Trouton’s Rule Applied to Dissociative and Associative Evaporation (33) D. R. Stull and G. C. Sinke, Adv. Chem. Ser., No. 18 (1956). (34) D. R. Stull and H. Prophet, “JANAF Thermochemical Tables”, 2nd ed, US. Government Printing Office, Washington, D.C., 1971. (35) K. K. Kelley, U.S.Bur. Mines, Bull., No. 584 (1960). (36) K. K. Kelley and E. G. King, U . S . Bur. Mines, Bull., No. 592 (1961). (37) D. D. Wagman, W. H. Evans, V. B. Parker, I. Haiow, S. M. Bailey, and R. H. Schumm, Natl. Bur. Stand. ( U . S . ) Tech. NoteNo. 270-3 (1968). (38) R. Hauge, Rice University, personal communication
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(39) G. Herzberg, “Molecular Spectra and Molecular Structure 11. Infrared and Raman Spectra of Polyatomic Molecules”, Van Nostrand-Reinhokl, New York, N.Y., 1945, p 168. (40) J. D. Tretjakow and H. Schmalzried, Ber. Bunsenges. Phys. Chem., 69, 396 (1965). (41) J. A. Kessler, Yu. D. Tretyakov, I. V. Gordeyev, and V. A. Alferov, J . Chem. Tbermodyn., 8, 101 (1976). (42) Yu. D. Tretyakov, I. V. Gordeev, and Ya. A. Kesler, J . Solid State Chem., 20, 345 (1977).
Generalization of Trouton’s Rule to Dissociative and Associative Evaporation. Calculation of Heat and Entropy of Evaporation for Group 2-6 Compounds and for Alkali Halides P. S. Vincett Xerox Research Centre of Canada Limited, Mississauga, Ontario, Canada L5L 1J9 (Received April 11, 1978) Publication costs assisted by Xerox Research Centre of Canada Limited
Trouton’s rule states that for many liquids the molar heat of evaporation, AH, at the boiling point, Tb, is approximately equal to llRTb,where R is the gas constant. This rule is often well-obeyed, but is generally considered to be of rather little value in predicting AH for high boiling compounds, or for other compounds which dissociate or associate on evaporation. As an example of dissociative evaporation, we consider the group 2b-6a compounds, most of which evaporate with essentially complete (but reversible) dissociation at temperatures approaching 2000 K; for these compounds, AH (where known) differs from the Trouton-predicted values by more than 50%. In this paper, we develop calculation methods, based on Trouton’s rule and utilizing simple statistical mechanical relations to take account of the dissociation,which permit aH to be estimated to an accuracy of 5510% even for these compounds. Similar calculations show why it is that the alkali halides, in which the ions present in the melt are largely associated into molecules in the vapor, nevertheless obey the simple form of Trouton’s rule: this is because of a fortuitous cancellation of two large contributions to the entropy of evaporation. The success of these calculations suggests that most of the deviations from Trouton’s rule exhibited by high boiling inorganic compounds are due merely to changes of molecular species on evaporation; these deviations should often be calculable by our techniques. This can be very useful for compounds for which no experimental values of A H are available, or for which there may even be no liquid range at ordinary pressures.
I. Introduction One of the most remarkable generalizations in the whole of physical chemistry is provided by Trouton’s rule. This rule1 states that for very many liquids, the molar heat of evaporation, AH,a t the normal boiling point, Tb, is given approximately by
AH ~
Tb
- 10.5
-
11
where R is the gas constant. This is equivalent to stating that the entropy of evaporation, AS, is approximately constant for different liquids a t Tb. This rule is most commonly applied to compounds (especially organics) having Tb within about a factor of 2 of room temperature and it then usually holds to within about *lo%, provided that little molecular association occurs in the liquid or gas phase. Although Trouton’s rule is usually stated to need considerable correction for higher boiling materials,2 it remains reasonzbly accurate for many common compounds even a t high temperatures. If, for example, AH/(RTb)is calculated for those compounds (not elements) for which AH is listed in “Lange’s H a n d b ~ o k ”and ~ for which Tb.k greater than about 1000 K,3,4the ratio so obtained is within about 10% of 11 for more than two-thirds of the compounds; the exceptions are almost exclusively fluorides and oxides, for which complex liquid molecular structures would not be unexpected. The majority of the high boiling 0022-3654/78/2082-2797$01 .OO/O
compounds for which 4 H is listed3 are halides. However, using recent data it is possible to calculate values of AH/(RTb) for other relatively simple high boiling compounds, such as many of the sulfides, selenides, and tellurides of lead, tin, zinc, and ~ a d m i u r n . The ~ , ~ group 4a-6a compounds (which vaporize largely without dissociation5) again yield values of AH/(RTb)within about 10% of 11. The group 2b-6a compounds, which dissociate’ (congruently and reversibly) on vaporization, on the other hand, give values5 in the region of 17. The quite widespread applicability of Trouton’s rule suggests that most of the basic considerations underlying the rule (see section 11) may be valid even a t very high temperatures, and that the deviations may merely be due to the fact that most high-boiling materials are not molecular liquids (in the organic sense) and therefore often undergo, on boiling, drastic changes in the number and type of molecules. I t is therefore of interest to attempt t o calculate AH for typical processes of the latter kind using Trouton-like considerations, but correcting for the molecular changes using very simple statistical-mechanical expressions. To do this, we need to consider boiling processes in which the molecular changes are substantial but reasonably well-defined; we therefore consider the dissociative boiling of the group 2b-6a compounds, hereafter referred to as the 2-6’s. (Most of these compounds vaporize7 according to the scheme 0 1978
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MX(1) M(g) + ‘/zXzk) (2) where M is the metal and X the chalcogen. The boiling points are generally -1500 K and upward. The liquid species is not precisely known, as we shall see.) We shall develop the principles of the simple “Trouton-like” calculations discussed above, and shall show that the values of AH/(RTb)calculated in this way for the group 2-6 compounds are within 5 1 0 % of the experimental values (where the latter are known). The success of the calculation for these rather extreme cases (for which AH/(RTb)deviates from the simple Trouton’s rule value by around 50%) is striking evidence that the underlying mechanisms of Trouton’s rule are still valid a t very high temperatures and for boiling processes drastically different from the simple ones normally considered. Furthermore, this gives us considerable confidence that AH may be calculated in a similar way for other complex-boiling compounds for which there are no experimental values, or for which there may even be no liquid range a t ordinary pressures. This was in fact the original reason for undertaking the calculation^.^ Finally, the procedures developed throw considerable light on why it is that compounds like the alkali halides obey Trouton’s rule, even though their apparently simple boiling process involves, as we shall see, a change of molecular species. +
11. Principles of Trouton-Like Calculations Trouton’s rule states that the entropy of evaporation (As) a t Tb is similar for most liquids. In general, the entropy of most liquids or gases may contain five terms? translational, rotational, vibrational, and electronic entropy and entropy of mixing (in order of descending importance), so we have A S = Stg + S,P + S,g S,g S,g - S,‘ - S,’ - S,’ s,‘-s,’ (3)
+
ecules present in the liquid. The electronic contribution, Se,llis normally the same in liquid and gas, because the chemical nature of the compound is normally unaltered on boiling. This assumption is no longer valid in the present situation; moreover, the diatomic chalcogens, unlike most other molecules, have a triplet ground state.12 This adds R In 3 to the entropy of each mole of diatomic chalcogen. Thus
AS, = Y2R In 3
(6)
The largest contribution to the entropy is translational. For an ideal gas, the translational entropy, St,is given by the Sackur-Tetrode equationg as follows:
where m is the mass of each molecule and N is the number of molecules in volume V . Formally, when a liquid boils simply, i.e., without changes in the number and type of molecules, the only quantity in the above equation which varies is V . Thus, formally AS, = R In ( V g /V f ) (8) where V , is the volume of the gas a t Tb and atmospheric pressure, and V , is an effective free volume in the l i q ~ i d . ~ J ~ Since, in simple boiling, S, is the only entropy contribution which changes, Trouton’s rule is equivalent to the statement that In ( V g / V f ) 11
-
or that
-
+
V g / V f 60000
where Stg and S,‘ are the translational entropies of the gas and liquid, S,P and S,1 are the corresponding rotational entropies, and so on. Since Trouton’s rule holds equally well for monatomic or polyatomic molecular liquids, it is generally assumedl that both rotational and vibrational motions are sufficiently free a t Tb that the respective entropies are essentially unaltered as the liquid boils. This is no longer true for dissociative boiling, since entirely different molecules are present in the liquid and the gas; however, it is simple to calculate the rotational and vibrational entropy contributions ( S , and Sv)for diatomic molecules in the gas phase, using the well-known relation^:^
Since V , is easily calculable, V fcan be calculated from this equation and is generally a few percent of the total liquid volume. In our case, where the number of molecules changes on boiling, we are forced to calculate actual values of S , using eq 7, rather than mere differences. This is obviously straightforward for gaseous molecules; for the liquids, our Trouton-like calculation will simply use eq 9 to give an estimate of the effective free volume to be substituted into eq 7. In other words, we are assuming that Trouton’s rule would hold if the liquid did not dissociate on boiling. Our gases consist of a mixture of molecules. The calculation, therefore, takes account of the entropy of mixing; it is easy to show that the sum of translational and mixing entropies for our 1.5 mol of gas (generated from 1 mol of liquid) is given by
(
S R = R l+ln-----
8.rr21kT ah2
(4)
In these equations, I is the moment of inertia of the molecule, E is the spacing between adjacent vibrational energy levels, h is Boltzmann’s constant, T is the temperature of the gas, and h is Planck’s constant; u is a symmetry number equal to 1 for heteronuclear diatomic molecules and 2 for homonuclear molecules.1° As discussed above, the normal form of Trouton’s rule implies that molecules in a boiling liquid have similar values of SRand Sv to those of the same molecules in a gas; our Trouton-like calculations will therefore also use eq 4 and 5 to calculate SRand Sv for the diatomic mol-
(9)
In this equation, ml and mz are the masses of one atom of metal and one molecule of chalcogen, respectively, and V , is the total volume accessible to each molecule (i.e., the volume of 1.5 mol of gas at atmospheric pressure and T b ) . The final term, and the use of V, rather than the volume per mole, represent the entropy of mixing. In summary, our calculations will assume that the underlying principles of Trouton’s rule are still valid in these unusual boiling processes. This leads to the assumptions (i) that S , and s, in the liquid can be calculated as if the molecules were essentially free, and (ii) that St
Trouton’s Rule Applied to Dissociative and Associative Evaporation
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TABLE I: Calculation of the Translational (St),Rotational (SR),Vibrational (S,,), and Electronic (S,) Contributions to the Entropy of Evaporation, AS,of Several Liquid Compounds“ complete dissociation in liquida
complete association in liquida
ZeSe
ZnTe
CdSe
CdTe
ZnSe
ZnTe
CdSe
CdTe
S t / R (liquid) S t I R (gas) AStIR S R / R (liquid) S R / R (gas) S,/R (liquid) ( + _ 0 . 2 ) SvIR (gas) (ASR+ AS,)/R ASeIR
27.56 37.45 9.89 0 4.92 0 1.10 6.02 0.55
27.91 37.52 9.61 0 5.29 0 1.27 6.56 0.55
28.08 38.04 9.96 0 4.89 0 1.07 5.96 0.55
28.40 38.10 9.70 0 5.26 0 1.24 6.50 0.55
14.05 37.45 23.40 10.64 4.92 2.16 1.10 -6.78 0.55
14.32 37.52 23.20 10.92 5.29 2.37 1.27 -6.70 0.55
14.35 38.04 23.69 10.98 4.89 2.38 1.07 -7.40 0.55
14.49 38.10 23.61 11.30 5.26 2.48 1.24 -7.28 0.55
A S I R (total)
16.46
16.72
16.47
16.75
17.17
17.05
16.84
16.88
a Two extreme assumotions (comolete dissociation and complete binary association) are made regarding the structure of the liquid. See text fo; further explanation.
can be calculated for both the liquid and the gas using the Sackur-Tetrode equation, and employing the effective liquid free volume given by eq 9. We also take account of the change of electronic entropy where chalcogen molecules are created, and of the entropy of mixing in the gas. Mixing in the liquid is discussed below. In the case of the elements involved in our calculations, experimental entropy values14are available for the gaseous state. Entropies calculated on our basis agree with these experimental values to within about 0.1 YO. 111. S t r u c t u r e of t h e Liquid Group 2-6 Compounds The structure of the liquid group 2-6 compounds is not known precisely, and this causes a further complication in the calculation. Data relating to the liquid structure of the compounds of Zn and Cd with (at least) Se and Te are generally interpreted in terms of a mixture of M and X ions or atoms,15-17with a t least some association into the binary compound MX. Since the degree of association is unknown, we shall perform the calculation for the two limiting cases of no association and complete association, and show that the two results are very similar.lB This leaves us with two problems. (i) If the liquid is dissociated, we do not know how perfectly the ions or atoms are mixed. If they are perfectly mixed, we must allow an entropy of mixing term; if, on the other hand, each ion or atom tends on average at any given instant to be surrounded by unlike ions or atoms, then such a term is nearly absent. To express the problem another way, the effective volume accessible to each atom or ion in the two cases is either the whole free volume or essentially half of it. The entropy difference is 2R In 2, or 1.39R. Since there appears to be little evidence on this point, we will take the mean of the two extreme entropies, and accept that we have an uncertainty of f0.69R. (ii) If the liquid is partly associated and partly dissociated, we cannot merely combine the entropies of the two pure phases (with appropriate weighting factors) but must also allow a further entropy of mixing term; it can be shown that this term is between zero and about R , depending on the proportion of the two species. Again, we can take the midpoint, and accept an uncertainty of k0.5R. Combining the above uncertainties, we clearly have a t most an uncertainty of --1.2R, which as we shall see is just 7% of our final result for the entropy of evaporation. In fact, if we assume that the liquid is at least 10% associated, which seems rather safe, the possible uncertainty from the above causes can be shown to drop to kO.8R, or less than 5% of our final result. Given the rather substantial uncertainties regarding the nature of the liquids, this seems to be a quite satisfactory situation,
TABLE 11: Final Estimates of the Entropy of Evaporation, A S , for Several Compounds, together with the Experimental Values of A Sand Experimental Information which Yields These Valuesa
__-
ZnSe AH(sub) (exptl), kcal/mol AH(fusion) (exptl), kcal/mol A H (exptl), kcal/mol Th,b (K) Hence A S (exptl)/R AS(calcd)/R (this work)
17.3
ZnTe
CdSe
CdTe
71.5
73
62
15.6
10.5
12
56
62.5
50
1675 16.8 (tl.0)
1650 18.9 (i1.5)
1600 15.8 (+1.0)
17.4
17.2
16.4
a See text for further explanation.
Reference 6.
IV. Calculation of t h e Entropy of Evaporation of t h e Group 2-6 Compounds Table I shows the results, and the intermediate stages, of calculations performed as discussed above for ZnSe, ZnTe, CdSe, and CdTe. The two extreme assumptions of complete dissociation and complete (binary) association in the liquid are shown separately. T, is estimated as mentioned previously,6while the other data required are taken from standard h a n d b o o k ~ . ~ >The ~ J ~vibrational frequencies of the group 2-6 compounds have been estimated from data on other compound^;^^ the rather large uncertainties in these estimates produce a very small effect on the calculated entropies. It is clear from Table I that the differences between the final results obtained on the two assumptions of complete association and dissociation are very small, considerably less than the uncertainties in the calculation. Table I assumes a Trouton “constant” of 11 (eq 9); different assumptions would of course alter the final entropy values. For example, if a value of 10.5 had been assumed, the dissociated values of A S I R would fall by 0.5, while the associated values would fall by 1.0. We do not know the exact constitution of the liquids, but we assume as discussed in section 111that they can be represented as some kind of mixture of dissociated atoms or ions and binary associated molecules.15-17 Then we may obtain a final estimate of the entropy of evaporation of the actual liquid by taking the mean of the dissociated and associated values (accepting an uncertainty equal to half of their difference), and adding an approximate value for the entropy of mixing as discussed in section 111. The final estimates of AS/R are shown in Table 11. The overall uncertainty in A S arising from our lack of knowledge of the liquid structure (assuming a t least 10% association)
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is then roughly &LOR or less than 6% of the overall value of AS. This obviously has to be added to any other errors arising from the principles of the calculation (section 11) and in particular from the assumed value of the Trouton constant. Table I1 also shows the experimental values for AS/R. These are given by AHIRT,, where Tb is inferred as discussed previously,6 and AH is deduced by subtracting the experimental heats of fusionz0from the experimental heats of ~ u b l i m a t i o n . ~T~o ~do~ this, - ~ ~ the heats of sublimation must be slightly corrected to allow for their variation with temperature, using data in the above refe r e n c e ~ . ' , ~The ~ , ~heat ~ of fusion of ZnSe is apparently not available. The quoted uncertainties for the other compounds are approximate estimates, and are due to uncertainties in the determinations both of vapor and of the heat of fusion.20
V. Discussion While both the experimental and theoretical estimates of A S in Table I1 are subject to uncertainties in the region of l R , the agreement between the two sets of values is clearly rather satisfactory. This shows that using very simple assumptions based directly on Trouton's rule, and making reasonable assumptions about the structure of the liquid, it is possible to predict the heat of evaporation for a set of dissociable liquids to an accuracy comparable to that with which Trouton's rule holds for most other compounds. The calculated values of A S are rather insensitive to the masses of the constituent atoms. Assuming, for example, that our assumptions about the structure of the liquids are still reasonably valid for ZnS, CdS, and even ZnO, calculations similar to those given above predict their A S / R to be in all cases within a couple of percent of 17. It is of interest to ask why it is that Trouton's rule holds well both for molecular liquids such as most organics, and also for liquids such as the alkali halides which (while they consist largely of simple molecules in the gas phase2*)are largely ionized in the The change in number of independent entities on boiling seems to present just the same kind of complication as is present in hydrogenbonded liquids such as water' (which does not obey Trouton's rule) or in the group 2-6 compounds, namely, simultaneous change of S,, SR,and Sv. Table I shows why this complication is of little importance: for diatomic molecules of typical inorganic mass and size, the translational entropy created on dissociation in the liquid (- 14R) is almost exactly balanced by the simultaneous reduction of rotational and vibrational entropy (-11R and 2R, respectively). This cancellation is clearly not exact, and in fact appears to be a pure numerical coincidence; it is nevertheless the reason why liquid dissociation does not cause Trouton's rule to break down completely for a very large class of high boiling materials. One can, in fact, calculate the entropy of evaporation for simple ionics such as KC1, making assumptions similar26 to those made above, and assuming that the boiling process is as follows27 K + + C1- KCl(gas) (11) N
-
The calculated A S is around 12, in good agreement with the experimental value3 of 11.6.
VI. Conclusion We have shown that the entropy of evaporation, AS, of a group of high-boiling dissociable compounds (the 2-6's) can be calculated to reasonable accuracy on the basis of the considerations underlying Trouton's rule, even though
the values of 4S/R differ from the usual value of -11 by more than 50%. Similar calculations show why the alkali halides, although they are dissociated in the liquid but not the vapor, roughly follow the simple rule. These results suggest that most of the basic considerations underlying Trouton's rule hold even for complex boiling processes occurring a t temperatures approaching 2000 K, and that the well-known deviations from the rule a t high temperatures may in most cases be due merely to changes of molecular species on boiling. Such deviations will in many cases be calculable. In addition to the intrinsic interest of this result, calculations of this type can be most useful when LH values are required5 for materials for which little experimental data exists or for which there may not even be a liquid range under normal pressures. We should perhaps caution however that calculations of this type generally cannot be used to distinguish between competing models of liquid structure, since as we have seen the entropy of the liquid tends to be fairly insensitive to the structure.
Acknowledgment. I thank Professor G. G. Roberts (University of Durham, England) for several helpful discussions. References and Notes See, for example, G. M. Barrow, "Physical Chemistry", 3rd ed, McGraw-Hill, New York, 1973, Chapter 19. J. R. Partington, "An Advanced Treatise on Physical Chemistry", Vol. 2, Longmans, London, 1951. J. A. Dean, Ed., "Lange's Handbook of Chemistry", 11th ed, McGraw-Hili, New York, 1973, p 9-72. R. C. Weast, Ed., "Handbook of Chemistry and Physics", 57th ed, Chemical Rubber Co., Cleveland, Ohio, 1976, p B 85. P. S. Vincett, W. A. Barlow, and G. G. Roberts, d . Appl. Phys., 48 3800 (1977), and references cited therein. AHis calculated by subtracting the measured heat of fusion (ref 20) from the heat of sublimation. Tb is estimated from knowledge of the vapor pressure, P,, over the liquid at the melting temperature, T, (when it is equal to the known P, over the solid), and initially assuming the rate of variation of P, over the liqua with Tis as predicted by Trouton's rule and the Clausius-Clapeyron equation; this yields an initial value of AHI(RT,), which gives a better rate of P, variation. Simple iteration yields final values which should be accurate to a few percent. For the 2-6's T, is occasionally below T,. This means that the solid sublimes at ordinary pressures. However, the liquid exists (and T, can be determined) at high pressures; T, is the temperature at which the supercooled liquid would have a vapor pressure of 1 atm. P. Goldfinger and M. Jeunehomme, Trans. Faraday Soc., 59, 2851 (1963). We assume that entropy contributions due to nuclear spin are generally the same in the liquid and the gas, even when intramolecular changes occur. Likewise, isotopic multiplicity usually adds equal entropy to the liquid and gas (see ref IO, however). Reference 1, Chapters 5 and 10. Although a molecule such as Te, will have a symmetry number, u, of 1 if the two atoms are of different isotopes (ref 9), the small correction caused by this effect is cancelled by an isotopic multiplicity term. I f u is taken as 2 for all homomolecules, the isotopic terms automatically cancel. R. H. Fowler and E. A. Guggenheim, ''Statistical Thermodynamics", MacMillan, New York, 1939, Chapter 3. F. A. Cotton and G. Wilkinson, "Advanced Inorganic Chemistry", 3rd ed, Interscience, New York, 1972, Chapter 15. The same result can be derived more formally by a simple extension of equations given in ref 11. D. R. Stull and G. C. Sinke, Adv. Chem. Ser., No. 18 (1956). M. Ilegems and G. L. Pearson, Annu. Rev. Mater. Sci., 5 (1975). (a) A. S. Jordan, Met. Trans., 1, 239 (1970); (b) J. Steininger, A. J. Strauss, and R. F. Brebrick, d. Nectrochem. Soc., 117 1305 (1970). M. R. Lorentz in "Physics and Chemistry of 11-VI Compounds", M. Aven and J. S. Prener, Ed., North-Holland Publishing Co., Amsterdam, 1967, Chapter 2. We make the reasonable assumption that if the liquid is dissociated (or associated) V, is obtained from eq 7 by substituting the volume which would be occupied at T, and 1 atm by 1 (or 2) mol of gas respectively. D. E. Gray, Ed., "American Institute of Physics Handbook", 3rd ed, McGraw-Hili, New York, 1972. (a) B. M. Kulwicki, Ph.D. Thesis, University of Michigan, Ann Arbor, 1963, p 49; (b) N. K.Abrikosov, "Semiconducting 11-VI, IV-VI and V-VI Compounds", Plenum Press, New York, 1969, p 104. The values
Solubility of Gases in Liquids
(21) (22)
(23) (24) (25) (26)
The Journal of Physical Chemistry, Vo/. 82, No. 26, 1978
of AH(fusion) were determined at the melting points, T,. The difference between T , and T, is small enough that the variation of &/(fusion) with temperature can be neglected. W. T. Lee and Z. A. Munir, J . Nectrochem. Soc., 114 1236 (1967). R. A. Reynolds, D. G. Stroud, and D. A. Stevenson, J . Electrochem. Soc., 114 1281 (1967). A. G. Sigai and H. Wiedemeier, J. Electrochem. Soc., 119, 910 (1972). L. Torpor, J . Chem. Thermodyn., 4, 739 (1972). See, for example, (a) L. V. Woodcock, Chem. Phys. Lett., 10, 257 (1971); (b) J. W. E. Lewis, K. Singer, and L. V. Woodcock, J . Chem. Soc., Faraday Trans. 2, 71, 301 (1975). It is probably reasonable to assume here that each ion is on average surrounded by ions of the opposite charge. See section 3 and ref
2801
25. (27) For most of the alkali halides, only -60-70% of the total mass in the vapor is monomeric (e.g., KCI), the rest being largely dimeric (e.g., [KCI],); for the sodium salts the proportion of monomer is around 35-50% (see ref 24). The bH(per gram formula weight) for boiling to the dimer is probably just about ' / 2 of that for boiling to the monomer (ref 28); this means that the slopes of the In p vs. T-' line will be similar for either process. Thus literature (e.g., ref 3) values of AH, which were presumably deduced from vapor pressure data assuming a monomeric vapor, will still be a good estimate of the true A H f o r boiling to the monomer. (28) See, for example, I. G. Murgulescu and L. Torpor, Rev. Roum. Chim., 13, 1109 (1968); 15, 997 (1970).
Solubility of Gases in Liquids from the First-Order Perturbation Theory of Convex Molecules Tomas Boublikt and Benjamin C.-Y. Lu" Department of Chemical Engineering, University of Ottawa, Ontario, Canada K I N 6N5 (Received June 19, 1978; Revised Manuscript Received October 2, 1978) Publication costs assisted by the University of Ottawa
The perturbation theory of convex-molecule systems was applied to determine the Henry's law constant, KH, partial molar volume, V2,and heat of solution, m2, at infinite dilution of Ar, CH4,N2,02,and C 0 2 in benzene and in CCl, at 298.15 K. The Kihara acentric pair potential was used to characterize the intermolecular interactions of the given molecules, hard cores of which were assumed to be points in the case of Ar and CH4, rods in the case of N P ,02,and C02,a hexagon for benzene, and a tetrahedron for CC14. The method of separation of the predominantly repulsive-forcesregion as that proposed by Barker and Henderson was followed to define the reference system. A simple approximation of the dependence of the average correlation function on the shortest surface-to-surface distance, based on Monte Carlo data, enabled the determination of the first-order perturbation term. The calculated values of KH and V 2agree well with the experimental values available in the literature, while the AH, values agree only qualitatively.
Introduction In recent years, we have witnessed an increasing interest in the solubilities of gases in liquids. As a result, several reviews and original papers have appeared dealing with the solubility both from the standpoint of the experimental determination and the theoretical calculation. The first attempt to estimate the solubility of simple fluids within the framework of modern theories, which employ distribution functions to the characterization of the system structure, was connected with the derivation of a hard sphere equation of state from the scaled particle theory (SPT)by Reiss et a1.l However, in their application of the derived equation to gas solubility, the effect of attractive forces was neglected; Pierotti2 improved this approach by adding an attractive contribution term in a physically intuitive way. Wilhelm and Battino3 employed this intuitive method too (where a solution of a solute molecule in a solvent is considered as a two-step process, the creation of a cavity for the solute molecule in the solvent and the introduction of the solute molecule into the cavity plus the full coupling of intermolecular forces) and in addition to dispersion forces, they also considered interactions of dipole moments etc. Several oversimplifying approximations were introduced within this approach; thus, the success in its application to a variety of systems, including that with water (see Wilhelm et aL4),obviously results from I n s t i t u t e of Chemical Process Fundamentals, Czechoslovak Academy of Science, Prague, Czechoslovakia.
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the fortuitious cancellation of errors. The use of experimental values of solvent volumes, compressibilities, and thermal expansion coefficients seem to be inevitable with this method. Snider and Herringtons employed a modified van der Waals equation of state, in which the repulsive SPT term was combined with the vdW attractive term, for both the determination of the excess functions and the Henry's law constant. An exact treatment of the solubility of simple fluids (interacting via the Lennard-Jones pair potential) was given by Neff and McQuarrie6 who derived expressions for the Henry's law constant, KH,the partial molar volume of solute, V2,and the heat of solution, AHz, from the Lennard-Henderson-Barker (LBH) theory of simple fluid mixtures,' with use of the Percus-Yevick equation of state (P-Y(c)) and radial distribution functions. Derived relations were successfully applied to describe the behavior of the system neon in argon. Goldmans used the same expressions as Neff and McQuarrie for the contribution of attractive forces but he also used an improved hard-sphere term as obtained from the extended Carnahan-Starling equation of state. In addition, he considered quantum corrections for the systems Hz in N2,Ar, and CH4,He in N2, Ar, and CH4, and Ne in N2 and Ar. A similar approach was followed in his application of the variational method to the interpretation of gas solubilities for the same binary r n i x t ~ r e s .In ~ addition, Goldmanlo employed the Weeks-Chandler-Andersen perturbation theory of simple fluidsll as extended to mixtures by Lee and LevesquelZand applied it to gas
0 1978 American
Chemical Society
