5917
J. Phys. Chem. 1986, 90, 5917-5921
complexed CyS' a t low pH. However, in alkaline solution 12.does oxidize CyS-. By contrast Br2+ was capable of oxidizing cysteine at all pHs (Le., in both protonated and deprotonated forms). Taking E40 = 1.42 V for a typical complex mercaptan, E l 0 = 0.92 V for cysteine calculated from this by using AGS0 = 11.32 kcal mol-' and the values of EZofor Iz' and Br2' in Table 11, one can see that these observations are as predicted. Registry No. HS', 13940-21-1; CH3S', 7175-75-9; C2HSS*,14836(36) Packer, J. E. J . Chem. Soc., Perkin Tram. 2 1984, 1015. Note that overall oxidations proceed through equilibria with p ~ t*h r ~ - e l ~ t r o n - ~ n d ~22-7; C3H7S0, 4985-58-4; CH$SCH3-, 34527-95-2; C2H$SC2HS-, 91603-20-2; C3H7SSC3H7-, 34525-27-4;HSSH-, 91523-24-9; CH,SSCcomplexes such as RS:.I- and RS:.Br-. H3,624-92-0; C~HSSSC~HS, 110-81-6;C&I7SSC3H7,629-19-6; HSSH, (37) Adam, G. E.; Aldrich, J. E.; Bisby, R. H.; Cundall, R. B.;Redpath, J. L.; Wilson, R. L. Radiat. Res. 1972, 49, 218. 13465-07-1.
The more complex and unusual behavior of RSSR- has made it difficult to design equilibria from which potentials for RS' systems can be obtained by experiment. However, data in the recent literature can be explained rather well by the present calcultions. Thus Packer36has confirmed the earlier observation3' that 12'- does not oxidize protonated cysteine (CysSH) to un-
'.*
Thermodynamic Propertles of Empty Lattices of Structure I and Structure I1 Clathrate Hydrated Y. Paul Handa* and John S. Tse Division of Chemistry, National Research Council, Ottawa, Ontario, Canada K l A OR6 (Received: May 19, 1986)
Thermodynamic properties of empty lattices of structure I and structure I1 clathrate hydrates have been obtained by analyzing the thermophysical properties of structure I xenon and structure I1 krypton hydrates in terms of the ideal solid-solution model of van der Waals and Platteeuw. The heat capacities, in the temperature range 100-270 K, of the empty lattices are found to be essentially the same as that of ice. For the process ice empty lattice, the change in free energy is 1287 J mol-' for structure I and 1068 J mol-' for structure 11; the corresponding changes in enthalpy are 931 and 764 J mol-', respectively.
-
Introduction Clathrate hydrates are nonstoichiometric compounds in which water molecules form a host lattice. Most hydrates exist in one of the two cubic structures termed structure I and structure 11. The unit cell in structure I is made up of 46 water molecules and in structure I1 of 136 water molecules. Each structure has small cages and large cages. The small cages in both structures are pentagonal dodecahedra, the large cages in structure I are tetrakaidecahedra consisting of 12 pentagons and two hexagons, and the large cages in structure I1 are hexakaidecahedra consisting of 12 pentagons and four hexagons. Structure I is formed' by molecules with diameters in the range 410-580 pm whereas structure I1 is formed'*2by molecules with diameters less than 410 pm or greater than about 550 pm. Thus xenong and methane2 form structure I hydrate whereas krypton* forms structure I1 hydrate. Table I summarizes some of the structural properties of the hydrates taken from the review article by Davids0n.l Subscripts S and L refer to the small and the large cages, respectively, N is the number of cages per unit cell, v is the number of cages per water molecule, z is the number of water molecules forming a cage, and a is the average radius of the cage determining the space available to the guest molecule. The free radius of a cage is approximately given by its average radius minus the van der Waals radius of a water molecule. Guest molecules with radii close to or slightly larger than the free radius of a cage can be enclathrated but a t the expense of distorting the cage. The empty lattice of either clathrate hydrate structure is unstable and has never been synthesized in the laboratory. The empty lattice serves as a reference state (though a hypothetical one) for the hydrate and a knowledge of its thermophysical properties is important in understanding the lattice dynamics, testing water-water interaction potentials, prediction of the phase equilibria in water (or ice)-hydrate-former systems, and analysis 'Issued as NRCC No. 26156.
TABLE I: Some Structural Prowrties of Clathrate Hvdrates
structure I
Ns NL "s VL 2 6 1/23 3/23
structure I1 16
8 2/17 1/17
2s
ZL
20
24 (20) (21)
as/Pm aL/Pm 391 433 (388) (415)
20 28 390 (20) (28) (387)
468 (470)
-
TABLE 11: Changes in Thermodynamic Properties for the Process Empty Lattice at 273.15 K and 1 bar Ice
Ap(i-e)/ (J mol-I)
structure I
structure I1
1287 1235 1291 1297
AH(i-e)/ (J mol-') 931
1684 1451
AC,(i-e)/
(J K-I mol-') 0.0
0.56 0.65
ref this work 7" 8'
1389
1l b
1123
2gC 31d
1297 1264
1150
1068
764
937
1025
1l b
808
43e 44"
883
44' 0.0
this work
'Values based on analysis of dissociation pressures of hydrates in terms of the ideal solid-solution model and using Kihara potential. Values based on analysis of experimentally determined composition of structure I cyclopropane hydrate in terms of the ideal solid-solution model and using Kihara potential. CValuesbased on sorption isotherms of Xe in the small cavities of structure I1 CHC13 hydrate and dissociation pressures of xenon hydrate. dExperimentalvalue based on Os and Or of Xe determined by using calorimetry and '29Xe NMR. Experimental value based on composition of SF6hydrate derived from dissociation pressures. of the motion of the encaged molecule. In the past, the thermodynamic properties of the empty lattice have usually been
This article not subject to U.S. Copyright. Published 1986 by the American Chemical Society
5918 The Journal of Physical Chemistry, Vol. 90, No. 22, 1986
Handa and Tse
obtained from analysis of phase equilibrium and/or composition results in terms of some form of the ideal solid-solution model for clathrates as originally proposed by van der Waals? The first estimations of the chemical potential difference between ice and the empty lattice of cubic structure I hydrate AN(i--e) was based on bromine hydrate which later was found to form a tetragonal structure.6 Recent estimate^^^^ of Ab(i-+e) and the enthalpy difference between ice and the empty lattice iW(i-e) have been based, among others, on argon and krypton hydrates which however now have been found to form structure II.2 Cyclopropane forms both structure I and structure I1 hydrate^,^,'^ structure I1 being skble within a narrow temperature and pressure region only. The calculations by Dharmawardhana et al." and Holder et a1.12 for structure I were based on composition results and for structure I1 on phase equilibrium results for cyclopropane hydrates. The thermodynamic properties of the empty lattices at 273.15 K and 1 bar reported by various workers are given in Table 11. There is good agreement among the &(+e) values for structure I; the rest of the results show considerable variation. This is partly due to the reasons noted above and partly due to the fact that in order to force an agreement between the model and the experimental results, the energy and the distance parameters for the water-water potential have been treated as adjustable parameters which, sometimes, leads to the use of rather unrealistic values for these parameters. l 3 To a first approximation, the heat capacity C, of an encaged guest molecule can be obtained from
For these systems C, was estimated from eq 1 with the heat capacity of the empty lattice assumed to be the same as that of ice. This is a reasonable approximation in view of the similarity of the three-dimensional hydrogen-bonded network of water molecules in the hydrates and in ice, provided there is no distortion of the lattice by the guest and the guest-host interactions are weak. The assumption is least valid in the case of hydrates of large, polar polyatomic molecules. C, can be obtained from a knowledge of calculated C, and measured C,,. However, in the case of polyatomic guests, C cannot be calculated reliably because of the likely presence of strong coupling between the translational (rattling) and rotational degrees of freedom.19 However, in the case of Xe and Kr hydrates the guests undergo only the rattling motion and are small enough to be encaged without any distortion of the lattice, and the guest-host interactions are relatively weak. Xe and Kr hydrates respectively provide the simplest examples of structure I and structure I1 hydrates for a theoretical evaluation of C,. In this paper, the recent results on the compositions and thermal properties of hydrates of Xe,U' Kr,zoand C H t l are combined with the relevant literature results and analyzed in terms of the ideal solid-solution model without recourse to the use of adjustable parameters. The thermodynamic properties of structure I and structure I1 empty lattices so calculated are compared with previous estimates.
c, = c, - c,
a. Classical Statistical Model. The ideal solid-solution model proposed by van der Waals4 is based on classical statistics. The model assumes that (1) each cage can hold only one guest molecule and that the free energy contribution from an enclathrated guest molecule is independent of the free energy contribution made by the water molecules and by the other enclathrated molecules, and ( 2 ) the guest can rotate freely in its cage. The cell theory for liquids and compressed gases22v23developed by Lennard-Jones and Devonshire is used to evaluate the grand partition function of the system and the partition function of the guest in the cage. A spherically symmetrical Lennard-Jones 12-6 potential is used to model the water-guest interactions. The essential equations of the theory are summarized below; a detailed account has been given by van der Waals and P l a t t e e ~ w . ~For ~ the reaction at temperature T
where ch and C, are the isobaric or isochoric heat capacities of the clathrate and the empty host-lattice, respectively. If there is no coupling between the guest and the lattice modes, no distortion of the lattice by the guest, and negligible interaction between the encaged guest molecules, eq 1 can be considered exact. In the case of c!athrates of hydroq~inone,'~.'~ C, for a number of guests have been obtained from experimental determinations of ch and C,. The evaluation of C, provides a useful technique for studying the motion of an isolated molecule in a well-defined space. However, direct experimental determination of C, of a clathrate hydrate is not possible because of the unstability of the empty lattice and because the clathrate hydrates exist over a very narrow range of composition which renders it impossible to arrive at C, by measuring ch as a function of composition. In the past few years, heat capacities of clathrate hydrates of a number of relatively large, polar polyatomic guests have been reported.I6l8
(1) Davidson, D. W. In Water-A Comprehensive Treatise; Franks, F., Ed.; Plenum: New York, 1973; Vol. 2, Chapter 3. (2) Davidson, D. W.; Handa, Y. P.; Ratcliffe, C. I.; Tse, J. S.; Powell, B. M. Nature (London) 1984, 311, 142. (3) Bertie, J. E.; Jacobs, S. M. J. Chem. Phy,F. 1982, 77, 3230. (4) van der Waals, J. H. Trans. Faraday SOC.1956, 52, 184. (5) Platteeuw, J. C.; van der Waals, J. H. Mol. Phys. 1958, 1, 91. (6) Allen, K. W.; Jeffrey, G. A. J. Chem. Phys. 1963, 38, 2304. (7) Holder, G. D.; Corbin, G.; Papadopoulos, K. D. Ind. Eng. Chem. Fundam. 1980, 19, 282. (8) Barakov, S. P.; Sawin, A. Z.; Tsarev, V. P. Russ. J . Phys. Chem. 1985, 59, 608. (9) Hafemann, D. R.; Miller, S. L. J . Phys. Chem. 1969, 73, 1392. (10) Majid, Y. A.; Garg, S . K.; Davidson, D. W. Can. J . Chem. 1969,47, 4697. (11) Dharmawardhana, P. B.; Parrish, W. R.; Sloan, E. D. Ind. Eng. Chem. Fundam. 1980, 19, 410. (12) Holder, G. D.; Malekar, S. T.; Sloan, E. D. Ind. Eng. Chem. Fundam. 1984, 23, 123. (1 3) Tse, J. S.; Davidson, D. W. Proc. 4th Can. Permafrost Conf., Calgaty, Alberta Canada 1981, 329. (14) Parsonage, N. G.; Staveley, L. A. K. In Inclusion Compounds; Atwood, J. L., Davies, J. E. D., MacNicol, D. D., Eds.; Academic: London, 1984; Vol. 3, Chapter 1 . (15) Parsonage, N. G.; Staveley, L. A. K. In Disorder in Crystals; Clarendon: Oxford, U.K., 1978; Chapter 1 1 . (16) Leaist, D. G.; Murray, J. J.; Post, M. L.; Davidson, D. W. J . Phys. Chem. 1982,86, 4175. (17) Handa, Y. P. J . Chem. Thermodyn. 1985, 17, 201.
Theory
empty cage
+ guest molecule * occupied cage
(2)
the equilibrium constant K (also called the Langmuir constant) is given by Ki = ( 2 x a i 3 g i / k r )exp(-wi(o)/kT)
(3)
where k is the Boltzman constant, i = S,L is the cage type, and g is a dimensionlessfree uolume integral for the guest in the cage and is given by gi = S y i 1 i 2exp(Fi) dy
Z(y) = ( 1
(4)
+ 12y + 2 5 . 2 ~+ ~12y3 + y4)(l -y)-'O m(y) = (1 + y ) ( l - y ) - 4 -
y = r2/a?
1
- 1 (6) (7)
(8)
(18) White, M. A.; MacLean, M. T. J . Phys. Chem. 1985, 89, 1380. (19) van der Waals, J. H. J . Phys. Chem. Solids 1961, 18, 82. (20) Handa, Y. P. J . Chem. Thermodyn., in press. (21) Handa, Y. P. J. Chem. Thermodyn., in press. Curtis, C. F. J . (22) Wentorf, R. H.; Buehler, R. J.; Hirschfelder, J. 0.; Chem. Phys. 1950, 18, 1484. (23 Fowler, R.; Guggenheim, E. A. In Statistical Thermodynamics; University Press: Cambridge, U.K., 1960; Chapter 8. (24) van der Waals, J. H.; Platteeuw, J. C. Adu. Chem. Phys. 1959, 2, 1 .
The Journal of Physical Chemistry, Vol. 90, No. 22, 1986 5919
Thermodynamic Properties of Clathrate Hydrates
= a:/(u321/2)
(9)
where r is the distance from the center of the cage; e and u are the energy and the distance parameters in the Lennard-Jones (LJ) potential function and are given by the usual combination rules = (e,eg)’/2
(10)
= 0.5(uw + as)
(11)
e
u
where the subscripts w and g denote water and the guest, respectively. The potential energy a t the center of the cage wi(o) in eq 3 is given by Wi(0)
= ziE(CYi-4 - 2ai-2)
(12)
and the probability Oi of finding a guest in a cage of type i is given by ei = KJ-/(~ if) (13)
+
where f is the fugacity of the guest species in the gas phase with which the hydrate is in equilibrium. For the hydrate-ice-gas equilibrium, the difference between the chemical potentials of a water molecule in the empty lattice and in ice is given by Ap(i-e)
= pe - pi- = -kT i
q In (1
- Oi)
(14)
The molar change in internal energy for eq 2 when one mole of guest M in the ideal gas state reacts with an empty lattice containing c moles of water to form the hydrate M.cH20 is AU(e+g-h) = U,, - U,- U,
+
-
= CL C ~&zie((l g//gi)~ri-~ 2(1 I
+ gm/gi)ai-2]
(15)
where L is the Avogadro’s number. Differentiating eq 15 with respect to temperature (assuming Oi to be independent of temperature) we get AC,(e+g-h)
= cL
C ~ ~ O ~ ( ( ( q t / T ) ~ a ~ ~X/ ( g ~ k ) )
g m m = Jyi1’2(mCv))2 exp(Fi) d~
(20)
g / m = j’yi1/21Cv) m ~ v exp(Fi) ) d~
(21)
The limits of integration for the g integrals are 0 and 0.25 because the contributions to the integral become negligible at values of y greater than 0.25. Assuming the volume of the hydrate to be the same as that of the empty lattice, the enthalpy change for eq 2 is obtained from AH(e+g-h) = AU - RT (22) and the heat capacity of the guest due to its rattling motion in the cavity is obtained from C, = AC,(e+g-h) + 1.5R (23) where R is the gas constant. In the derivation of eq 23, the rotational and the intramolecular vibrational heat capacities of the guest in the clathrate phase have been taken to be the same as in its ideal gas state as a consequence of the assumption number 2. This is not strictly valid in many clathrate hydrate systems but is entirely justifjed in the case of monatomic species like Xe and Kr. It should be noted that as a consequence of eq 22, both
the isochoric and the isobaric heat capacities of the encqged guest are the same. In other words, in eq 1, the difference between the heat capacities of the hydrate and the empty lattice is the same whether taken at constant volume or constant pressure. b. Quantum Statistical Model. For an oscillator in a potential box, the symmetrical case of the one-dimensional Poschl-Teller potential (as applied to a hydrate cage) is given by25,26 h2Xi(Xi - 1) V(x) = (24) 8mdF sin2 ((?r/di)(x- 0.5di)) where h is the Planck’s constant, m is the mass of the guest molecule, Xi is a measure of the restoring force for small displacements and thus a parameter reflecting the stiffness of the cage-the larger the Xi the more stiff the cage walls, d, is the distance through which the guest molecule can move in the cage and is given by (diameter of the cage - van der Waals diameter of the guest - van der Waals diameter of water), and -0.54 I x I0.5di. The energy levels for the translational or rattling mode for this potential are given by E, = (h2/8mdt)(X, n)2; n = 0, 1, 2, ... (25)
+
where n is the quantum number for the rattling motion. The parameter can be evaluated from eq 25 provided the frequency for the n = 0 n = 1 transition is known for each cage.26 The partition function for the guest in the cage is taken as the cube of the partition function
-
q, = Ce-EnIkT n
(26)
because of the cubical symmetry. The molar heat capacity of the guest in the cage i is obtained from
and, for a hydrate with full cage occupancy, the heat capacity of the encaged guest species is given by
Results and Discussion For use in the ideal solid-solution model, the most extensive information available on a structure I hydrate is for xenon hydrate. For the hydrate-ice-gas equilibrium at 273.1 5 K, the fugacity of xenon is 1.467 bar.27 Barrer and Edge28 measured sorption isotherms of xenon for the small cages of structure I1 CHC13 hydrate in the temperature range 233-268 K. On extrapolation their results give Ks = 1.75 bar-’ at 273.15 K. From 129XeNMR, Ripmeester and D a v i d s ~ obtained n ~ ~ ~ ~Os/BL ~ = 0.75 f 0.03 for the Xe-D20 hydrate prepared under the conditions similar to those for the equilibrium at 273.15 K. A recent 129XeN M R and calorimetric study” on a sample of Xe-H20 hydrate prepared and extensively conditioned under the equilibrium conditions at 273.15 K gave &/eL = 0.73 f 0.02, c = 23/(OS + 30L) = 6.29 f 0.03, Os = 0.7161 f 0.0161, and OL = 0.9809 f 0.0070. The ideal solid-solution model reproduces all these results, within their experimental uncertainties, using the LJ parameters for water of ew/k = 162 K and uw= 305 pm. For krypton hydrate, the same set of LJ parameters also predicted c = 6.10 at 273.15 K based on f = 13.92 bar for the hydrate-ice-gas e q ~ i l i b r i u m . ~This ~ result is in complete agreement with the literature value of 6.10 (25) Neece, G . A.; Poirier, J. C. J . Chem. Phys. 1965, 43, 4282. (26) Burgiel, J. C.; Meyer, H.; Richards, P. L.J . Chem. Phys. 1965, 43, 4291. (27) Aldijk, L.P h B . Thesis, Technical University, Delft, 197 1 . (28) Barrer, R. M.; Edge, A. V. J. Proc. R . SOC.A 1967, 300, 1. (29) Ripmeester, J. A.; Davidson, D. W. J . Mol. Struct. 1981, 75, 67. (30) Davidson, D. W.; Ripmeesrer, J. A. In Inclusion Compounds; Atwood, J. t.,Davies, J. E. D., MacNicol, R. D., Eds.; Academic: London, 1984; Vol. 3, Chapter 3. (31) Davidson, D. W.; Handa, Y. P.; Ripmeester, J. A. J . Phys. Chem., in press. (32) De Forcrand, R. Compr. Rend. 1923, 176, 3 5 5 .
5920
The Journal of Physical Chemistry, Vol. 90, No. 22, 1986
Handa and Tse
TABLE III: Vibrational Frequencies of the Encaged Guests Calculated from the Quantum Statistical Model, QS, and from Molecular Dynamics Simulations, MD
MD
QS guest
Xe
Kr CHI
dvdw/pm
dslpm
dLlPm
440 400 400
42 80 82
126 236 166
w,/cm-' 40 13
75
wL/cm-I
ws/cm-'
29 12 41
47 34 18
wL/cm-' 11, 35 9
35, 54
frequencies derived was a unique one. The vibrational frequencies derived for xenon and krypton are given in Table 111 and compared with those obtained from molecular dynamics (MD) simulations on xenon35 and krypton36 hydrates. The guest-host interactions in MD simulations were calculated by using the LJ 12-6 potential and the same LJ parameters as used in this work. The MD simulations were done on systems with 8, = 1. Also given in Table I11 are the van der Waals diameters, dvdw, and di's used for xenon and krypton. The vdw diameter of water was taken to be equal to the vdw diameter of the oxygen atom,37300 pm, since the hydrogen atoms of the water molecule do not point toward the cage center. The agreement in vibrational frequencies for xenon is quite good. For krypton, the agreement in vibrational frequencies for the large cage is quite good but is poor for the small cage. Within the context of a given potential model, the predictions of MD simulations are exact. The small cage provides a more severe test of the choice of the potential model than the large cage because of the stronger repulsive interaction experienced by the guest. AH(i-e) = AH(i+g-+h) - AH(e+g-+h) (29) Moreover, the LJ parameters for water used in the classical model were obtained from results on structure I hydrate and then aswhere -AH(i+g-+h) is the enthalpy of dissociation of the hydrate sumed to be applicable to structure 11. We take the overall into ice and gas. For structure I, the results are based on calagreement of the frequencies to be satisfactory and thereby the orimetric value of AH(i+g+h) = -26.50 kJ mol-' for Xe.5.90H20 C, for xenon and krypton calculated from the classical model to reported previously20and AH(e+g+h) calculated from the ideal represent the heat capacity contributions to their hydrates. solid-solution model as discussed below. For the Xe.5.90H20 The molar heat capacity contributions C, from xenon and sample on which calorimetric measurements were made, lz9Xe krypton to their hydrates of composition Xe.5.90H20 and KP NMR gave 8J8, = 0.90 and thus BS = 0.8996 and eL = 0.9996. 6.10H20 were calculated from the heat capacities of the guest The molar enthalpies of encagement of Xe in each cage type were in the two kinds of cages weighted according to 0, in the manner calculated from eq 15 and 22 assuming Bi = 1 and found to be described above, eq 30, for AH(e+g+h). In the temperature AHs = -30.219 kJ mol-' and AHL= -32.528 kJ mol-'. These range 100-270 K, these values are of the order of 2.5R and 2.3R were then weighted according to the cage occupancies to obtain for xenon and krypton, respectively, and decrease slightly with AH(e+g-+h) from temperature. The isobaric heat capacities of the structure I and AH(e+g-+h) = zNi8,AHi/CNifli structure I1 empty lattices, in the temperature range 100-270 K, i i were obtained from eq 1 by subtracting the C, of the guest from the isobaric heat capacities of Xe.5.90H20 and Kr.6.10H20 reFor structure 11, the results are based on the calorimetric value ported elsewhere." The heat capacities of structure I and structure of AH(i+g+h) = -19.54 kJ mol-' for Kr.6.10H20 reported I1 empty lattices were found, within experimental errors, to be previously20and M(e+g-.h) calculated from eq 30. For Kr, we the same as that of ice reported p r e v i o ~ s l y . ~Thus ~ ACp(i-e) calculated the equilibrium cage occupancies as BS = 0.9233 and r 0 for both structures over the temperature range 100-270 K. BL = 0.9425 and molar enthalpies of encagement as AHs = As seen in Table 11, the present result for structure I is slightly -26.033 kJ mol-' and AHL = -20.606 kJ mol-'. Previous applications of the c l a ~ s i c a l 'and ~ * ~the ~ q u a n t ~ m ~ ~ *different ~ ~ from the values suggested in the literature whereas that for structure I1 supports the widely used that statistical models to predict heat capacities of monatomic guests ACJi-e) = 0. in hydroquinone clathrates showed that the two models work In order to test our conclusion on the C, of structure I empty remarkably well, especially for temperatures greater than 100 K. lattice, we analyzed the recently reported2' heat capacities of There are no infrared or Raman spectra available to give the methane hydrate in the range 100-260 K. Subtracting the heat vibrational frequencies of xenon and of krypton in the hydrate capacity of ice from the heat capacity of the hydrate, we get the cages. Thus the heat capacity contribution from encaged xenon contribution from CH4to the hydrate heat capacity. NMR39 and or krypton cannot be evaluated by using the quantum statistical MD simulationsM indicate that CH4 can rotate freely in the small model. Consequently, the guest heat capacities in the range and the large cages of structure I. For the temperature range 100-270 K were calculated from the ideal solid-solution model 100-260 K, we assign 1.5Rfor the heat capacity associated with assuming BS = BL = 1. These were then combined with the the three rotational degrees of freedom. The heat capacity due quantum statistical model to obtain the vibrational frequencies to intramolecular vibrations in the temperature range of interest ws and wL of the guest in the two cages. The procedure used a least-squares program which minimized, over the temperature (35) Tse, J. S.; Klein, M. L.; McDonald, I. R. J . Chem. Phys. 1983, 78, range 100-270 K, the differences in the heat capacities predicted 2096. by eq 28 for a given set of wi and those obtained from eq 23. The (36) Tse, J. S., to be published. summation in eq 26 was carried over 150 vibrational levels. The (37) Huheey, J. E. In Inorganic Chemistry; Harper & Row: New York, starting guesses for eq 28 were varied to ensure that the set of 1978; Chapter 6. reported recently.20 The LJ parameters for water thus derived are in excellent agreement with those obtained from molecular beam e ~ p e r i m e n tew/k , ~ ~ = (163.9 f 10.9) K and uw= (329 f 25) pm, and with those, t w / k = 187.7 K and uw = 304 pm, obtained from the L J parameters for the neon-neon and the neon-water interactions by using eq 10 and 1l.I3 The neon-water interaction parameters were obtained by fitting an L J function to an accurate theoretical neon-water potential curve.13 The LJ parameters obtained in this work were used for the subsequent calculations on structure I and structure I1 hydrates. The LJ parameters used for xenon (e,/k = 225.3 K, u, = 406.9 pm), krypton (t.,/k = 166.7 K, u, = 367.9 pm), and methane ( e / k = 142.7 K, ug = 381.0 pm) were taken from the The differences in chemical potentials of water in the empty lattice and in ice calculated from eq 14 with calculated 8's for structure I xenon hydrate and structure I1 krypton hydrate are given in Table 11. Also given in Table I1 are the enthalpy differences for structures I and I1 obtained from
(33) Bickes, R. W.; Duquette, G.; van der Heijdenberg, C. J. N.; Rulis, A. M.; Scoles, G.; Smith, K. M. J. Phys. B 1975, 8, 3034.
(34) Hirschfelder, J. 0.;Curtiss, C. F.; Bird, R. B. In Molecular Theory of Gases and Liquids; Wiley: New York, 1954.
(38) Handa, Y. P.; Hawkins, R. E.; Murray, J. J. J . Chem. Thermodyn. 1984, 16, 623. (39) Garg, S . K.; Gough, S. R.; Davidson, D. W. J. Chem. Phys. 1975, 63, 1646. (40) Tse, J . S.; Klein, M. L.; McDonald, I. R. J . Chem. Phys. 1984, 81, 6146.
J. Phys. Chem. 1986, 90. 5921-5927 is negligible!] Thus we arrive a t the heat capacity of CH4 due to its rattling motion in the cages by subtracting 1.5R from the total heat capacity contribution from CH4 to the hydrate. The resulting C, was then treated in terms of the quantum statistical model as described above to obtain ws and wL. The dvdwand d[s used for CHI are given in Table 111. The vibrational frequencies obtained for CH4are given in Table I11 and compared with those obtained from the M D simulation^.^^ The agreement is quite satisfactory. The literature values for AH(i+e) and ACp(i+e), and sometimes for Ap(i--e), reported in Table I1 are actually the parameters required to fit the ideal solid-solution model to the phase equilibrium results. The present results for each property were obtained by combining the results for eq 2 obtained from the ideal solid-solution model with the corresponding experimental results for the formation of hydrate from ice. Moreover, the results are based on xenon and krypton hydrates whose structures are now correctly known. Both xenon and krypton fit comfortably in the hydrate cages and thus the properties of the empty lattices reported are essentially those of the unperturbed lattices. Xenon and krypton hydrates are the systems which can be expected to be most adequately represented by the ideal solid-solution model. However, the use of a spherically symmetrical potential for describing the guest-host interactions is questionable. The 12-hedra, which are present in both structures, and the 16-hedra, which are present in structure I1 only, are nearly spherical whereas the 14-hedra, which are present in structure I only, deviate up to 14% from (41) JANAF Thermochemical Tables, 2nd 4.; National Bureau of Standards: Washington, DC, 1971.
5921
spherical symmetry.' John and Holder12 imposed the spherical potential over the potential obtained by discrete summation and arrived at a new set of z and a values for the various cages in the two structures. These values are given in parentheses in Table I. It was proposed4* that the use of this new set of values in the classical model should account for the departure from the spherical symmetry. The calculations for AC,(e+g+h) performed by using the parameters suggested by John and Holder yielded essentially the same results as obtained with the original parameters but yielded rather poorer values for Ap(i+e) and AH(i+e). The values of AH(i+e) and AC,,(i+e) for the two structures obtained are based on the assumption that the enthalpy of encagement and the heat capacity change are linearly dependent on 6, a direct consequence of assumption number 1 in the ideal solid solution model. This assumed ideality of the guest-host solution has never been verified experimentally. Xenon is enclathrated easily and its hydrates of different compositions can be prepared readily. Determinations of enthalpies of dissociation and heat capacities of xenon hydrates of different compositions would help in evaluating the extent of nonideality of these guest-host systems and thus in improving the estimates for the empty lattice properties reported here. Spectroscopic determinations of the vibrational frequencies of the guests dealt with in this work shall also be of interest. Registry No. Water, 7732-18-5. (42) John, V. T.; Holder, G. D. J. Phys. Chem. 1981,85, 1811. (43) Sortland. L. D.: Robinson. D. B. Can. J. Chem. E m . 1964. 42. 38. (44j Parrish, W. R.;Prausnitz,'J. M. Ind. Eng. Chem. Process d e s . be". 1972, 11, 26.
Behavior of Dilute Mixtures near the Solvent's Critical Point R. F. Chang* and J. M. H. Levelt Sengers Thermophysics Division, National Bureau of Standards, Gaithersburg, Maryland 20899 (Received: May 27, 1986)
In the limit of infinite dilution at the critical point of the solvent many thermodynamic properties such as excess properties and partial molar quantities exhibit remarkable anomalies. A striking effect is that finite properties such as the partial molar volume of the solvent exhibit dependence on the path of approach to the critical point. Using the Leung-Griffiths model of mixtures we are able to calculate these thermodynamic properties. The properties considered are partial molar volume, partial molar enthalpy, osmotic susceptibility, isothermal compressibility at constant composition, heat capacity at constant pressure and composition, partial molar heat capacity, osmotic coefficient, and activity coefficient. The Leung-Griffiths thermodynamic potential is nonclassical and is of a scaled form. By the use of the model, we are able to analyze the path dependence of many of these properties and to obtain their explicit x dependence (where x is the mole fraction of the solute) as well as asymptotic expressions along various paths leading to the pure solvent's critical point.
Introduction The thermodynamic behavior of dilute mixtures near the solvent's critical point has undergone a revival of interest in recent years. New experiments have revealed large anomalies in the partial molar volume of the solute,I4 apparent molar specific and apparent heats of dilution8 of dilute salt solutions (1) van Wasen, U.; Schneider, G. M. J. Phys. Chem. 1980, 84, 229. (2) Paulaitis, M. E.; Johnston, K. P.; Eckert, C. A. J .Phys. Chem. 1981, 85. 1770. (3) Eckert, C. A.; Ziger, D. H.; Johnson, K. P.; Ellison, T. K. Fluid Phase Equilib. 1983, 14, 167. (4) van Waser, U.; Schneider, G. M. Angew. Chem., Int. Ed. Engl. 1980,
-19- , -5 1. 5- .. (5) Smith-Magowan, D.; Wood, R. H.J . Chem. Thermodyn. 1981, 13,
1047. (6) Wood, R. H.; Quint, J. R. J . Chem. Thermodyn. 1982, 14, 1069. (7) Gates, J. A.; Wood, R. H.; Quint, J. R. J. Phys. Chem. 1982.86.4948.
in near-critical steam. Furthermore, extraordinarily large excess enthalpies of mixing have also been o b ~ e r v e d . ~ Many J ~ of these anomalies, however, can be explained in terms of critical point phenomena."J2 For instance, the behavior of the partial molar volumes on the isotherm-isobar can be seen from the molar volume (8) Busey, R. H.; Holmes, €1. F.; Mesmer, R. E. J. Chem. Thermodyn. 19a4,16,343. (9) Christensen, J. J.; Walker, T. A. C.; Schofield, R. S.;Faus, P. W.; Harding, P. R.; Izatt, R. M. J . Chem. Thermodyn. 1984, 16, 445, and references therein. (10) Wormald, C. J. Ber. Bunsenges. Phys. Chem. 1984, 88, 826, and references therein. (1 1) Chang, R. F.; Morrison, G.; Levelt Sengers, J. M. H. J. Phys. Chem. 1984,88, 3389. (12) Levelt Sengers, J. M. H.; Chang, R. F.; Morrison, G. In Equations of S t a t e T h e o n e s and Applications; Chao, K. C., Robinson, R. L., Jr., Ms.; ACS Symposium Series 300; American Chemical Society: Washington, DC, 1986; p 110.
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