Physica 91A (1978) 619-624 © North-Holland Publishing Co.

T H E E X C E S S T H E R M O D Y N A M I C P R O P E R T I E S OF S O L U T I O N S OF I S O T O P I C I S O M E R S , ONE IN T H E O T H E R

GSbor JANCSO* and W. Alexander VAN HOOK

Chemistry Department, University of Tennessee, Knoxville, Tennessee 37916. USA

Received 2 August 1977 Received in final form 4 October 1977

An analysis of the excess free energy of solutions of isotopic isomers is given. It is demonstrated that consideration of the contribution of internal degrees of freedom is essential and in the case of solutions of C6HdC6D6 this term predominates.

An analysis of the excess t h e r m o d y n a m i c propert ies of solutions of isotopic isomers , one in the other , has been repor ted by Prigogine and co-workers~-4). To begin with, we adopt their me thod and consider the He lmhol t z free energies, fA and fa, of the two isotopic isomers , A and B, in terms of expans ions about their respect ive equilibrium volumes , V ° and V °, at pres- sures pO and pO:

i ,, ( l a ) . tA (v ) = .IA( v ° ) + .f;,( v - v ° ) + ~"A( v - v ° ? + . . . ,

/ . ( v ) = / B ( v °) + / ~ ( v - v~) ' . . . . . + ~fB(V V°)2+ ". ( lb)

In the equat ions f ' = (df /dV) , f " = (d2f/dV2), etc. The cont r ibut ion of the vo lume change to the excess f ree energy of solution, re, is given by

i f ( V ) = X d A ( V) + XBfB( V) - SARA( V°,O - XsfB( V°). (2)

The vo lume of the solut ion may be expressed in terms of the s tandard state and excess partial molal volumes , V ° and Q~-, and the mole f rac t ions Xi,

V = XA( V ° + I7~) + XB( V ° + I7~), (3)

but for solut ions of isotopic i somers the terms in I7"~, and I7"~ are negligible in compar i son to V ° and V °, respectively2'5). Subst i tut ion of eqs. (1) and (3) into (2) yields

i f (V) = s a g a ( V ° - V°)(f6 - f~) + ~XAXa( V ° - V°)2(XB/'~, + XAf~). (4)

N o t e in par t icular that the fac to r in the second term (XAf~ + Xaf~,) cor rec t s the

* Permanent address: Central Research Institute for Physics, Hungarian Academy of Sciences, Budapest, Hungary.

619

620 G. J A N C S 0 AND W.A. VAN HOOK

one (XAf '~ -- X B f D originally repor ted in the t rea tment of the ze ro-pressure case (neglect of first term)6), and e m p l o y e d in the calculat ion of reduced part i t ion funct ion ratios f rom isotope f rac t iona t ion dataT). Differentiat ing and recall ing

and

p , ~ , ( V ) = f ¢ ( V ) - v { d f ( V ) ' ~ " B k d XB J

d e ~_ , . { / ( v ) ~

yields expressions for the excess partial molal free energies*:

p,.~(V) -- X2[( V ° - V ° ) ( f ~ - f ; , ) + ½( V ° - V°)2[(2XB - l ) ( f~ - f~) + f~]],

~ ( V ) = X2[( V ° - V ° ) ( f 6 - f ; , ) - ½( V ° - V°)2[(2XA -- 1)(f;( -- f~) -- f~]l.

(5a)

(5b)

The excess chemical potentials at infinite dilution are given by

t . t A ( V ° ) = A V ( f ~ - f ~ ) + ½(AV)2f'~ ", XB~-- !, (6a)

and

t ~ ( V °) = a V ( f ~ - f ; ) + ½ ( a v ) 2 f ~ ; XAm 1, (6b)

where A V = V ° - V °. Neglec t ing the first term in (5) it is a simple mat ter to obtain the relat ion be tween the separa t ion factor , In a, at infinite dilution, XB------ 1, and the logari thmic part i t ion func t ion ratios, in terms of the compres - sibility, /3A, and the molar vo lume isotope effect ( V ° - V°)/VOA,

liq 0 liq 0 • [ Q A ( V A ) / Q B (VB)] ( V ° - V ° ) :

In ,~ = m [ ~ j - 2I~A VO R T . (7)

This equa t ion has been widely e m p l o y e d s) in conver t ing isotopic f rac t ionat ion data to the more theoret ical ly useful part i t ion funct ion ratios. In a later deve lopmen t Phillips et al. 9) cons idered the case of finite pressure as a part of their analysis of argon isotope f rac t iona t ion data. They pointed out the impor tance of due cons idera t ion of the concen t ra t ion d e p e n d e n c e of the excess f ree ene rgy and cited an equat ion connec t ing the concen t ra t ion dependences of the Gibbs and He lmho l t z free energies

( d g ¢ / d X A ) z r - ( d f e / d X A ) e . r = -~(Xs - XA)(/~ Vm)-1( V O -- V ° ) 2, (8)

where Vm refers to the mixture volume. The authors point out that the term ( d f f / d X ) prev ious ly neglected j-4) is small with respec t to the isotope separa- tion fac to r for argon isotopes but made no numerical evaluat ions .

Our interest in this ques t ion was initiated by the availabili ty of new data on

* In deriving expressions of the type (5), which can be obtained by a variety of routes, it is necessary to keep in mind that the total volume, V, is an explicit function of X [see eq. (3)].

THERMODYNAMICS OF SOLUTIONS OF ISOTOPIC ISOMERS 621

the free energies of mixing of benzene-deu te robenzene and cyc lohexane- deuterocyclohexane solutions1°). The detailed analysis has been restricted to the benzene case. At 25°C, tt~6, 6 = 2.3-+ 0.2 J/mole. Following Prigogine 2) we calculate the excess free energy of mixing in a two-step process. In the first step the separated components are compressed (dilated) to the molar volume of the solution. In the second step the components are mixed at constant volume. The assumption is made that the excess free energy of mixing in step II is zero") .

For the moment we restrict attention to the infinitely dilute solution. For complicated molecules it is convenient to introduce an oscillator model in the pseudo-harmonic approximationm3). The partition function, Q, of an average molecule in the condensed phase with 3 n - 6 internal and 6 external degrees of f reedom is related to the free energy f = - R T In Q ( /al l , lJ 2 . . . . ) . According to the process described in the last paragraph the free energy change cor- responding to the expansion (dilation) of isotopic isomer A to the volume of isomer B, elf, is equal to /z~(V °) and is expressed

Af =p , ~ ( V° ) = - ~ d V = - R T ~ (dlnQ~[d~'i'~dV i \ dl"i ]\dV] " (9)

v o v o

The condition for use of this equation is a detailed knowledge of the volume dependence of the 3n vibrational frequencies. However , the available data refer exclusively to the pressure coefficients of the frequencies (c~vi/cgP), which can be converted to the desired volume coefficients using the compressibility. In the present approach the drawbacks of the slowly con- verging Taylor series (1)are avoided.

We have carried out calculations for solutions of H2 in D2 and C6H6 in C6D6 using eq. (9). For the H2-D2 system the molar volume isotope effect has been reported as 4.85 cm3/mole at 20.3 K and f~: = 1/(V°2/3.2)= 0.44-+0.02 2). The shift in the internal vibrational f requency with pressure was taken as 2.5 cm -1 kbar -114) which is equivalent to 0.220 cm -1 when H2, at its equilibrium molar volume (28.36cma/mole), is compressed to the equilibrium molar volume of deuterium (23.51 cm3/mole). This is equivalent to a free energy change of 1.32J/mole. The value of /z~2(V ~) obtained from mixing data at X = 0.5 is 30-+ 13 J/molem'2), while that calculated from the second term of (6) is 22-+ 1 J/mole. Thus, in hydrogen the effect is dominated by the compres- sibility which is principally determined by the intermolecular potential.

In the second example, benzene in deuterobenzene, the contribution of the internal degrees of f reedom is relatively more important. Details of the normal coordinate calculation have been reported as part of an analysis of benzene vapor pressure isotope effects1°). The VPIEs were fitted to experi- mental accuracy with a set of 3n pseudo-harmonic frequencies consistent with the available spectroscopic data. The six external translations and rotations and the six internal CH stretching motions were taken as volume dependent to conform with the experimental measurementslS). The contribu-

622 G. JANCS0 AND W.A. VAN HOOK

t ions of the d i f fe ren t k inds of f r e q u e n c i e s a re de t a i l ed in t ab le I. T h o s e f rom the in te rna l d e g r e e s of f r e e d o m p r e d o m i n a t e . I t is in t e res t ing tha t the con t r i - bu t ion c a l c u l a t e d f rom the e x t e r n a l deg ree s of f r e e d o m is a b o u t the s ame as tha t e v a l u a t e d f rom the c o m p r e s s i b i l i t y , ind ica t ing (once again) tha t the c o m p r e s s i b i l i t y is d o m i n a t e d by the i n t e r m o l e c u l a r po ten t i a l . The role of the c o m p r e s s i b i l i t y can be seen in a n o t h e r way . In the c o n t e x t of an osc i l l a to r m o d e l Q is a p r o d u c t o v e r (vo lume d e p e n d e n t ) h a r m o n i c o sc i l l a to r pa r t i t ion f unc t i ons ,

e -~"'/2' 1 = RTV ~ [02 In Qi~ Q = l - I Q,; Qi = (1 - e-",------~' ~ k~ff-Q ~--] T"

To the e x t e n t that the f r e q u e n c i e s con t r i bu t i ng to Q show a l inear v o l u m e d e p e n d e n c e , O2u/dV 2 = 0 and

e-Ui 1 r e", (,o) -~=~i \OV] I _ l - e - ' k -

w h e r e ui = hv~/kT. The c o m p r e s s i b i l i t y is d e t e r m i n e d by the t he rma l e xc i t a t i on f ac to r s . F o r mos t m o l e c u l e s wi th in te rna l s t ruc tu re at t e m p e r a t u r e s o f in te res t on ly the e x t e r n a l d e g r e e s of f r e e d o m c o n t r i b u t e s igni f icant ly to the sum. ( F o r b e n z e n e the l owes t in te rna l v ib ra t iona l f r e q u e n c y is 408 cm -~ which g ives a b r a c k e t e d c o n t r i b u t i o n of on ly 0.2 uni ts as c o m p a r e d to the 8 uni ts f rom the

TABLE I Calculated and observed excess chemical potentials of benzene at infinite dilution in

deuterobenzene

~2~ Vo2) (J/mole)

Contributions T(K) to ~ , (V°2) calc5 obs) °)

298 internal d.f? 1.32 external d.f. b 0.21 2.3_+0.2 eq. (6) 0.33

323 internal d.f. 1.03 external d.f. 0 .18 2.2_+0.2 eq. (6) 0.26

353 internal d.f. 0.66 external d.f. 0 .12 1.9_+0.2 eq. (6) 0.14

" Degrees of freedom. b A temperature independent value of 2

was used for the pseudo-Griineisen constant~6.~v).

~The difference between the calculated and observed values has been attributed to anharmonicityL°).

THERMODYNAMICS OF SOLUTIONS OF ISOTOPIC ISOMERS 623

largest external mode.) We have calculated/3 at 25°C for C6H6 using the set of external frequencies previously reported (54.7, 54.7, 71.6, 50.5, 41.0 and 27.3 cm -I) ~0) together with a pseudo-Griineisen constant of 2 17) obtaining 1//3 = 1 × 104atm in fortuitously good agreement with the experimentally observed value, 1//3 = 1.03 x 104 atmlS).

It now becomes clear that the earlier Prigogine-Bingen-Bellemans ap- proach 1-3) and its subsequent modifications 4"~9'2°) are appropriate only for those cases where the isotope effects are essentially independent of contributions from the internal degrees of freedom. In previous applications to solutions of monatomic species, or to isotopic hydrogen, this was the case which obtained. The present analysis demonstrates that for more complicated molecules where the internal contribution is relatively important-even dominating-the simple relations fail. The dependence of the internal part of the partion function on the overall system volume must be given due consideration.

In addition to the intrinsic importance of a proper understanding of the excess properties of solutions of isotopic isomers, the present considerations can be employed in correcting fractionation factors to yield partition functions ratios. The present results indicate that such corrections are not at all straightforward unless data on the volume dependence of the internal degrees of freedom are available. Corrections on the order of a few percent, even more in some cases, are to be expected.

Acknowledgements

Partial supports from the US National Science Foundation and the Hungarian Institute of Cultural Relations as part of the cooperative research program between the Central Research Institute for Physics and the Uni- versity of Tennessee, the NSF Chemical Thermodynamics Program, and the National Institutes of Health are gratefully acknowledged.

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624 G. J A N C S O AND W.A. VAN H O O K

11) Since the molecules themselves are s t ructured there may be an isotope effect on their intrinsic size. V, as well as on the total volume occupied per mole, V. There is no reason to expect that

(o h _,: ± (oh aP ] ~ (/ g \aP ] ~

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