The basic relation connecting the Gibbs energy to the temperature and pressure in any closed system:

(nG) (nG)d(nG) dP dT (nV)dP (nS)dTP T

and pressure in any closed system:

T,n P,nP T

applied to a single-phase fluid in a closed system wherein no h i l tichemical reactions occur.

Consider a single-phase, open system:

Chemical potential

i(nG) (nG) (nG)d(nG) dP dT dn

P T

jT,n P,n i P,T,niP T n

Define the chemical potential:

j

ii P,T,n

(nG)n

The fundamental property relation for single-phase fluid systems of constant or variable composition:

i ii

d(nG) (nV)dP (nS)dT dn

When n = 1,

dG VdP SdT dT,x

GVP

i i

idG VdP SdT dx

GST

P,xT

C id l d t i ti Consider a closed system consisting of two phases in equilibrium:

i ii

d(nG) (nV) dP (nS) dT dn

i ii

d(nG) (nV) dP (nS) dT dn

nM (nM) (nM)

i i i ii i

d(nG) (nV)dP (nS)dT dn dn

i i i ii i

d(nG) (nV)dP (nS)dT dn dn

The two phase system is closed thus:

d(nG) (nV)dP (nS)dT

i i i ii i

dn dn 0

i idn dn

ii

M lti l h t th T d P i ilib i h

i i ii

( ) dn 0

Multiple phases at the same T and P are in equilibrium when chemical potential of each species is the same in all phases.

Define the partial molar property of species i:

j

ii P,T,n

(nM)Mn

The chemical potential and the particle molar Gibbs energy are

i iG

The chemical potential and the particle molar Gibbs energy are identical:

1 2 inM M(P,T, n , n ,..., n ,...)

For thermodynamic property M:

M M i i

iT,n P,n

M Md(nM) n dP n dT M dnP T

i iiT ,n P ,n

M Md(nM) n dP n dT M dnP T

M M

i i id n x dn n dx d nM n dM M dn

i i iiT ,n P ,n

M MndM Mdn n dP n dT M (x dn ndx )P T

i i i ii iT n P n

M MdM dP dT M dx n M x M dn 0P T

i iT,n P,nP T

0

dxMdTMdPMdM 0

,,

i

iinPnT

dxMdTT

dPP

dM

0 iiMxM0 iiMnnMSummability i

ii

Calculation of mixture properties from partial properties

i

iiyrelations

i iiT P

M MdM dP dT M dx 0P T

iT,n P,nP T

i i i ii i

dM x dM M dx

i iiT,n P,n

M MdP dT x dM 0P T

The Gibbs/Duhem equation

This equation must be satisfied for all changes in P, T and the caused by changes of state in a homogeneous phase

iMchanges of state in a homogeneous phase.

For the important special case of changes at constant T and P:and P

i ii

x dM 0

2211 MxMxM For binary systems 2211 MxMxM

Const. P and T, using Gibbs/Duhem equation

y y

22221111 dxMMdxdxMMdxdM

dMdMdM dM2211 dxMdxMdM

121 xx

211

MMdxdM

21 ddMxMM 12 dx

dMxMM 1 12

M x MM (A) (B)

121 dx 1dx

2x (A)

The three kinds of properties used in solution thermodynamicsthermodynamics

Solution properties M, for example G,S, H, Up p p , , ,

G ,S , H , Ui i i iPartial properties, , for exampleiM

G ,S , H , Ui i i iPure-species properties Mi, for example

, , ,i i i ii

, , ,i i i ii

EXAMPLEDescribe a graphical interpretation of equations (A) and (B)Describe a graphical interpretation of equations (A) and (B)

2dM M I

1 1d x x

1 21 2

dM I I I I 1 2

1I I

d x 1 0

dMI M x2 1

1I M x

d x

1 11

dMI M 1 x

d x

1

1 1I M 2 2I M

The need arises in a laboratory for 2000 cm3 of an antifreeze solution consisting of 30 mol-% methanol in water. What volumes of pure methanol and of pure water at 25°C must be mixed to form the 2000 cm3 of antifreeze at 25°C? The partial and pure molarmust be mixed to form the 2000 cm3 of antifreeze at 25 C? The partial and pure molar volumes are given.

3 -1 3 -11 1V 38.632 cm mol V 40.727Methan cm ool : m l 1 1

3 -1 3 -12 2V 17.765 cm mol V 18.06Water 8 cm m: ol

2211 VxVxV molcmV /025.24)765.17)(7.0()632.38)(3.0( 3

molVVn

t

246.83025.24

2000

moln 974.24)246.83)(3.0(1

moln 272.58)246.83)(7.0(2

3111 1017)727.40)(974.24( cmVnV t

31053)06818)(27258( cmVnV t222 1053)068.18)(272.58( cmVnV

The enthalpy of a binary liquid system of species 1 and 2 at fixed T and P is:

)2040(600400 212121 xxxxxxH

Determine expressions for and as functions of x1, numerical values for the pure-species enthalpies H1 and H2, and numerical values for the partial enthalpies at infinite dilution and

1H 2H

1H

2Hinfinite dilution and1H 2H

)2040(600400 212121 xxxxxxH

12 1x 1 x

311 20180600 xxH

21dHxHH

31

211 4060420 xxH 3

12 40600 xH

121 dx

xHH

01 x 11 x01x

molJH 4201

11x

molJH 6402

mol mol

i

iidnGdTnSdPnVnGd )()()(

iG (nS) iG (nV)

jP ,n i P,T,nT n

jT ,n i P,T,nP n

i

xP

i STG

, i

iT ,x

GV

P

dTSdPVGd iii

H U PV

nH nU P (nV)

(nH) (n U) (n V)Pn n ni i iP,T,n P,T,n P,T,nj j j

, , , , , ,j j j

H U P Vi i i

كه رابطه اي خطي بين خواص ترموديناميكي يك محلول با تركيب ثابت هر معادلهاز ك اظ ا خ ك ا ث ال ل ا ك ا ا ا ن ا ق

For every equation providing a linear relation among the thermodynamic properties of a constant-composition برقرار نمايد، داراي يك معادله المثني است كه خواص جزيي متناظر هر يك از

.هم مربوط مي نمايد اجزاء مخلوط را بهthermodynamic properties of a constant composition solution there exists a corresponding equation connecting the corresponding partial properties of each species in the p g p p p psolution.

nRTPt

p ni i yi tVyiP n

n RTipi tV p y P i 1,2,... , Ni i

V

Gibbs’s theoremA partial molar property (other than volume) of a constituent آل دهنده مخلوط گاز ايده يك جزء تشكيل) غير از حجم ( هر خاصيت جزيي مولي

آل در دماي مخلوط ولي صورت گاز ايده برابر همان خاصيت مولي جزء خالص بهم مخلوط در آن جزي فشار معادل فشاري باشددر

A partial molar property (other than volume) of a constituent species in an ideal-gas mixture is equal to the corresponding

molar property of the species as a pure ideal gas at the mixtureباشد در فشاري معادل فشار جزيي آن در مخلوط مي.molar property of the species as a pure ideal gas at the mixture temperature but at a pressure equal to its partial pressure in the

mixture.ig

iigi VM ),(),( i

igi

igi pTMPTM

ig ig(nV ) (nRT P) RT n RTigVi n n P n Pi i iT,P,n nT,P,n j jj

j jjig igV Vi i

Since the enthalpy of an ideal gas is independent of pressure:

ig ig igH (T P) H (T ) H (T P) ig igH Hig ig igi i i iH (T, P) H (T, p ) H (T, P) ig ig

i ii

H y H

i iM y M i ii

yig ig ig

i ii

H H y H 0 M M y M i i

iM M y M

Property change of mixingEnthalpy change of mixing

ig igU y U Similarly

Property change of mixing

g gi i

iU y U Similarly

ig ig igU U y U 0 i ii

U U y U 0 Internal energy change of mixing

The entropy of an ideal gas does depend on pressure

dT dPig igP

dT dPdS C RT P

igidS Rd ln P (T const.)

ig igi i i i

i i

P PS (T, P) S (T, p ) R ln R ln R ln yp y P

ig igi i iM (T, P) M (T, p )

ig igi i iS (T, P) S (T, P) R ln y

ig ig ig igi i i i

i iS y S R y ln y

ig ig igi i i i

i iS S y S R y ln y

ig ig igi i iG H TS

ig ig igi i i iG H TS RT ln y ig ig

i iH (T, P) H (T, P)

ig igS (T P) S (T P) R ln y

ig ig igi i i iG G RT ln y

i i iS (T, P) S (T, P) R ln y

RTig igdG V dP dP RT d ln P (const. T)i i P

igi i i(T) RT ln y P ig

i iG (T) RT ln P

igi i i i

i iG y (T) RT y ln y P

Chemical potential:Chemical potential:

id f d t l it i f h ilib i– provides fundamental criterion for phase equilibria

– however the Gibbs energy hence μi is defined in relation tohowever, the Gibbs energy, hence μi, is defined in relation to the internal energy and entropy - (absolute values are unknown).

igi iG (T) RT ln P i i ( )

Fugacity: a quantity that takes the place of μi

G (T) RT l fi i iG (T) RT ln f

With units of pressureWith units of pressure

i i iG (T) RT ln f ig i

i ifG G RT lnP

igi iG (T) RT ln P

i i P

ii

fP

RG RT ln

PFugacity coefficient

i iG RT ln

ig

Ri

iGlnRT

Ri 0G For ideal gases ig

i 1

P dPl (Z 1) ( t T) i i0ln (Z 1) (const. T)

P

When the compressibility factor is given by two terms virial equation

iii

B PZ 1

RT

P Biiln dPi 0ln dPi RT

P

i i0

dPln (Z 1) (const. T)P

B Piiln i RT

v vi i iG (T) RT ln f Saturated vapor:

l li i iG (T) RT ln f Saturated liquid:

vv l ii i l

i

fG G RT ln

f v l

i iG G

vili

fRT ln 0

f

v l satf f fi v l sati i if f f

v l sati i i= =

For a pure species coexisting liquid and vapor phases are in equilibrium p p g q p p qwhen they have the same temperature, pressure, fugacity and fugacity coefficient.

f

The fugacity of pure species i as a compressed liquid:

i i iG (T) RT ln f sat i

i i sati

fG G RT ln

f

sat sati i iG (T) RT ln f

i i iG (T) RT ln f

Psati i iPsat

iG G V dP (const. T) dG = V dP – S dT

Pi

isat

f 1ln V dPRTf

Since Vi is a weak function of P

satPi sati

RTf l satf V (P-P ) sat sat sat

i i if = P l satV (P P )i i isati

f V (P-P )ln =f RT

i i if P l satsat sat i i

i i iV (P-P )f = P exp

RT

For H2O at a temperature of 300°C and for pressures up to 10,000 kPa (100 bar) calculate values of fi and i from data in the steam tables and plot them vs. P.

For a state at P: iii fRTTG ln)(

For a low pressure reference state: ** ln)( iii fRTTG

)(1ln *i GGf

iii TSHG

)(1ln **iii SSHHf

)(ln * iii

GGRTf

)(ln * iii

SSTRf

kPaPfi 1**

The low pressure (say 1 kPa) at 300°C: gJHi 8.3076*

gKJSi 3450.10* gK

i if H 3076.818.015ln (S 10 345)

iln (S 10.345)1 8.314 573.15

For different values of P up to the saturated pressure at 300°C, one obtains the values of fi ,and hence . Note, values of fi and at 8592.7 kPa are obtainedi i

P = 4000 kPaT = 300 ºC

Hi = 2962 J/gS 6 3642 J/gKT = 300 C Si = 6.3642 J/gK

if 18.015 2962 3076.8ln (6.3642 10.345)1 8.314 573.15

if 3611 kPa

i 3611/ 4000 0.9028

T = 300 ºC Psat = 8592.7 kPa

Hi = 2751 J/gSi = 5.7081 J/gKVi = 1.403 cm3/g

satif 18 015 2751 3076 8 i 18.015 2751 3076.8ln (5.7081 10.345)1 8.314 573.15

satf 6738 9 kPaif 6738.9 kPa

sati 6738.9 / 8592.7 0.7843 i

Values of fi and i at higher pressure

l satsat sat i i

i i i

V (P P )f P exp

RT

l 3iV (1.403)(18.015) 25.28 cm / mol

i25.28(P 8592.7)f 6738.9exp

8314 573.15

P = 10000 kPa

i25.28(10000 8592.7)f 6738.9exp 6789.4 kPa

if 6738.9exp 6789.4 kPa8314 573.15

i 6789.4 /10000 0.6789 i

Fugacity and fugacity coefficient: species in solution

igi i i(T) RTln y P

For species i in an ideal gas mixture:

For species i in a mixture of real gases or in a solution of liquids:

i i i(T) RTln y P

i i iˆ(T) RT ln f

Fugacity of species i in solution (replacing the partial pressure)

Multiple phases at the same T and P are in equilibrium when the fugacity of each constituent species is the same in all phases:

ˆ ˆ ˆi i i

ˆ ˆ ˆf f ... f

R igM M M The residual property:

R igi i iM M M The partial residual property:

R igR igi i iG G G

ig ii i

i

f̂G G RT ln

y P

i i i iˆG (T) RT ln f

ig igi i i iG (T) RT ln y P i i i i( ) y

ii

i

f̂ˆy P

The fugacity coefficient of species i in solution

Ri i

ˆG RT ln

For ideal gas RiG 0 ig

ig if̂ˆ 1 Ri i

ˆG RT ln i

i1

y P

igi if̂ y P

Fundamental residual-property relationrelation

2nG 1 nGd d(nG) dTRT RT RT

i ii

d(nG) (nV)dP (nS)dT dn G H TS

ii

i dnRTGdT

RTnHdP

RTnV

RTnGd 2 ),,( inTPf

RTnG

G/RT as a function of its canonical variables allows evaluation of all other thermodynamic properties, and implicitly contains complete property information.

ii2

i

GnG nV nHd dP dT dnRT RT RTRT

igig ig ig GG V H

RR R Ri

i2i

GnG nV nHd dP dT dnRT RT RTRT

igig ig igi

i2i

GnG nV nHd dP dT dnRT RT RTRT

R R R

i i2i

nG nV nH ˆd dP dT ln dnRT RT RT

or

R R R

i i2i

nG nV nH ˆd dP dT ln dnRT RT RT

R R

T x

V (G / RT)RT P

T,x

R RH (G / RT)TRT T

P,xRT T

R

i(nG / RT)ˆln

n

j

ii P,T,n

n

Develop a general equation for calculation of values form compressibility-factor data.

iˆln

j

R

ii P,T,n

(nG RT)ˆlnn

P (nZ n) dPˆl

j

PR

PdPnnZ

RTnG

0)( j

i0 i P,T,n

( ) dlnn P

(nZ)i

i

(nZ) Zn

n 1

i1

n

d

P

ii PdPZ

0)1(ˆln Integration at constant

temperature and composition

Generalized correlations for the fugacity coefficient

rP dP )()1(l r

rr

rii Tconst

PdPZ

0).()1(ln

10 10 ZZZ

r rP P

TdPZdPZ 10 )()1(l r r

rr

r

r

r TconstP

dPZP

dPZ0 0

10 ).()1(ln

rP dP00 r

r

r

PdPZ

0

00 )1(ln

rP rdPZ 11ln

10 lnlnln

rP

Z0

ln

For pure gas

Table E13-E16

p g0 1( )

Estimate a value for the fugacity of 1-butene vapor at 200°C and 70 bar.

1271T 0127.1rT

731.1P

0 0.627

1 1.096

Table E15 and E16

731.1rP

0.191

1.096

0 1( )

0.638

f P (0.638)(70) 44.7 bar

Generalized correlations for the fugacity coefficient

BPP0 1rPZ 1 (B B )

cr

r c

BPPZ 1T RT

P dP

r

( )T

0 1cBP B BRT

rP

ri r0

r

dPln (Z 1) (const. T )P

cRT

0 1rPln = (B +ωB )0

1.6r

0.422B 0.083T

r

ln (B +ωB )T

1

4.2r

0.172B 0.139T

For pure gasr

Fugacity coefficient from the virial E.O.SThe virial equation

BPRTBPZ 1

The mixture second virial coefficient B

i j

ijji ByyB

For a binary mixture

2222211212211111 ByyByyByyByyB

Fugacity coefficient from the virial E.O.S

nBPZRT

nnZ

22 ,1,,11

)(1)(

nTnTP nnB

RTP

nnZZ

22 ,,, nTnTP

)()(1ˆlP nBPdnB

22 ,10

,11

)()(1lnnTnT n

nBRTPdP

nnB

RT

1 1 11 1 2 12 2 1 21 2 2 22B y y B y y B y y B y y B

1 2 11 1 2 12 2 1 21 2 1 22B y (1 y )B y y B y y B y (1 y )B

1 11 2 22 1 2 12 11 22B y B y B y y (2B B B )

12 12 11 222B B B

1 11 2 22 1 2 12 11 22B y B y B y y (2B B B )

nny /

1 11 2 22 1 2 12B y B y B y y 1 2 1 2

11 22 12n n n nB B Bn n n n

nny ii /

1 21 11 2 22 12

n nnB n B n B

2 1 2n n n n(nB) B

1 11 2 22 12nB n B n Bn

2

2 1 211 122

1 T,n

( ) Bn n

2

2 1 211 122 2

1 T,n

n n n n(nB) B ( )n n n

2,

2 1 211 12

1

n n n(nB) B ( )n n n n

21 T,n

n n n n

11 2 1 2 12(nB) B (y y y )n

11 2 1 12(nB) B y (1 y )n

21 T,n

n 21 T,nn

2(nB) B

P (nB)ˆ 2

1 11 2 12Pˆln = B +y δ

RT2

211 2 12

1 T,n

( ) B yn

2

11 T,n

P (nB)ˆlnRT n

RT

PˆSimilarly: 22 22 1 12

Pˆln B yRT

For multi-component systems

k kk i j ik ijP 1ˆln B y y (2 )

RT 2

i jRT 2

Where: 2B B B Where: ik ik ii kk2B B B

cij 0 1ij ij

RTB = (B +ω B ) i jω +ω

ω =ij ijcij

B (B ω B )P ijω

2

cij ci cj ijT = (T T )(1-k )

E i i l i t ti t

cij cijZ RTP ci cjZ +Z

Z

Empirical interaction parameter

cij cijcij

cij

P =V

c cjcijZ =

2

31/3 1/3ci cj

cij

V +VV =

2 cij 2

Determine the fugacity coefficients for nitrogen and methane in N2(1)/CH4(2) mixture at 200K and 30 bar if the mixture contains 40 mol-% N2.

molcmBBB

3

22111212 6.200.1052.35)8.59(22

30P 0501.0)6.20()6.0(2.35)200)(14.83(

30ˆln 212

22111 yB

RTP

ˆ

18350)620()40(010530ˆl 22 BP

9511.01

1835.0)6.20()4.0(0.105)200)(14.83(

ln 212

21222 yB

RT

83240̂ 8324.02

Estimate and for an equimolar mixture of methyl ethyl ketone (1) / toluene (2) at 50°C and 25 kPa. Set all kij = 0.

1̂ 2̂

2ji

ij

cij

cijcijcij V

RTZP

2cjci

cij

ZZZ

33/13/1

2

cjci

cij

VVV

)1()( ijcjcicij kTTT

)( 10 BBP

RTB ij

cij

cijij 22111212 2 BBB

0128.0ˆln 1222111 yB

RTP

987.01̂

0172.0ˆln 1221222 yB

RTP

983.02̂

The ideal solution

• Serves as a standard to be compared:

idii

id MxM

iiid

i xRTGG ln

cf igig yRTGG ln

i

iii

iiid xxRTGxG ln

i

iiMxM

cf. iii yRTGG ln

iP

i

xP

idiid

i xRTG

TGS ln

ii

idi xRSS ln i

iii

ii

id xxRSxS ln PxP ,

iid

iidi P

GP

GV

iid

i VV ii

iid VxV

TxT PP ,

iiiiid

iid

iidi xRTTSxRTGSTGH lnln i

idi HH

i

ii

iid HxH

The Lewis/Randall Rule

iii fRTT ˆln)(

• For a special case of species i in an ideal solution:id

iiid

iidi fRTTG ˆln)(

iiid

i xRTGG ln

iii fRTTG ln)( iii fRTTG ln)(

iiid

i fxf ˆThe Lewis/Randall rule

iidi ˆ

The fugacity coefficient of species i in an ideal solution is equal to the fugacity coefficient of pure species i in the same physical state as the solution and at the same T and P.

Excess properties

• The mathematical formalism of excess properties is analogous to that of the residual properties:

– where M represents the molar (or unit-mass) value of i h d i (

idE MMM

any extensive thermodynamic property (e.g., V, U, H, S, G, etc.)

– Similarly we have:Similarly, we have:

ii

Ei

EEE

dnRTGdT

RTnHdP

RTnV

RTnGd 2

The fundamental excess-property relation

The excess Gibbs energy and theThe excess Gibbs energy and the activity coefficient

• The excess Gibbs energy is of particular interest:idE GGG

iii fRTTG ˆln)(

iiiid

i fxRTTG ln)(

ii

iEi fx

fRTGˆ

ln

ii

ii fx

f̂

The activity coefficient of species i in solution.A factor introduced into Raoult’s law to account for liquid-phase non-idealities.

iE

i RTG lnq p

For ideal solution,

c.f. iR

i RTG ̂ln

1,0 iE

iG

ii

Ei

EEE

dnRTGdT

RTnHdP

RTnV

RTnGd 2

iii

EEE

dndTRTnHdP

RTnV

RTnGd ln2

EE

PRTG

RTV )/(

xTPRT ,

xP

EE

TRTGT

RTH )/(

Experimental accessible values:activity coefficients from VLE data,VE and HE values come from mixing experimentsxP ,

jnTPi

E

i nRTnG

,,

)/(ln

V and H values come from mixing experiments.

i

ii

E

xRTG ln

Important application in phase-equilibrium thermod namics),.(0ln PTconstdx

iii thermodynamics.