This article was downloaded by: [Stony Brook University]On: 31 October 2014, At: 22:22Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registeredoffice: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

Journal of Nuclear Science andTechnologyPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/tnst20

Isotope Effect in Molecular Rotation ofHydrogen Chloride in Condensed PhaseYumio YATO aa Power Reactor and Nuclear Fuel Development Corp. , Tokai-mura,Ibaraki-ken , 319-11Published online: 15 Mar 2012.

To cite this article: Yumio YATO (1992) Isotope Effect in Molecular Rotation of HydrogenChloride in Condensed Phase, Journal of Nuclear Science and Technology, 29:8, 768-778, DOI:10.1080/18811248.1992.9731593

To link to this article: http://dx.doi.org/10.1080/18811248.1992.9731593

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the“Content”) contained in the publications on our platform. However, Taylor & Francis,our agents, and our licensors make no representations or warranties whatsoever as tothe accuracy, completeness, or suitability for any purpose of the Content. Any opinionsand views expressed in this publication are the opinions and views of the authors,and are not the views of or endorsed by Taylor & Francis. The accuracy of the Contentshould not be relied upon and should be independently verified with primary sourcesof information. Taylor and Francis shall not be liable for any losses, actions, claims,proceedings, demands, costs, expenses, damages, and other liabilities whatsoever orhowsoever caused arising directly or indirectly in connection with, in relation to or arisingout of the use of the Content.

This article may be used for research, teaching, and private study purposes. Anysubstantial or systematic reproduction, redistribution, reselling, loan, sub-licensing,systematic supply, or distribution in any form to anyone is expressly forbidden. Terms &Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

Journal of NucLEAR SciENCE and TECHNOLOGY. 29i81. pp. 768-778 (August 1992).

Isotope Effect in Molecular Rotation of Hydrogen Chloride in Condensed Phase

Yumio YATO

Power Reactor and Nuclear Fuel Development Corp.*

Received February 26, 1992

Infrared spectra of HCI and DC! are observed in solid and liquid states to investigate the change in rotational motion of these molecules due to the displacement of the center of mass on isotopic substitution. The moment analysis of measured spectra indicates strongly hindered rotational motion and the mean square torque measured about the center of mass of a DC! molecule is found to be smaller by about 10% than that of a HCI molecule both in ,a-solid and liquid phases. The difference in mean square torques on isotopic substitution is interpreted as resulting from translation-rotation interaction based on the analysis combined with isotopic vapor pressure data. Discussion is also made on the effective root-mean-square oscillation am­plitude associated with molecular rotation from the aspect of corresponding states.

KEYWORDS: hydrogen chloride, deuterium chloride, infrared spectra, molecular rotation, moment analysis, intermolecular forces, torque, rotation-translation interac­tion, phase equilibrium, vapor pressure, isotope effects, rotational oscillation, cor­responding states

I. INTRODUCTION

The study of isotope effect in phase equili­bria has been an important source of informa­tion about intermolecular forces and molecular dynamics in condensed phase0 '-'

5'. For mo­

natomic substances, the isotopic vapor pres­sure difference is mainly due to the difference in the zero point energy of the motion of a molecule as a whole in the field of its neigh­bors, and then the difference is well repre­sented in terms of the mean square force on an atom 03'. This means that the lighter isotope of monatomic substances always has the higher vapor pressure. By the effort of Bigeleisen and his coworkers, numerous cases have been known for diatomic and polyatomic substances, where the heavier isotopic mole­cule has the higher vapor pressure than the lighter one''''')(7

)(12 H1''- 0 7l or the isotopic

molecules with the same mass have different vapor pressures<s)(<)(G)(7)( 11 )(

12 '. In these cases not only the effect arising from the transla­tional motion but the effects arising from

rotation, vibration and their interactions must be taken into account0 '-'"''

18)( 19'. Additional effects are due to the anharmonicity in ex­ternal and internal modes, which has been found important by more recent precise mea­surements<SJ-o2J. One of the successful con­clusions led by some of these studies is that the principle of corresponding states applies quite well to the translational oscillation am­plitude<8H2"''21', while there have been few studies which throw light on the molecular rotation in condensed phase from this aspect primarily because of the difficulties in its theoretical approach. The moment analysis of a rotation-vibration band, provided by Gordon'22 '-'24

', allows us to extract the in­termolecular torque which has an important role in isotope effects in phase equilibria. The present work is made for the difference in intermolecular torques measured about the center of mass of two isotopic hydrogen chlo­ride molecules HCI and DC! and for the com­parison of effective root-mean-square oscilla­* Tokai-mura, Ibaraki-ken 319-11.

-48-

Dow

nloa

ded

by [

Ston

y B

rook

Uni

vers

ity]

at 2

2:22

31

Oct

ober

201

4

Vol. 29, No. 8 (Aug. 1992)

tion amplitudes associated with molecular rota­tion of several diatomic and linear triatomic molecules from the aspect of corresponding states.

ll. EXPERIMENTAL

The HCl and DC! gases were prepared by dropping sulfric acid (Wako Pure Chemical Ind.,

Ltd.) and deuterated sulfric acid (Showa Denko.

K.K.), respectively, on hydrochloric acid (Wako

Pure Chemical Ind., Ltd.) and deuterochloric acid (Showa Denko K.K.), respectively. Chemical purification of each of the samples was ac­complished by the sample bulb to bulb sub­limation followed by passage through a helical trap immersed in a dry ice bath to eliminate trace of water. The condensed samples were opened to a vacuum pump at liquid nitrogen temperature for a short time to remove any volatile impurities following repetitive freeze­thaw cycles.

The spectra of liquid samples were meas­ured with a Perkin-Elmer I-125 spectrometer, using a pressure cell made of brass which was a slight modification to that described by West<25

l as shown in Fig. 1. The length of the absorbing layer could be altered by addi­tion or removal of spacer D. Quartz windows,

l....J..__l.__l

0 2 4 em

A: Window, B: Sliding piece, C: Screw, ]): Spacer, E. Valve

Fig. 1 Pressure cell used for measurements of spectra of liquid HCl and DC!

769

which were convex in a cross-sectional shape, 14 mm thick, 20 mm optically effective aper­ture and had little absorption in 2.57 pm due to water, were held in the sliding piece B made of phosphor bronze. The cell was kept tight by forcing the pieces B against the spacers by the screws C, even when the spacers became hard at low temperature. On filling the samples in the cell, the lower reservoir together with the joint were uns­crewed from the absorption chamber, and then attached to an outlet of the manifold. The lower reservoir was evacuated and cooled down in a liquid nitrogen bath. The purified sample in a reservoir of the manifold was transported and then condensed in the lower reservoir. After collecting the necessary quantity, the lower reservoir was screwed home after closing the valve E 2 • On observ­ing the liquid spectra, the cell was turned the lower reservoir up so that the liquid sample ran into the absorption chamber. The cell was cooled down in an ice box by con­tinuously introducing dry cold N2 gas which was evaporated from liquid nitrogen by blowing nitrogen gas directly into it. The temperature was monitored by a copper-con­stantan thermocouple soldered to the absorp­tion chamber. A heating wire wound around the chamber operated by a cryostat controller (Model CCR-302, Dan Electric Co. Ltd.) allowed to control within ±2 K over a relatively wide range of temperature desired above the boil­ing point of liquid nitrogen by carefully ad­justing the flow rate of the dry cold N2

gas into the ice box. Adequately dry liquid samples were found to produce little corrosion of the cell material under the conditions used here.

The solid spectra were measured with a Shimazu grating spectrometer IR-27G. The absorption cell used here was a conventional low temperature cel!<2'l. To monitor and control temperature, we used an insulated heating wire wound around the cooling block, operated by the cryostat controller, and a copper-constantan thermocouple clamped firmly between the copper cooling block and the front surface of a NaCl window. Temperature flue-

-49-

Dow

nloa

ded

by [

Ston

y B

rook

Uni

vers

ity]

at 2

2:22

31

Oct

ober

201

4

770

tuation was within ± 1 K under the conditions used here by carefully maintaining liquid nitrogen in equilibrium in the coolant reser­voir of the absorption cell. The vacuum line manifold was filled to a pressure of 2 to 3 Torr with the gas sample, which was bled from a glass nozzle through a fine needle valve of the manifold to produce a film of the sample on the cold NaCI window. The spectra was recorded several times to confirm the change of integrated intensities in the initial and the final bands to be small. The zero absorption lines were determined with the empty cells both for liquid and solid.

m. RESULTS AND DISCUSSION

1. General Features of Measured Spectra The spectra of HCI and DC! in the gas

and in the liquid observed at different tern-

0. 6 (a) HCI

"" 0. 4 :::: .:: '-

"' .Q 0. 2

]. Nucl. Sci. Techno!.,

peratures and pressures are shown in Fig. 2. The general features of the spectra of liquid HCI are quite similar to those measured by West. As seen from Fig. 2(a), there still exist the two maxima corresponding to the P and R branches separated by the zero gap in gaseous HCI even at high pressures. In liquid HCI at 180 K, the band consists of a single rather sharp maximum, while the vestige of the doublet of a gaseous diatomic molecule remains slightly in the band at room temperature. The liquid band is largely dis­placed to lower frequency from the gas origin. The spectra of DC! also observed in the first overtone region are shown in Fig. 2(b). Even at room temperature only a single maximum is observed in liquid DC!, probably because the separation between individual rotational lines is much smaller in DC! than in HCI.

0. 6 (b) DC!

""0.4 l :::: .:: '-

~ ~ 0.2

5200 6000 3600 4400 Wavenumber (cm· 1) Wavenumber (em·')

A: liquid at 180 K, B: liquid at 298 K, C: gas at -40 atm, D: gas at -20 atm

Fig. 2 First overtone of gaseous and liquid HCI(a) and DCI(b) at various temperatures and pressures

The spectra observed at 80 K, below the phase transition temperature, which are not shown in the present paper, resembled the doublet predicted by a classical theory for the rotation-vibration spectrum of a diatomic molecule in the gas phase, in good agreement with the previous studies'29Jcao). This indicates

that both HCI and DC! molecules rotate almost freely in the a-solid state. The band profiles of ,8-solid HCI and DC! are shown in Fig. 3. As seen from Fig. 3(a), the spectra of ,8-solid HCI observed at 130 K is very similar to that obtained by Hettner<2'l in the band origin and the shape of a rather broad single band, while

-50-

Dow

nloa

ded

by [

Ston

y B

rook

Uni

vers

ity]

at 2

2:22

31

Oct

ober

201

4

Vol. 29, No. 8 (Aug. 1992)

0. 6,--,--,---,,----,,----,-----,---.---,----,

""' 0. 4 ::::: ~ '-

~ '0.2

(a) HCI

2600 2800 3000 Wavenumber (cm'1)

0. &.---....--....--,r---r----,---.--.---,

""0 .4 ::::: ~ ~ '0.2

(b) DCI

1800 2200 Wavenumber (em·')

(a) jl·solid HCI at 1~10 K, (b) jl·solid DC! at !30 K

Fig. 3 Infrared spectra of f3-solid hydrogen chloride

the band wings, which are most important to extract the intermolecular torque from the moment analysis, are lost in his measurement. The observed band origins and band widths are summarized in Table 1. According to the analysis by Gordon<""'-<"•', the band width has little dependence on intermolecular force and hence the difference in band widths is con­sidered to be due to the difference in the

771

moment of inertia between HCI and DC! mole­cules, in good agreement with the present measurement.

2. Intermolecular Torque Acting on Isotopic Hydrogen Chloride Molecules

Profiles of the infrared band in condensed phase contain detailed information of inter­molecular forces. Although the interpretation of band profiles in terms of molecular dy­namics has been difficult, the moment analysis provided by Gordon<22l-<2

•' makes it possible to extract the effect of intermolecular forces. In this method the n-th order moment M(n) is defined as

M(n)= )C1/1J)(1J-1J,)n log(!./ l)diJ

;)(1/IJ) log(f0/ J)diJ, ( 1)

IJ,= )log(!./ l)dlJ/)Cl/IJ) log(!./ I )diJ-Mo,,

( 2)

M0,=2Bv+(kT I hc)IJ-Z(Dv! Bv)(kT /he),

( 3)

where IJ, is the band ongm in condensed phase, Io! I the band intensity, L1=(Bv-B 0)/ B0 ,

B refers to the rotational constant, D to the rotational constant representing centrifugal effects, v to the v-th excited state, k is the Boltzmann constant, h the Planck's constant, c the velocity of light, and T the absolute

Table 1 Infrared spectra of HCI and DC! in condensed phase

Substance

HCI (gas) HCl (gas) HCI (liquid) HCl (liquid) HCI (j3-solid)

DC! (gas) DC! (gas) DC! (liquid) DC! (liquid) DC! (j3-solid)

T (K)

298 298 298 180 130

298 298 298 180 130

p (atm)

~20

~40

~20

~40

! 1 Values taken from Ref. (37)

Band

2<--0 2<--0 2<--0 2<--0 1<--0

2<--0 2<--0 2<--0 2<--0 1+-0

,_,, Halfwidth (cm- 1) (em-')

5,651 5, 668. 05! 1

5,660 243 5,604 215 5,542 193 2,798 185 2, 885.821 1

4,053 4, 063.2712

4,058 171 4,020 154 3,973 137 2,003 130 2, 068.7612

12 Calculated values from Ref. (37) based on the anharmonic oscillator model.

-51-

Dow

nloa

ded

by [

Ston

y B

rook

Uni

vers

ity]

at 2

2:22

31

Oct

ober

201

4

772

temperature. The values of M(2) and M(4) calculated from the experimental band profiles

]. Nucl. Sci. Techno! ..

are given in Table 2.

Table 2 Moment analysis of infrared bands of HCl and DC! in condensed phase

T M(Z)~1ynam

(L11J )' h' M(2)1~eor M(2)obs M(4)obs Substance (K) N 12(k T) 2 M ( 4)

(103 cm-2) (103 em-') (103 cm-2) (103 cm-2) (103 cm-2) (107 cm-4)

HCI (liquid) 298 7.49 0.34 0.01 7.84 8.17 21.18

HCl (liquid) 180 4.45 1. 31 0.02 5.78 5.70 17.70

HCl (,8-solid) 130 3.46 0.64 0.03 4.13 3.92 14.23

HCl (liquid) 298 3.99 0.16 0.00 4.15 4.27 6.50 HCI (liquid) 180 2.34 0.68 0.01 3.03 3.05 4.57 HCI (,8-solid) 130 1. 79 0.36 0.01 2.16 2.05 3.48

" M(2)dynam is the sum of the first three terms of Eq. (2) in the text. t2 M(2)theor=M(2)dynam+Cdv) 2

/ N+ {li'!12(kT)'}M(4).

Theoretical expression of M(2) is given by<22)-(24)

M(2)=4(kT I hc)Bv+8(kT I hc)BvL1

+2(kT I hc)'Ll'+(Llv)' IN

+(1112)(/il kT)~ M(4), ( 4)

where li=hl2rr, Llv=vg-l.lc, the subscripts g and c refer to the gas and condensed phases, respectively, and N is assumed to be 12, the number for a close packed sphere. The reasonable agreement between experimental and theoretical second moments shown in Table 2 indicates that no appreciable portion

of the band intensity is lost in the wings of the spectra.

The fourth moment is related to the mean square torque by<22l-<••J

M(4)=32B~(kT I hc)'(l+6L1)

+48Bv(kT I hc)3 L1 2(3+4L1)

+4m(1 +4L1)(( OU)'>, ( 5 )

where OU is the angular gradient of inter­molecular potential energy or the torque acting on a molecule and the bracket refers to a statistical average. The values of mean square torque calculated from the experi-

Table 3 Mean square torque and effective root-mean-square rotational oscillation amplitude of HCl and DC! molecules in condensed phase

Substance T <COU)') <(0UfkT) 2 ) <O')''' J<(OU) 2)t 1

(K) (105 cm-•rad-2) (rad-2) (de g) lOO <COU) 2)

HCI (liquid) 298 2.56 5.96 23.4 DCI (liquid) 298 2.44 5.69 24.0 4.7

HCl (liquid) 180 4.09 26.1 11.2 DC! (liquid) 180 3.73 23.8 11.7 8.8

HCl (,8-solid) 130 3.09 37.9 9.3 DC! (,8-solid) 130 2.80 34.3 9.8 9.5

HCl (,8-solid) 155 (31. 8)12 DC! (,8-solid) 155 (28. 8)t2 (9. 4)12

"J((OU)'>=<CO'U )2)-((0U ) 2 ).

" Derived from the observed values at 130 K.

-52-

Dow

nloa

ded

by [

Ston

y B

rook

Uni

vers

ity]

at 2

2:22

31

Oct

ober

201

4

Vol. 29, No. 8 (Aug. 1992)

mental fourth moments are given in Table 3, in which are also included the values of an effective root-mean-square torsional oscillation amplitude calculated from the relationship

(8 2)

1t 2 =(1801n)l { <( OU I k T)'>} 112

• ( 6)

Experimentally determined (( OU I kT)2) and

(8 2)

112 indicate that the rotational motion of a hydrogen chloride molecule about the center of mass is strongly hindered in the liquid and fi-solid phases and that the rotational freedom in condensed phase becomes smaller at lower temperature. We also note that the mean square torque acting on a DC! molecule is by some 10% smaller than that on a HCI mole­cule in condensed phases. Since there is a significant displacement of the center of mass on isotopic substitution of this molecule, this difference is considered to be due to the rotation-trans Ia tion interaction.

3. Isotope Effect in Intermolecular Torque

When the center of force is defined as the point on the molecular axis through which the force passes, the torque measured about this point is zero. Therefore the torque measured about the center of mass is given by

- OU = -(z-ztV'U X z),

where -f7U is the gradient of the intermole­cular potential energy or the force arising from neighboring molecules in condensed phase, z; the center of force, z the center of mass and z a unit vector along the molecular axis. If the center of force depends only on the electronic charge in space, it is isotope independent within the framework of the Born-Oppenheimer approximation. The torque measured about the center of mass of the asymmetrically substituted molecule includes a part of the force, and thus the mean square torque is given by a quadratic function of the displacement of the center of mass (z' -z)

<( O'U I kT?>=<( OU I kT)2)

+2g1(z' -z)+g2(z' -z)', ( 7)

where g1=<(0UikT)(f7UxzlkT)),

773

g2=<(f7V Xzl kT)2)

=<sin28)((f7U I kT)2),

and the prime refers to the substituted mole­cule. Because of a very small contribution of the quadratic term in Eq. ( 7 ), we shall take the value of ~=<sin28)=0.8, which is obtained based on a small vibration model with central forcesc 21 J. We shall also use the isotopic vapor pressure data for further dis­cussions and for deriving the mean square force <WV I kT)2

) as well as the rotation­translation interaction term g 1=((0U I kT)(f7U XzlkT)).

According to Bigeleisen's general statistical mechanical theory of the isotope effect in phase equilibria Cll, the vapor pressure ratio for a pair of isotopic molecules is given by

in(P' I P)=ln{(QciQ~)I(QciQ~)cd

-in{(QuiQ;)I(QuiQ;)ct}, ( 8)

where Q u is the melecular partition function for the vapor phase, Qc the average molecular partition function for the condensed phase, and the subscript cl refers to the classical partition function. Since the internal vibra­tion is almost all in its ground state in the present case, it contributes essentially a zero point energy shift on condensation. Accord­ing to the conventional manner0 l, the vibra­tional contribution to the isotopic vapor pres­sure ratio is given by

(hcl2vkT){(11~-11c)-(11;-11g)}. ( 9)

The rotational quantum correction to the parti­tion function was calculated by Kirkwoodc3ll,

QrotiQrot.cl=1 +h2 16kT fe

-{h2124kTI.}<(OUikT)2), (10)

where Ie is the moment of inertia. The translational term was derived by Wignerc32

J,

Qtrans/Qtrans.cl

=1-{h2124kTM}((f7U I kT)2). (11)

By collecting these expressions, we obtain the final expression for the isotopic vapor pres­sure ratio

-53-

Dow

nloa

ded

by [

Ston

y B

rook

Uni

vers

ity]

at 2

2:22

31

Oct

ober

201

4

774

ln(P' I P)=(hcl2vkT){(v~-vc)-(v~-v 0 )}v

+(hcl12kT){B'<( O'U I kT)">

-B<(OU lkT)')}c

+W 124kT)(11 M'-11 M)

(12)

where we neglect the forces and torques between molecules in vapor phase because they are very small in comparison with those for the condensed phase. Since the first and second terms in Eq. (12) are given from ex­perimentally determined band origins and mean square torques for isotopic hydrogen chlorides, the mean square force <(li'U I kT)"> can be determined if the isotopic vapor pres­sure data are available. The vapor pressure of HCI was measured by Giauque & Wiebe< 14l,

while the measurement of the vapor pressure of DC! was made by Lewis, McDonald & Schutz<15

). By combining the spectroscopic data with the vapor pressure data, we obtain the mean square force acting on a hydrogen

]. Nucl. Sci. Techno! ..

chloride molecule in the liquid phase

<(li'U lkT)')=l.lxlD'A- 2 at 180 K.

For comparison the mean square force on liquid Ne, Ar and Kr molecules are derived from the isotopic vapor pressure data of 22Nei2"Ne by Bigeleisen & Roth<2

l, ••Ari'"Ar by Lee et at.<•) as well as Boato et a/_<2°), and 84Kr I""Kr by Lee et at. <9

J by using the relationship

ln(P'IP)

=(h2 124kT)(11 M'-11 M)<(V'U I kT)2)c. (13)

The results are given in Table 4, where we see that the mean square force and hence the effective root-mean-square translational oscil­lation amplitude of a HCI molecule is quite similar to those of rare gas molecules if com­pared at the same reduced temperature relative to the critical temperature Tcr. This in­dicates that the principle of corresponding states still applies well as it does for other equilibrium properties.

Table 4 Mean square force and effective root-mean-square translational oscillation amplitude of HCl and some monatomic molecules in condensed phase

Liquid at T* =0. 555t1 Solid at T*==0.478F

Substance Tcr ----

(K) ((flU!kT) 2) (i2).lf2 t2 ((PU/kT)2) (l">:'2t2

cwA. -•) (A) cwA. -•) (A) -------- -----~--···--~------

HCl 325 1.1±0.5 0.095 1.4±0.4 0.085 Ne 44.5 1. 3±0.1t3a 0.090 1.7±0.1 0.077

Ar 151 { 1. 1 ±0. 2t3b 0.095 1. 34 0.086 0. 95t3C 0.103 Kr 210 1. 07t3d 0.097 1. 07 0.085

t• T*=T/Tcr. t' The value (l2) 112 is the effective root-mean-square translational amplitude. t• Derived from the isotopic vapor pressure data by Reb. a: (2), b: (20), c: (8)

and d: (9), respectively.

Although the mean square torques are not measured for ,8-solid hydrogen chlorides in the temperature range in which the vapor pressure data are available, the following estimation will be applicable. Since the average angular Laplacian of intermolecular potential energy < 0 2U) is considered to be temperature independent, the mean square torque at temperature T can be estimated

from the known value ((OU I kT)">. at T=T 0

from the relationship<'!)

Table 3 also includes the mean square torques at 155 K estimated from Eq. (14) using the experimental value at 130 K. The mean square force on a ,8-solid HCl molecule is found to be

-54-

Dow

nloa

ded

by [

Ston

y B

rook

Uni

vers

ity]

at 2

2:22

31

Oct

ober

201

4

Vol. 29, No. 8 (Aug. 1992)

at 155 K.

This larger value in the solid reflects the greater translational freedom in the liquid. For comparison the mean square force on solid rare gas molecules are also given at the same reduced temperature in Table 4.

Based on the experimental mean square torques and the derived mean square force, we now can determine the torque function expressed by Eq. ( 5 ). The torque functions are

<(OU I kT)2)=26-70(z-zo)+88(z-zo)2

for liquid HCI at 180 K,

<(OU I kT)')=32-90(z-zo)+ llO(z-zo)'

for .a-solid HCI at 155 K,

where z is the position about which the torque is measured, Zo the position of the center of mass of the normal HCI molecule, and the distances are measured from the chlorine nucleus. As expected that the center of force lies closer to the larger atom, the above functions both have their minimum value at a distance of about 0.4 A from the center of mass of the normal HCI toward the hydrogen atom. In the loaded sphere model by Friedmann & Kimmel<30l, the intermole­cular force is assumed spherical about the center of force, in other words the mean square torque curve passes through zero at the minimum. As shown in Fig. 4, the mean square torque functions obtained here are always positive about any point on the mole­cular axis. This indicates that the orienta­tiona! effect is not negligible in the intermole­cular interactions of hydrogen chloride mole­cules in condensed phase. Since the deter­mination of the torque functions is based on the experimentally determined slight change in mean square torques on isotopic substitu­tion, the value of (z;-zo)~0.4 A may probably include some error. However the error thus introduced may not alter the general features derived here that the center of force zi lies closer to the chlorine atom and the orienta­tiona! effect is not negligible. It should be also noted that the error introduced here is

775

to d

chlorine 0.5 1.0 hydrogen

Z(JI)

a: Center of mass of HCl, b: Center of Mass of DC!, c: Center of mass of TCl, d: Center of force, e: Midpoint of the molecular axis of HCl

Fig. 4 Mean square torque acting on a molecule in liquid and .S-solid hydrogen chloride, as a function of position about which torque is measured

not so serious to the discussion of the vapor pressure isotope effect because the contribu­tion of rotational part is mainly due to the large difference in the moment of inertia in the case of hydrogen isotope effect and on the contrary the translational contribution is most important in the case of chlorine isotope effect as discussed in the next section.

4. Vapor Pressure Isotope Effect of Hydrogen Chloride

The mean square torque functions and the mean square force derived above allow us to predict the vapor pressures for other isotopic hydrogen chlorides such as TCI and H37Cl. Substitution of Eq. ( 5) to Eq. (12) gives

ln(P' I P)=(hcl2vkT){ (v;-vg)(v~-v~)lv~~ v

+(hcl12kT)(B'- B)<(O'U I kT)'>

-55-

Dow

nloa

ded

by [

Ston

y B

rook

Uni

vers

ity]

at 2

2:22

31

Oct

ober

201

4

776

-(hc/6kT)Bg1(z-zo)

+{(h2/24kT)(1/M' -1/M)

-(~hcj12kT)B(z-z0)2 }<(VU /kT)2).

(15)

In arriving at Eq. (15), we assume v~/vg=

vUv. 0 l, and here the prime refers to the lighter isotope. The contribution of each term to the vapor pressure ratio of isotopic species to H35Cl is given in Tables 5 and 6. As seen in these Tables, the translation-rota­tion interation gives a small but not negligible contribution to the vapor pressure isotope effect of hydrogen chloride. The mean square force (translational effect) gives a dominant contribution to the chlorine isotope effect, while the mean square torque (rotational effect)

and the internal vibrational frequency shift on condensation are important in the hydrogen isotope effect. We should also mention the interesting facet of vapor pressure isotope effect of hydrogen chloride that the isotopic molecules with the same mass have different vapor pressures and the heavier isotopic mole-

]. Nucl. Sci. Technlo.,

cule has higher vapor pressure than the lighter one: P(H'7Cl)> P(D35Cl)> P(T35Cl).

Gverdtsiteli et al. <tsJ measured P(H35Cl)/

P(H37Cl) at the boiling point (188 K) by dis­tillation technique. For comparison an ex­trapolation is made from the estimated vapor pressure ratio at 180 K given in Table 5 to the value at 188 K by using the Bigeleisen's two-constant equation°l

ln(P' I P)=a/P-b/T. (16)

The second term in Eq. (16) is equivalent to the first term (frequency shift term) in Eq. (15). The assumption that the frequency shift on condensation is small within such a narrow temperature range as considered here yields the vapor pressure equation

In{ P(H"Cl)/ P(H37Cl)} =39/T2 -8.1 X 10-2 /T,

(17)

which gives the value of 7 x 10- 4 at 188 K in fairly good agreement with the experimental value of 4"'6 X 10- 4

•

Table 5 Contribution of internal vibration, rotation and translation to measured and predicted vapor pressure ratios of isotopic hydrogen chlorides at 180 Kt'

ln(P0/ P)t2 Internal Mean square torque Molecule vibration

Obs. Cal. shift Constant (z-z 0 ) (z-z 0 ) 2

H37Cl 0. 05-0. 07!3 0.07 -0.02 0.03 -0.09 -o D35Cl 2.5 2.5 -7.1 8.7 0.8 -o T 35Cl 2.6 -10.2 11.6 1.1 -0.1

11 Vapor pressures and the contributions of various terms are multiplied by 102 •

t' Po=P(H35Cl). 13 Extrapolated value based on Eq. (17) in the text.

Mean square force

0.15 0.1 0.2

Table 6 Contribution of internal vibration, rotation and translation to measured and predicted vapor pressure ratios of isotopic hydrogen chlorides at 155 Kt

ln(P0/P)t Internal Mean square torque Mean Molecule vibration square

Obs. Cal. shift Constant (z-z 0 ) (z-zo) 2 force

H37Cl 0.10 -0.03 0.04 -0.13 -o 0.22 D35Cl 3.5 3.5 -11.5 14.0 1.0 -0.1 0.1 T 36Cl 4.7 -16.5 19.5 1.6 -0.1 0.2

t For P0 , see the footnote of Table 5.

-56-

Dow

nloa

ded

by [

Ston

y B

rook

Uni

vers

ity]

at 2

2:22

31

Oct

ober

201

4

Vol. 29, No. 8 (Aug. 1992)

5. Effective Root-Mean-Square Oscillation Amplitude Associated with Rotational Motion of Some Linear Molecules

Theoretical and experimental investiga­tions on the vapor pressure isotope effect of monatomic solids and liquids have led to a remarkably successful conclusion that the principle of corresponding states applies quite well to the average Laplacian of the inter­molecular potential energycs)czo)(zl). It has been also mentioned that the mean square force acting on such diatomic and linear triatomic molecules as HCl, by the present work, and N,O, by Yato et at. (II), are much the same as those on rare gas molecules when compared at the same reduced tem­perature. However, there have been little studies about the rotational motion of a con­densed molecule from this aspect primarily because of the difficulties in its theoretical approach. Three ways are applicable for the experimental determination of the intermole­cular torque in condensed phase. The first one is the Gordon's moment analysis of a rotation-vibration band as employed in the present work. Secondary, the observed libra-

777

tiona! frequencies can be used to estimate the mean square librational amplitude and torque C:"'JC 3") As mentioned in Sec. 111-4, the measurement of the isotopic vapor pressure ratios of two or three isotopic pairs also suffices to derive the mean square torque and force acting on diatomic and linear triatomic molecules.

Shown in Table 7 are the values of inter­molecular torques on several linear molecules, which are obtained either from the isotopic vapor pressure ratios or from spectroscopic data. We note that the root-mean-square linear displacements of the end atom per­pendicular to the molecular axis a<fJ2) 112 are much the same among these diatomic and linear triatomic molecules if compared at the same reduced temperature, and can be ex­pressed as follows:

a{<(;f2)/T*} 112 ~1/3 A, (18)

where a refers to the length of the molecular axis. In Table 8 we also see a similar trend for solid molecules. This relationship, when combined with the one for translational os­cillation amplitudes as described in Sec. 111-3,

Table 7 Intermolecular torque and root-mean-square oscillation amplitude associated with molecular rotation for diatomic and linear triatomic molecules in liquid phase

Molecule T T* Methodtl <JOU/kT';'> (02)1/2 11 a{(Oz)/T*)'Iz (K) (rad-2) (deg) cA) cA)

-~---

HCI 180 0.56 A 26.112 11 1. 28 0.33 co 80 0.60 A 19.41 115 13 1. 13 0.34 co 78 0.58 A 16. 8t't6 14 1.13 0.35 co 78 0.58 A 16.0tH7 14 1.13 0.36 co 74 0.56 B 20. 2t't" 13 1.13 0.35 Nz 70 0.55 B 13. 6t1t9 16 1.10 0.41 NzO 184 0.60 c 78±2213!0 6.7±0.9 2.31 0.33±0.03 NzO 182 0.69 B 81. Ot3t'l 6.4 2.31 0.33 co, 217 0.71 B 67. 3t't 10 7.0 2.32 0.34

t 1 Mean square torque is detennined from A: moment analysis, B: isotoic vapor pressure ratios and C: liquid-vapor isotopic fractionation factor.

1' This work.

t:1 Derived frotn isotopic vapor pressure ratios in this work.

t• Determined by Gordon in Ref. (24).

!'• As shown in Ref. (24), mean square torque is determined by the moment analysis of !' (1<--0),

t' (2<-0) and t' (3<--0) bands.

!'Vapor pressure data are taken from Ref. (17) by Johns, Ref. (35) by Clusius & Schleich, Ref. (11)

by Yato, e(al. and Ref. (36) by Bigeleisen & Ribnikar, respectively.

110 Distillation data are taken from Ref. (38) by Bilkadi et a!.

-57-

Dow

nloa

ded

by [

Ston

y B

rook

Uni

vers

ity]

at 2

2:22

31

Oct

ober

201

4

778 ]. Nucl. Sci. Techno!.,

Table 8 Intermolecular torque and root-mean-square oscillation amplitude associated with molecular rotation for diatomic and linear triatomic molecules in solid phase

Molecule T T* Methodt 1 ((OUjkT) 2> (0')'1' (] a{(U2)/T*) 112

(K) (rad- 2 ) (deg) cA) (A) ------- -- -~-- --. ----------

19-HCl 130 0.40 A 37.912 9.3 1. 28 0.33 p-CO 67 0.50 A 1611 14 1. 13 0.36 p-CO 68 0.51 A 2214 12 1. 13 0.33 p-CO 68 0.51 A 271'1 11 1. 13 0.28 p-CO 64 0.48 B 26t3t5 11 1. 13 0.30 N.o 124 0.40 B 128t3j5 5.0 2.31 0.32

N.o 124 0.40 D 8516 6.2 2.31 0.40

C02 124 0.44 D 9416 5.9 2.32 0.34

11 For A and B see the footnote of Table 7. In D, mean square torque is derived fron1

the librational frequencies. !'This work. ta Derived frotn isotopic vapor pressure ratios in this work. t• See the footnotes t• and t> of Table 7. t> Vapor pressure data are taken from Ref. (17) by Johns and Ref. (ll.l by Yato ct a{. respectively. 16 Determined by Cahill & Leori in Refs. (33) and (34).

is especially useful in predicting the separa­tion factor in phase equilibria. We only mention here that the separation factors of "Oz/160 180 and 35Clz/35CP7Cl are well reprodu­ced by these relationships.

ACKNOWLEDGMENT

The author wishes to thank Prof. H. Kaki­hana and Y. Takashima for many helpful suggestions and comments. The encourage­ment and suggestions of Prof. ]. Bigeleisen was vital to the completion of the work.

---REFERENCES---(!) BIGELEISE:'\, J.: ]. Chern. Phys., 34, 1485 (1961). (2) BrGELEISE;-;, ]., Ronr, E.: ibid., 35, 68 (1961). (3) STERN, M. ]., et at.: ibid., 39, 3179 (1963). (4) BrGELEISE'>, ]., et a!.: f. Chim. Phys., 60, 60

(1963). (5) BIGELEISE:'-1, ]. : ibid., 61, 87 (1964). (6) BIGELEISE'\, ]. : ]. Chern. Phys., 39, 769 (1963). (7) ISHIDA, T., BIGELEISEN, ]. : ibid., 49. 5498

(1968). (8) LEE, M. W., et al.: ibid., 53, 4066 (1970). (9) LEE, M. W., et al.: ibid., 56, 4585 (1972). (1~ 8ILKADI, Z .. et a/.: ibid., 62, 2087 (1975). (11) YATO, Y., et a!.: ibid., 63, 1555 (1975). M BIGELEISE'\, ]., FLTKS, S., RIHNIKAR, S., YATO,

Y.: ibid., 66, 1689 (1977). (1~ HERZFELD, K. F., TELLER, E.: Phys. Rev., 54,

912 (1938). M GIAUCWE, W. F., WIEBE, R.: ]. Am. Chern.

Soc., 50, 101 (1928). (1$ LEwis, G. N., et a!.: ibid., 56, 494 (1934).

M GvERDTSITELI, I. G., et a!.: Sov. ]. At. Energy, 4, 390 (1958).

(1~ ]or-INS, T.F.: Proc. of Int. Symp. on Isotope Separation, p. 74 (1958), North Holland Pub.

M BABLOYANTZ, A.: Mol. Phys., 2, 39 (1959). M FRIEDMA'\N, H.: Adv. Chern. Phys., 4, 225

(1962). ~~ BoA TO, G., et at.:]. Chern. Phys., 31, 201(1962). ~D GoRDON, G.: ibid., 44, 576 (1966). ~~ idem: ibid., 39, 2788 (1963). ~~ idem: ibid., 40, 1973 (1964). ~~ idem: ibid., 41, 1819 (1964). ~$ WEST, W. : ibid., 7, 795 (1939). ~~ YAMADA, H.: jasco Rep., Vol. 2, 1 (1965). ~~ HETTNER, G.: Z. Phys., 89, 234 (1934). ~$ FRIEDRICH, H. B., PERSON, W.B.: ;. Chern.

Phys., 39, 811 (1963). ~~ ITo, M., et at.: ibid., 50, 2949 (1969). (3~ FRIEDMANN, H., KI~IMEL, S.: ibid., 42, 3327

(1965). (31) KIRKWOOD, ].G.: ibid., 1, 597 (1933). (3~ WIGNER, E.: Phys. Rev., 40, 749 (1932). (3~ CAHILL, J.E., LEROI, G. E.: ]. Chem. Phys.,

51, 97 (1969). (3~ idem: ibid., 51, 1324 (1969). (3$ CLUsrus, K., ScHLEICH, K.: Proc. of 2nd U.N.

Int. Con/. on the Peaceful Use of Atomic Energy, Vol. 4, 455 (1955).

(l~ BIGELEISEN, ]., RmNIKAR, S.: f. Chern. Phys., 35, 1297 (1961).

(l'i) HERZBERG, G.: "Molecular Spectra and Mole­cular Structure (1), Spectra of Diatamic Mole­cules", (1950), D. Van Nostrand.

(3$ BILKADI, Z, et al.: ]. Chern. Phys., 62, 2087 (1975).

-58-

Dow

nloa

ded

by [

Ston

y B

rook

Uni

vers

ity]

at 2

2:22

31

Oct

ober

201

4