J . Phys. Chem. 1985,89, 5738-5743
4
2
17 0
I
22
I
I
I
I
,
1
t 6
I
I
I
I
I
8
IO
12
14
16
I
18
I
20
i7qa”], IO3 c m l
Figure 8. Plots of energy location of absorption maximum of retinal radical anion vs. that of solvated electron at (A) room temperature and (B) 77 K. For numbers identifying the solvents, see Table I. The arrows with some of the points indicate that the true energies for e; absorption maxima are possibly lower in these cases.
acid cation in hexane ,A,(, = 590 nm),32acetone ,A,(, = 575 = 585 nm). Furthermore, nm),8cand 1,2-dichloroethane,,A,( in micelles containing CTAB, while the Ama;s of retinal anion are significantly blue-shifted (as in alcohols, see later), those of the cation appear to be located in the same long-wavelength region (580-600 nm) as in nonprotic solvents (Table I). Figure 7 shows retinal anion maximum as a function of solvent dielectric constant. Clearly, relative to nonpolar solvents, the polar-protic ones (althan the polarcohols) cause much larger blue shifts in, , A, nonprotic ones. Within the alcohols, a systematic variation is indicated for, , A, with respect to polarity (dielectric constant). The interactions of the ground and excited states of retinal anion with solvents are expected to be similar to those of solvated electron. Figure 8 shows the plots of energies corresponding to
retinal anion maxima against those corresponding to solvated electron at room temperature and 77 K. Reasonably good correlation is suggested by either of the plots. This is reminiscent of a similar comparison33between absorption maxima of e; and of iodide ion (charge-transfer-to-solvent band system) in ethylenediamine and concentrated aqueous electrolytes. Some specific points in this context are as follows. (i) Me,SO, the polar-nonprotic solvent with high dielectric constant, proves to be weakly solvating for both solvated electron and retinal anion. (ii) Although the primary anionic species seen in the course of pulse radiolysis in acetonitrile at room temperature is recogni~edl’~ to be the solvent anion (CH3CN--) rather than e;, its, , ,A correlates very well with that of retinal anion. (iii) In contrast, the, , ,A (650 nm) assignedZoto e; in DMF at room temperature deviates significantly from the general parallelism (Figure SA) in Amax’s of the two anionic systems. The Amax’sof retinal anion in the aqueous micelles are all in the short-wavelength spectral region (448-460 nm, Table 11). Rather unambiguously, this establishes the polar/hydrogenbonding, alcohol-like nature of the micellar interiors where the substrate is located. In the cationic micelles, e.g., those consisting of CTAB, the ion pairing of retinal anion with cationic head groups The magnitudes of these can possibly cause blue shifts in.,,A, shifts would be smallMin a relatively polar environment. However, the ion-pairing interaction can explain the small progressive red shifts on going from CTAB to mixed CTAB Triton X-100 to Triton X-100 micelles. Interestingly, the retinal anion decay rate representing its protonation is the slowest in the CTAB micelle. This is not expected if the polyenal anion is strongly ion-paired with the cationic head groups, because the protonation reactivity is usually enhanced upon cation pairing as observed with several carbanionsllCand aromatic radical anions.35 Registry No. 2, 98921-61-0; 4, 83868-27-3; C6Hll+,22499-63-4; all-trans-retinal, 116-31-4; retinal radical anion, 34504-14-8; retinal radical cation, 67529-90-2;cyclohexane, 110-8277,
+
(33) Anbar, M.; Hart, E. J. J . Phys. Chem. 1965.69, 1244-1247. (34) In tetrahydrofuran, the hypsochromic shifts in ,,A upon the pairing of retinal anion with tetraalkylarnmonium ions are typically 20-40 nm.14 (35) Bank, S.; Bockrath, B. J . Am. Chem. SOC.1971, 93,430-437; 1972, 94, 6076-6083.
Solvation Therrnodynamlcs of Inert-Gas Molecules in I nert-Gas Liquids A. Ben-Naimt Department of Physical Chemistry, The Hebrew University of Jerusalem, Jerusalem 91 904, Israel (Received: December 18, 1985; In Final Form: May 17. 1985)
The solvation process as defined and discussed in previous articles is applied to inert-gas liquids. The thermodynamics of the solvation process are computed for the following systems: pure inert-gas liquids above the triple points, including neon, argon, krypton, and xenon; mixtures of inert-gas liquids, argon-krypton and krypton-xenon.
1. Introduction Recently a new approach to the study of solvation thermodynamics has been suggested and The main novelty of this approach consists of a new definition of the solvation process and the corresponding thermodynamic quantities. These quantities reflect the change that occurs in the surroundings of a given molecule when it is placed in a fluid. The fluid may be a “solvent”, in the conventional sense of this concept, or may be a liquid consisting of the same species as the “solute” molecule. Thus we Present address: NIH, Bldg 10, Room 4B-56,Bethesda, MD 20892.
0022-3654/85/2089-5738$01.50/0
may be speaking about the solvation of “argon in xenon” as well as the solvation of “argon in argon”. The latter is essentially a new concept that broadens the range of systems studied under the term of ”solvation”. Furthermore, the solvation quantities as defined in this and in previous papers, truly reflect the changes in the environment of the molecule being transferred from one phase to another. (1) Ben-Naim, A.; Marcus, Y . J . Chem. Phys. 1984, 80, 4438. (2) Ben-Naim, A,; Marcus, Y. J . Chem. Phys. 1984, 81, 2016. (3) Ben-Naim, A. J . Phys. Chem. 1985, 89, 3791.
0 1985 American Chemical Society
The Journal of Physical Chemistry, Vol. 89, No. 26, 1985 5739
Solvation Thermodynamics of Inert-Gas Molecules
TABLE I: Values of AAG,,*/kJ mol-’ as a Function of T/K for the Inert Gases along the Saturation Curve” helium-3 neon argon krypton T AAGa* T AAGa* T AAG,* T AAGA* 1 .o
1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.1 3.2 3.3 3.33
-0.041 15 -0.044 10 -0.046 11 -0.047 36 -0.047 78 -0.047 44 -0.046 3 1 -0.044 37 -0.041 49 -0.037 47 -0.03 1 91 -0.028 28 -0.023 75 -0.017 66
24.55 25 26 27 28 30 34 38 42 43
-1.1435 -1.1419 -1.1226 -1.0999 -1.0762 -1.0187 -0.8918 -0.7288 -0.4889 -0.4084
83.78 84 86 88 90 95 100 1I O 120 130 140 148
-4.0849 -4.0774 -4.0128 -3.9569 -3.8507 -3.7 133 -3.5640 -3.3206 -2.9889 -2.5041 -1.9305 -1.1854
116 118 120 122 124 130 140 160 180 200 208
xenon
-5.7326 -5.6698 -5.6120 -5.5682 -5.5 166 -5.3201 -5.0074 -4.3799 -3.5227 -2.3185 -1.0582
T
AAGA*
161.4 200 210 220 230 240 250 260 270 280 285
-7.9906 -6.5985 -6.4464 -6.1414 -5.7804 -5.3553 -4.9158 -4.4143 -3.7560 -2.8456 -2.2025
0.0
Based on data from ref 5.
This paper is concerned with the solvation quantities of inert-gas “solutes” in inert-gas “solvents”. In the next section we present a brief outline of the theoretical basis. Section 3 summarizes our calculations for one-component systems, e.g., solvation of argon in argon, for which the most complete set of data is available. In section 4 we present some further results on two-component systems: argon-krypton and krypton-xenon. Section 5 concludes with some comments on the significance of the results obtained in this work.
in (2.4); both may be obtained from simple experimental data. Once we have evaluated the solvation free energy AGA*, we may obtain all the other thermodynamic quantities of solvation by standard procedures, i.e.
2. Theory In this section we define briefly the quantities we shall be dealing with in subsequent sections. A more complete representation may be found The cornerstone of the present approach is the expression for the chemical potential of a “solute” A in a liquid 1, which may be written as
Of course, if the data is of sufficient accuracy, one can take also second derivatives of the free energy to obtain the solvation contribution to the heat capacity, isothermal compressibility, etc. The free energy of solvation may be obtained from the densities of A in the liquid (1) and the gaseous (8) phase along the saturation (on the orthobaric) curve. Applying eq 2.1 to the two phases at equilibrium, we have
where we have assumed that A lacks any internal degrees of freedom, otherwise a factor qA, the internal partition function of A, should be included in (2.1). The two terms in (2.1) correspond to two “steps” by means of which we introduce an A molecule into 1. First we place A at a fixed position in 1. The corresponding work is denoted by W(AI1). If the process is carried out at a given temperature T and pressure P, this work will be equal to the Gibbs free energy of the process. We shall therefore refer to this quantity as the solvation (Gibbs) free energy of A in 1, and denote it by The second term on the right-hand side has been referred to as “the liberation free en erg^".^ It involves the (number) density of A, p A = NA/Vand the momentum partition function d ~This ~ . quantity arises when we release the molecule A from the fixed position and let it wander in the entire volume V. In this study we focus attention only on the first quantity (2.2) and all the derived thermodynamical quantities. If the liquid 1 consists of a mixture of A and B, we shall use the following notation for the two limiting cases: AGA*O = lim W(AI1) = W(AIB)
(2.3)
PA4
AGA*P = lim W(AI1) = W(AIA)
(2.4)
PA-PAP
The first, (2.3), coincides with the conventional standard free energy of solution of A in B. This has been the only quantity that was amenable to evaluation from thermodynamic measurements. The new quantities are the general solvation free energy as defined in (2.2) and the particular limiting case of A in pure A, as defined (4) Ben-Naim, A. “Hydrophobic Interactions”; Plenum Press: New York,
1980.
M A * = -dAGA*/aT AHA* = AGA*
+ TMA*
AVA* = aAGA*/dP
(2.5) (2.6) (2.7)
Hence,
AAGA* = “(All) - W(Alg) = kT In
( ~ A g / p A l ). ~
(2.10)
If the density of the gas is low enough, so that interactions between molecules may be neglected, we may put W(Alg) = 0, and relation 2.10 will give the solvation free energy of A in A. However, as we proceed along the equilibrium curve the density of the gas increases and we can no longer neglect W(A(g). In these cases all we can obtain from the equilibrium densities is the differences in the solvation free energies of A in the two phases.
3. One-Component Systems The densities of the liquid and gaseous phases along the equilibrium line are available for helium-3, neon, argon, krypton, and These data may be used directly in eq 2.10 to obtain AAGA*. Table I reports these quantities for all the five gases above their triple points. Note that AAGA* reduces to AGA*P at the lower temperature for which the density of the gas is very low. More recent P, V, T measurements on krypton and xenon were carried out by Theeuwes and Bearmam8 Here we have the densities of the liquid phase and the total pressure of the system, (5) Cook, G. A,, Ed. “Argon, Helium and the Rare Gases’’; Interscience: New York, 1961; Vol. I. (6) Fastovskii, V. G.; Rovinskii, A. E.; V. Petrovskii, Yu.“Inert Gases”; Translated from Russian by J. Schmorak, IPST Cat. # 1876, Jerusalem, 1967. (7) Rowlinson, J. S.; Swinton, F. L. “Liquids and Liquid Mixtures”; Butterworths: London, 1982. (8) Theeuwes, F.; Bearman, R. J. J . Chem. Thermodyn. 1970, 2, 171, 179, 501, 507, 513.
5740 The Journal of Physical Chemistry, Vol. 89, No. 26, 1985
Ben-Naim
TABLE 11: Various Factors Involved in the Computation of ASA* for Argon and Methane"
T/K
AGA*/
pg/(mol/cm3) p'/(mol ~ m - ~ (kJ ) mol-')
hAGA*/AT/ (J mol-' K-I) Argon
83.80 85 87.28 90 95 100 105 110 120
9.889 X 1.1179 X 1.396 X 1.7882 X 2.7057 X 3.9056 X 5.4242 X 7.2881 X 1.2159 X
0.035 40 0.03522 0.034 85 0.03443 0.03364 0.031 82 0.031 96 0.031 04 0.02904
-4.0974 -4.0656 -4.0058 -3.9363 -3.8096 -3.6842 -3.5585 -3.43 13 -3.1661
26.5 26.228 25.551 25.34 25.08 25.14 25.44 26.52 29.42
90.68 100 110 11 1.631 120 130 140 150
1.5546 X IOe5 4.1474 X lo-' 9.6653 X 1.0917 X 1.9235 X 3.4049 X 5.515 x 10-4 8.3396 X
0.028 13 0.027 36 0.026 48 0.026 34 0.025 55 0.024 57 0.023 50 0.022 33
-5.6551 -5.3975 -5.1336 -5.0915 -4.8781 -4.6249 -4.3 6 7 8 -4.1001
27.6394 26.3900 25.8 120 25.4958 2 5.3200 25.7100 26.7700 29.0200
kT&'I
AVA*/
AVA*(dP/dT)/ J mol-' K-'
MA*/
(cm3mol-')
(cm3mol-')
(J mol-' K-I)
1.382 1.437 1.547 1.698 2.057 2.554 3.222 4.097 7.132
26.858 26.953 27.143 27.342 27.663 27.916 28.068 28.113 27.298
0.2156 0.2409 0.2937 0.3664 0.5256 0.7230 0.9571 1.2286 1.8345
-26.284 -25.987 -25.257 -24.974 -24.554 -24.417 -24.483 -25.291 -27.586
1.08 1.413 1.91 1 2.012 2.633 3.737 5.476 8.478
34.47 35.137 35.849 35.958 36.497 36.963 37.064 36.302
0.052 05 0.126 15 0.269 24 0.300 26 0.495 63 0.81692 1.23060 1.706 20
-27.587 -26.264 -25.543 -25.196 -24.824 -24.893 -25.539 -27.3 14
Methane
"Based on data from ref 7. TABLE 111: Values of ACA*/kJ mol-', TAsA*/kJ mol-', AHA*/kJ mol-', and A V,*/cm3 mol-' for Neon, Argon, Krypton, Xenon, and Methane Near Their Corresponding Triple Points" T,,IK AGa* TLiSA* AHA* AVA*
neon argon krypton xenon
methane
24.55 83.80 115.95 161.3 90.68
-1.144 -4.097 -5.733 -7.991 -5.6551
(-0.44) -2.202 (-2.66) (-3.94) -2.502
(-1.58) -6.299 (-8.53) (-11.89) -8.157
(16.18) 26.86 (34.22) (44.29) 34.47
"Approximate values, computed as described in section 3 and Appendix A, are included in parentheses. to T a t constant pressure. The relation between the required and the available derivatives is
Figure 1. Values of AAGA*P/kJmol-' as a function of (T - T,,)/K for the inert gases in their own liquids. The triple points for neon, argon, krypton, and xenon were takenSas 24.55, 83.78, 115.95, and 161.3 K, respectively. The full points correspond to computations based on eq 2.10 using data from ref 5. The open circles correspond to computations based on eq 3.1 using data from ref 7 and 8.
given as an analytical function of the absolute temperature T. In order to use these data, we must convert the vapor pressure into gas densities. We assume that the gas behaves as ideal gas at or below 1 atm. Hence the solvation free energy may be obtained from AGA*P = k T In (P/kTp,')
(3.1)
In Figure 1 we plot values of AAGA* as a function of T - T,,, where T,, is the triple point temperature for neon through xenon. The curves were computed from the densities by using eq 2.10 whereas the straight lines are computed from the more recent data8 but involve the assumption of ideality of the gaseous phase. Clearly the two sets of computed values of AAGA*merge at low pressures (as T - Tt,). At higher temperatures only the curves based on eq 2.10 truly represent the variation of AAGA* as a function of T. Note also that all the curves must approach AAGA* = 0 at the critical temperature, at which the densities of the two phases coincide. In order to compute the entropy of solvation and thereafter the enthalpy of solvation, we need the derivative of AGA* with respect
-ASA*
+ AVA*(
g)
(3.2)
eq
The straight differentiations in eq 3.2 are along the equilibrium curve. Both of them can be computed from the experimental data, provided we can transform between densities and vapor pressure. A detailed evaluation of all the factors in eq 3.2 is possible only for argon, for which a complete set of data is available. Table I1 shows the various factors for argon at temperatures close to the triple point (we added the data for methane for comparison). AVA* may be evaluated from AVA* = V,' - k T p 2
(3.3)
where VA1is the molar volume of A and p; is the isothermal compressibility of the liquid (see Appendix A). The full computational procedure is as follows. First, we evaluate AGA* from eq 3.3, then use this value together with dP/dT to estimate ASA* by eq 3.2, and finally AHA* is computed from AHA* = T M A * AGA*. The full procedure may be carried out only for argon (and for comparison, also for methane). However, we note that the term kTB2 is quite small compared with VA' (e.g., for argon, at T,,, kT&' is about 5% of VA'). Furthermore, the term AVA* (dP/dT) in eq 3.2 is almost negligible compared with dAGA*/dT (e.g., for argon, near T,,, it is about 0.8%, for methane, about 0.2% of the MA* value). We may therefore conclude that ASA* may be approximated by the straight derivative in eq 3.2. The values of ASA* and AHA* for all the gases near their triple point are reported in Table 111. The figures in brackets are approximate values, involving the neglect of the second term on the right-hand side of either eq 3.2 or of eq 3.3.
+
The Journal of Physical Chemistry, Vol. 89, No. 26, 1985 5741
Solvation Thermodynamics of Inert-Gas Molecules TABLE I V Values of AAGA*, TAASA*, and AAHA* (all in J mol-’) for Argon in Mixtures of Argon and Krypton at T = 115.77 K XAa AAGA* TAASA* AAHA* 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.o
342.6 277.8 219.1 166.6 120.7 81.5 49.5 24.9 8.2 -0.2 0.0
143.7 68.6 22.3 -2.5 -12.3 -12.5 -7.9 -2.0 2.3 3.3
0
486.3 346.2 241.4 164.1 108.4 69.0 41.6 22.9 10.5 3.1 0
* XA is the mole fraction of argon.
4. Solvation in Binary Systems Once we have a complete set of thermodynamic quantities of solvation of A in A, it is of interest to follow the change in these quantities as the “solvent” changes gradually from pure A to pure B. Such information may be obtained from either activity coefficients or excess functions. Some details on the transformation from the available data into solvation quantities are discussed in Appendix B. Two systems were examined in the present work: argonkrypton99l0and krypton-xenon.” The excess free energy and the excess volume as a function of the mole fraction are available in an analytical form as
g“ = XAXBRT[A
+ B(XA - XB) + C(XA - XB)’]
I
-1001 0
I 1
0.25
I
0.5
I
0 75
I
I .o
XA I
Figure 2. Values of AAGA*/J mol-’ for argon and for krypton as a function of the mole fraction of argon at T = 115.77 K. Based on data from Chui and Canfield.’O 600,
(4.1)
and similarity for the excess volume DE = xAXB[A’
+ B’ (XA - XB) + c’ (XA - XB)2]
(4.2)
For the argon-krypton system the coefficients in (4.1) are available at two temperatures (103.94 and 115.77 K). On the other hand, the coefficients for uE in (4.2) are available at only one temperature. These data are sufficient to compute an approximate set of thermodynamic quantities for the argon-krypton system in the entire range of compositions. The entropy and the enthalpy of solvation, in the mixtures relative to the pure liquid, may be computed from the following relations:
+ g“ + XB(a?/aXA)
AAGA* = k T In (PA~XA/PA)
A a A * = -a(AAG*)/dT AAHA* = AAGA*
+ TAUA*
(4.3) (4.4) (4.5)
In the numerical evaluation of A M , * we use the procedure outlined in Appendix B to evaluate the quantity pApXA/pA. Since this is available at only one temperature, we assume that the temperature coefficient of the volume of the mixture is approximately the same as that of the pure liquids. With this assumption, we computed both A M A * and AAHA* for argon in the argonkrypton system. The results are given in Table IV. Also, with a slight modification of eq 4.3, we may evaluate the values of AAGB* (Le., for krypton) relative to pure krypton. The pertinent relation is Figure 2 shows the variation of both AAGA* (argon) and AAGB* (krypton) as a function of the mole fraction of argon in the entire range of compositions. Similarly, we present in Figure 3 the corresponding quantities of solvation for the krypton-xenon system at T = 161.38 K. Since all the data are available at one temperature, we could not evaluate the corresponding entropies and enthalpies of solvation in these mixtures.
0 .
0.25
0.5
0.75
I .o
XK,
Figure 3. Values of AAGA*/J mol-’ for krypton and for xenon as a function of the mole fraction of krypton at T = 161.38 K. Based on data from Chui and Canfield.Io
5. Discussion and Conclusion In Table I11 we have summarized the solvation quantities for the various “solutes” near their triple points. We noted that at these temperatures, since the vapor pressure is quite low, we could safely assume that AAGA* = AGA*P, Le., the solvation free energy of A in its pure liquid. A similar conclusion holds true for all the other quantities of solvation. We also note that the enthalpy of solvation is by far the dominating quantity in the formation of AGA*. This is a typical result for the solvation of a simple solute in nonaqueous solvents. The contrary holds true for the solvation of, e.g., argon in water,12where T a A * is the dominating quantity. Furthermore, assuming that AHA* is practically equal to the (Le., neglecting PAVA* compared with energy of solvation MA* AHA*I2),we may interpret values of AHA* in terms of the average “binding energy” of A to its surroundings’* AEA* = j / 2 ( B ~ )
(5.1)
(9) Davies, R. H.; Duncan, A. G.; Saville, G.; Staveley, L. A. K. Trans.
where (BA) is the ensemble average interaction energy of a single
(IO) Chui, C. H.; Canfield, F. B. Trans. Faraday SOC.1971, 67, 2933. (1 1 ) Calado, J. C. G.; Staveley, L. A. K. Trans. Faraday SOC.1971,67, 289.
(12) Ben-Naim, A. “Water and Aqueous Solutions”; Plenum Press: New York 1974.
Faraday SOC.1961, 63, 855.
5742
The Journal of Physical Chemistry, Vol. 89, No. 26, 1985
TABLE V Lennard-Jones P~rameters'~ and Average Coordination (CN) of the Inert Gases in Their Liauids at the TriDle Points Ne
0.290 0.996 1.422 1.837 1.232
Ar
Kr Xe CH,
10.9 12.6 12.0 12.9 13.2
A molecule with all the other molecules in the system. This, in turn, may be used to estimate the average coordination number of A in the liquids, near the triple points. To do that we take the values of eLJ, the Lennard-Jones parameters for the inert gases, as determined from the second virial coefficients,13 and assume the equality (BA) = - ( C N ) ~ L J
(5.2)
From (5.1) and (5.2) approximate average coordination numbers may be derived. These are shown in Table V. All the values of ( C N ) fall between 11 and 13, which is quite consistent with the expected close packing of the spherical molecules at the triple point. It is interesting to note that the solvation free energies of all the solutes that were studied in this paper, decrease (in their absolute values) upon replacing the pure liquid by a mixture of two components. This is quite a surprising finding, since the argon-krypton interaction must be somewhere in between the argon-argon and krypton-krypton interactions. Thus, starting with AGA* of argon in argon, we would have expected that adding krypton should increase AGA* (in its absolute magnitude) if only the solute-solvent interaction would have been taken into account. However, even for the solvation energy we see from (5.1) and (5.2) that both tu and (CN) combine to determine the values of AHA* or AE,*. The situation is far more complicated with the free energy of solvation. Here we need to take into account not only the interaction of A with its surroundings, but also the entropy change due to the solvation process. The latter, however, cannot be easily estimated in terms of interaction parameters and average coordination number. Finally, we would like to comment on the well-known Trouton's law in light of the solvation entropy as studied in this paper. In Appendix A we have derived a general expression for the entropy of condensation, Le., the entropy change upon transferring an A molecule from the gaseous phase into the liquid. This read &S,,d
=S A *
+ k In ( p ~ ~ / p A-] )k ~+ k T a i
(5.3)
The four factors involved in (5.3) may be evaluated at the normal boiling point. For argon and for methane, the values are in the same order as they appear in (5.3): for argon at Tb = 87.3 K hs,,d/J
mol-' K-l = -25.26 - 45.6 - 8.314
+ 3.24 = -75.93 (5.4)
Ben-Naim boiling points of the gases, he suggested looking at the condensation entropies of the gases at temperatures at which the concentration of their vapors are equal. In view of eq 5.3 this suggestion amounts to the choice of a constant value of pAg. Hence, one expects that the variations in the term k In (pAg/pAl) will be less than if measured at the normal boiling points. This indeed was actually observed by Hildebrand.14
Acknowledgment. This work was partially supported by the United States-Israel Binational Science Foundation (Grant No 2240/80) for which I am very grateful. Appendix A
Relation between Conventional Entropy, Enthalpy, and Volume of Transfer and the Corresponding Solvation Quantities. The partial molar entropy of A in the phases 1 and g are obtained from (2.1)
Thus for the transfer of A from the gas into the liquid we have
where a: is the isobaric thermal expansion of the phase I. Practically, we shall apply this equation for the condensation of A from a very rarefied gas (say at the triple point, or even at the normal boiling point) where we may assume that SA*(g) 1
=
av
(A.4) 1
aPg = V dT=T
For this case, at an equilibrium point we have hScond
= ASA*(l)
+ k In ( p A g / ~ A l )+~ kTa;
-k
64.6)
The connection between the condensation enthalpy and the solvation enthalpy is (assuming again the ideality of the gaseous phase)
AHcon* = T&$cond = AHA*(l)
+ kPa,'
- kT
(A.7)
Similarly for the solvation volume we have the relation
for methane at Tb = 111.63 K hscond/J
mol-' K-' = -25.26
- 45.6 - 8.314 -I-3.18 = -75.99 (5.5)
We see that the dominating quantity in (5.3) is the term k In (pAg/pAl). We may conclude that the origin of the constancy of a W n d is essentially due to the constancy of two factors. One is the solvation entropy, which for simple liquids contributes about -25 J mol-' K-I. The second is the change in entropy due to compression of the gas from pAg to pA1. This again contributes a roughly constant value to It is interesting to note that in 191514Hildebrand suggested a modification of Trouton's law. Instead of looking at the entropy of condensation at the normal (13) Hirschfelder, J. 0.;Curtis, C. F.; Bird, R. B. "Molecular Theory of Gases and Liquids", 2nd ed.; Wiley: New York, 1963. (14) Hildebrand, J. H. J Am Chem. SOC.1915, 37, 970
If the liquid is pure A then V,] and 6: are the molar volume and the isothermal compressibility of A, respectively. On the other hand, if 1 is any mixture of components, then V,' is the partial molar volume of A in 1 and 6: is the isothermal compressibility of the entire liquid mixture. Appendix B
Extraction of Solvation Thermodynamics from Activity Coefficients and Excess Thermodynamic Functions. In the following, when we refer either to activity coefficient or to excess function we mean that the reference system is the symmetrical ideal solution; one must be careful when using deviations from other kinds of ideality.'* The symmetrical ideal (SI) solution is characterized by the following expression for the chemical potential of each component
J. Phys. Chem. 1985, 89, 5143-5149 PAP = W(A1A)
i, in the entire range of compositions X i pi
=
pip
+ kT In X i
(B.1)
= p,J'
+ k T In X i + kT In y i = pip + k T In Xi + piE i
(B.2)
(B.3)
or (B.4)
= 8 + XB
