84

.4

623.

1 91

.9

2001

47

7.8

55

7.2

99

6

8.9

1254

.9

522.

1

3850

.4

1077

.8

813.

2

o

z > N

GO

4~

co

-q

,.q

o'J


40 I . . . . .

"

log A(Bi) Bi " / " " - - ~ 2O

log A (Sr)

10 1 1 - t "

t

o

Sr t i SrO

- 10 I I I J I I " '

-60 -50 -40 -30 -20 -10 0

log P(O2)/atr n

Fig. 5 Chemical potential diagrams in the Bi-Sr-O system calculated [44, 47] for two temperatures 837 K (full) and 937 K (dashed lines)

it was shown that log(aM ~aM2) vs. logP(O2) plot is very convenient and the use of log(aM~aM2 ) makes it possible to treat the two metallic elements (1 and 2) equivalently. Figure 5 shows such a simplified chemical potential diagram consisting of the stability areas of elements (Bi, Sr), binary (SrO, Bi203) and selected double (Sr2Bi6Ou, SrsBi10023 and SrsBi2Oll) oxides and a similar treatment was already applied in the course of evaluation of the systems Y-Cu-O [31] or Ga-Sb-S [45].

Conclusions

Estimation methods show their importance for computing thermodynamical and thermochemical properties of many inorganic compounds [46] as documanted on the well screened Y-Ba--Cu--O system [20-33]. It certainly would be a useful complementary source of data for existing databases [31, 35, 38, 47].

The work was carried out under the project No. A 2010532 supported by the Grant Agency of Academy of Sciences of the Czech Republic and the grant No. 104/97/0589 financed by the Grant Agency of the Czech Republic.

References

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~EST~,K et al.: THERMODYNAMIC ASPECTS 1121

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