WILLIAMC. CHILD,JR.
1834
Thermodynamic Functions for Metastable Ice Structures I and I1
by William C. Child, Jr. The Edward Davies Chemical Laboratory, University College of Wales, Aberystwyth, United Kingdom (Received January 28, 1964)
Because doubt has recently been cast on the value of G, - G,, the difference between the free energies of metastable ice structure I and stable ice, employed in two theoretical treatments of the thermodynamics of gas hydrates, a reappraisal of this quantity for both the structure I and the structure I1 ice lattices has been undertaken. The dependence of the entropy of dissociation of a clathrate on free volume has been utilized to estimate S, - S, by comparing the entropies of dissociation of gas hydrates and quinol clathrates. H , - H a has also been estimated from the value for p-quinol relative to a-quinol. The final results are G, - G, = 300 and 190 cal./mole for the structure I and structure I1 ice lattices, respectively.
A theoretical calculation by Platteeuw and van der Waals' of the dissociation pressures of nine gas hydrates of structure I and a recent extension2 of this treatment are both based on a value of the free energy change accompanying the process H20(s) --j HzO(s, metastable, structure I)
(1)
of +167 cal./mole a t 273OK. This value was calculated from the experimentally determineda composition of bromine hydrate in equilibrium with ice a t 273'K. The recent X-ray study by Allen and J e f f r e ~however, ,~ has shown that bromine hydrate has a tetragonal crystal structure rather than the cubic lattice of structure I gas hydrates, Hence, the above free energy difference appears to be inapplicable to structure I hydrates. Glew,6 from an analysis of all the data available for equilibria involving methane hydrate, which has structure I, found that the fraction of the cavities occupied in the equilibrium clathrate a t 273OK. is 0.997 f 0.021 (standard error). This value suggests a fairly large difference between the free energies of the metastable and stable forms of ice but does not permit a meaningful calculation of this quantity because the fraction occupied is so close to unity. I n fact, the accuracy required for such a calculation is an order of magnitude greater than that which has yet been achieved in any experimental determination of the formula of a gas hydrate. It is the purpose of this article to suggest plausible values for the differences between the entropies of ice T h e Journal of Physical Chemistry
structures I and I1 and the entropy of stable ice by comparison of the entropies of dissociation of some gas hydrates with those of the quinol clathrates. In addition, estimates of the differences between the enthalpies of the several forms of ice can be made from the value of AH for the formation of p- from a-quinol. Thus, a complete set of approximate thermodynamic values can be obtained. Theoretical Entropies of Dissociation of Clathrates. A theoretical equation for the entropy of dissociation of a clathrate, based on the cell model, can be derived from the equations for chemical potential and internal energy given by van der Waals and Platteeuwae The result is
where AS,'
is the entropy change for the decomposition
A.nB(s) --+ nB(s, metastable form)
+ A($)
(3)
(1) J. C. Platteeuw and J. H. van der Waals, Mol. Phys., 1 , 91 (1958). (2) V. McKoy and 0. Sinan;glu, J. Chem. Phys., 38, 2946 (1963). (3) E. M. J. Mulders, Thesis, Delft University, 1937. (4) K. W. Allen and G. A. Jeffrey, J . Chem. Phys., 38, 2304 (1963). (5) D. N. Giew, J . Phys. Chem., 66, 605 (1962). (6) J. H. van der Waals and J. C. Platteeuw, Advan. Chem. Phys., 2, 1 (1959).
1835
THERMODYNAMIC FUNCTIONS FOR METASTABLE ICE
in which A and B stand for guest and host, respectively; V is the molar volume of the gaseous guest, assumed to be ideal; Vf is the free volume per mole, as defined by Platteeuw and van der Waals1; and y is the fraction of the cavities occupied. The contribution of R appears because the cavities are distinguishable, while the last two terms arise because there are a number of ways in which N guest molecules can be arranged in N/y cavities. For convenience in the discussion which follows, eq. 2 can be rewritten as AS,'=
R In-V Vf
,):F
-
RT (a___
+ R - S,
(4)
in which the last two terms of eq. 2 have been replaced by -So, the negative of the configurational entropy. Implicit in eq. 2 and 4 are two assumptions: (1) the lattice is not distorted by the guest molecules, and (2) the barrier to the rotation of the guest molecules in the cages is small. McKoy and Sinan6glu2have given a theoretical justification for the absence of lattice distortion in the gas hydrates. I n addition, X-ray measurements7J have not revealed any distortion of the lost lattices of the clathrates discussed here. However, Staveley and his collaborators,@on the basis of heat capacity measurements, have noted evidence of possible perturbation of the low vibrational frequencies of the quinol lattice by some of the occluded molecules. Regarding the second condition, again McKoy and SinanKgXu2 have argued against the existence of a high barrier to the rotation of the guest molecules. Staveley, et u Z . , ~ have found that there is free rotation of methane and a barrier of 1 kcal./mole to the rotation of nitrogen in the quinol clathrates. By measuring dielectric loss in ethylene oxide and tetrahydrofuran hydrates, Davidson, Davies, and William@ have found activation energies of 0.5 kcal./ mole or less for the rotation of the guest molecules. I n conclusion, it appears that eq. 2 can justifiably be applied to the gas hydrates and quinol clathrates discussed here, becau& in nearly all cases the van der Waals radii of the guest molecules are smaller than the free-space radii of the cavities in the undistorted host lattice. Briefly, the procedure to be followed is as follows: first, AS,' S , for four quinol clathrates is calculated from experimental data. Then, the value for one of them (argon) is equated to AS,' S, for ethane hydrate, which has a nearly equal free volume in the square-well approximation. Finally, A S for the procSo, S,, ess given by eq. 1 is calculated from AS,' and AS,' for ethane hydrate. AS,', which can be determined experimentally, is the entropy change accompanying the process
+
+
+
A.nB(s)
--
nB(s, stable form)
+ A(g)
(5)
A similar procedure is used to calculate the difference between the entropies of ice of structure I1 and stable ice. Entropies of Dissociation of Quinol Clathrates. Three values of SSB0 S , have been determined by Staveley,g and two values (one for a clathrate also studied by Staveley) of AS,' can be calculated from experimental dissociation pressures and enthalpies of dissociation found in the literature. Comparison of eq. 3 and 5 shows that AS," can readily be converted into AS," Soby use of G, - G,, H , - Ha,and y for equilibrium quinol clathrates, all of which have been determined experimentally." The results are given in Table I.
+
+
Table I : Entropies of Dissociation of Quinol Clathrates a t 298°K.
AS,"?
No.
Guesta
e.u./mole
1
Ar
20.1 f 0.6'
2 3 4
Kr Nz CH4
20.6 f 0.6'
+
Vf,d
vrse
AS," So,' e.u./mole
cm.s/ mole
cm.8/
23.9 f 0 . 6 24. 26 26.1* 24.4 f 0 . 6 27.1@
1.29
0.95
0.86 0.69 0.52
0.42
'
mole
a Listed in order of increasing molecular size. AS" for the process, A 4 . 8 3 CeH4(OH)z(s) + 8.83 ceH,(OH)z(s, 8-form) A(g). So,configurational entropy, equals -R In y - R( 1 - y)/y In (1 - y). Free volume calculated from eq. 6. Free volume calculated from Lennard-Jones and Devonshire potential, ref. 6. Calculated from AH," of Evans and Richards, ref. 11, Hg - Ha and Go - G,, ref. 11, and dissociation pressures measured by van der Waals and Platteeuw, ref. 6. ' From heat capacity measurements of Staveley, et al., ref. 9.
+
For the purpose of comparing the entropies of dissociation of the several types of clathrate, free volumes based on the square-well potential have been calculated and included in Table I. The appropriate equation is6 (7) Quinol clathrates: D. E. Palin and H. M. Powell, J. Chem. Soc., 815 (1948). (8) Gas hydrates: M. von Stackelberg and H. R. Mtlller, 2. Elektrochem., 58, 25 (1954). (9) L. A. K. Staveley, et aE., Mol. Phya., 2, 212 (1959); ibid., 3, 59 (1960); ibid., 4, 153 (1961); Advances in Chemistry Series, No. 39, American Chemical Society, Washington, D. C., 1963, p. 218. (10) D. Davidson, M. Davies, and K. Williams, to be published. (11) H g - Ha = 160 cal./mole at 298"K., D. F. Evans and R. E . Richards, Proc. Roy. Soc. (London), A223, 238 (1954); y = 0.34 and Gg - Ga = 82-84 oal./mole at 298"K., ref. 6 and J. N. Helle, D. Kok, J. C. Platteeuw, and J. H. van der Waals, Rec. trau. chim., 81, 1068 (1962).
Volume 68, Number 7
July, 1964
WILLIAMC. CHILD,JR.
1836
lit
=
4 -n(a - g ) 3 N 3
(6)
where a is the Sadius of the cage, u is the distance of “closest” approach between a guest molecule and a molecule in the wall of the cavity and is the average of the collision diameters of the guest and the wa1116and N is Avogadro’s number. In calculating the free volumes listed in both Tables I and 111, radii of the cages and wall thicknesses mere taken from ref. 6. Many of the collision diameters of the guest molecules were found in the book by Hirschfelder, et a1.I2; others were estimated from the values for similar molecules. AS,’ S , us. log V f has been plotted in Fig. 1. The line has been given a slope of -2.3R, the theoretical slope in the square-well approximation.
+
28
0 IS
26
d
clathrates. These two assumptions are equivalent to assuming that the potential function for the interaction of a guest molecule with its cage has approximately the same shape and crosses the r axis a t approximately the same value of r for the two ~1athrates.l~ S , for ethane hydrate is placed On this basis, AS,” on the line in Fig. 1 which passes through the value for the argon-quinol clathrate. It is believed that the error in doing so is no greater than *1.4 e.u., ie., that the actual free volumes of the two clathrates do not differ by more than 25% and the second term on the right of eq. 4, which is of the order of 1 e.u., does not differ by more that 100% for the two clathrates. The next step is to calculate S,, which is small for gas hydrates and need not be known accurately. By combining a first approximation to S , - S a with the estimated H , - H adiscussed in the next paragraph, y , the fraction of the cavities occupied in ethane hydrate in equilibrium with ethane gas and the stable form of ice, is calculated from the equation6
+
G,
z
3
2 24 a;
6
: 22
m Y
G,
=
- vRT In (1 - y )
(7) H Here G, - G, is the difference between the free energies of structure I ice and stable ice, and v is the ratio of the number of cavities to the number of water molecules per unit cell. After successive approximations, the average fraction of the two types of cavity occupied is found to be 0.96. S , is next calculated as S,, S,, and AS,’ 0.4 e.u. Combination of AS,” then allows calculation of n ( S , - S a ) , where n is the number of water molecules per ethane molecule in the equilibrium clathrate and is taken equal to 6.0 in this calculation. Finally, S , - Sa is found to be -0.42 f: 0.23 e.u./mole for structure I. In order to obtain G, - G,, a value of H , - H a is needed. In the absence of any experimental data, Platteeuw and van der Waals assumed that this quantity is approximately zero.’ This is unlikely to be strictly true, however, because holes are created in stable ice to produce structure I. A small, positive value of H , - H a similar to the 160 cal./mole required to convert a-quinol into p-quinol therefore seems more likely. An estimate of the difference between the enthalpies of metastable ice I and stable ice a t 273°K. is obtained by the following procedure, which is based on the measured difference between the enthalpies of a- and
+
2C
18 -0.6
-
-0.4
-0.2 0.0 log By, cm.s/mole.
0.2
0.4
0.6
+
Figure 1 . Correlation of AS,” S,with log Vr. Vf is based on the square-well potential function. The line has been given the theoretical slope of -2.3R. Points are identified in the first column of Tables I and 111.
Thermodynamic Functions $01 Ice, Structure I . The free volumes of a number of gas hydrates for which values of AS,’ are available13 have also been calculated from eq. 6. The values for the structure I gas hydrates are averages, weighted according to the number of each type of cavity. I t is found that the free volume of ethane hydrate, 1.2 cm.a/mole, is nearly equal to that of the argon-quinol clathrate, 1.29 crnea/ mole. The assumption is now made that the free volumes of these two clathrates, calculated on the basis of the Lennard-Jones or the Kihara potential functions, while not equal to those calculated from eq. 6, are nevertheless approximately equal to each other. In addition, it is assumed that the second term on the right side of eq. 4 is approximately the same for both The Journal of Physical Chemistry
(12) J. 0. Hirschfelder, C. F. Curtiss, and R. B. Bird, “Molecular Theory of Gases and Liquids,” John Wiley and Sons, Inc., New York, N.Y., 1954, pp. 1110-1112. (13) D. N. Glew, Can. J . Chem., 38, 208 (1960). (14) See ref. 2 for graphs of typical potential functions.
1837
THERMODYNAMIC FUNCTIONS FOR METASTABLE ICE
p-quinols. First, this enthalpy difference is assumed to be related to the heat of sublimation of the stable host, AH,,I,I , by the equation
H,
-
Ha
"Va -
= k(AHaub~a)(
(8)
in which V , - V , is the difference between the molar volumes of the two lattice structures and V , is the molar volume of the a-form. The proportionality constant, IC, is evaluated from data for the quinols1'J6 and is found to be 0.096. S e x t k is assumed to have the same value for the ice system, and H , - Ha is calculated from the lheat of sublimation and molar volumes of the ices* by use of eq. 8. The estimate obtained is 180 cal./mole. The difference between the free energies of structure I and stable ice is calculated from H, - H a - T ( S , Sa) and is found to bie 300 cal./mole at 273"K., considerably larger than the 167 cal./mole used by Platteeuw and van der Waals.1 No other value is available for further compasison. Some sort of lower limit can be placed on this quantity by use of Glew's analysis6 of the methane hydrate data. The value of y found, 0.997, minus the standard deviation, 0.021, gives 0.976, which when substituted into eq. 7 yields G, - G, = 350 cal./niole as a lower limit. The true value of this quantity could therefore be considerably larger than that proposed here; but if so, it would seem that H , - H awould have to be correspondingly large. For example, selection of the values, G, - G, = 700 cal./mole and H , - H a = 180 cal./mole, evens, equal to 11.7 and 15.3 tually produces AS," e.u./mole a t 273°K. for methane and ethane hydrates, respectively. These values appear unreasonable when compared with the values for the quinol clathrates. On the other hand, postulating a fairly large H, - H a means that this qhantity becomes an appreciable fraction of the heat of sublimation of ice. While this possibility cannot be ruled out, it appears improbable. The thermodynamic functions proposed here for ice structure I are given in Table 11, which summarizes all the values proposed for the metastable ice structures to date. The proposed value [of S, - Sa can now be applied to experimental entropies of dissociation reported in the literature13 to yield values of AS," S , for these clathrates. These are presented in Table 111,and some of them are plotted in Fig. 1. The uncertainties indicated for AS," S , are merely those estimated for AS,". The dependence of AS," S, on V f seems reasonable. Thermodynamic Functions for Ice, Structure I I . 8, - Saand H, - H afor structure I1 ice are obtained
+
+
+
+
Table I1: Suggested Thermodynamic Functions for the Two Forms of Metastable Ice Relative to Stable Ice a t 273'K.
a,
-
C, cal./mole
Structure I
300 167 Structure I1 190 196 88-128
Hg
-
Ha, Gal./ mole
180 -0 200
-
Sg Sa, e.u./mole
Ref.
-0.42 f 0.23 This article 1, 6 0 . 0 5 f 0.08 This article 6 16
Table 111: Estimated Entropies of Dissociation of Some Gas Hydrates a t 273°K. AS,' No.
5 6 7 8
Guesta
HzS CHa CzHs SO2
+
&,C e.u./mole
A S a a l b e.u./moIe
(Structure I, formula A.6.OH20) 20.2 f 0 . 7 22.3 f 0 . 7 20.5 f 0 . 7 22.6 f 0 . 7 24.1 f 0 . 7 26.2 f 0 . 7 26.5 f 1 . 1 24.4 f 1.1
9 10
(Structure I, formula A.'7.67&0) Clz 23.7 f 1 . 8 20.5 f 1.8 CHsBr 26.8 f 3 . 7 23.6 f 3 . 7
11 12 13 14 15
(Structure 11, formula A.17H20) CHaI 22.1 f 1 . 8 23.0 f 1 . 8 CzHsC1 29.2 f 1 . 8 30.1 f 1 . 8 CsHs 25.3 d= 0 . 7 26.2 f 0 . 7 CHClzF 27.3 f 1.5" 28.2 f 1 . 5 CBrCIFz 26.8 f 0 . 1 27.7 f 0 . 1
VrVd cm.a/moie
3.1 1.2
2.5
3.2
0.8 0.3
Listed in order of increasing molecular size. AS" for the A(g). Obtained from process, A.nH,O(s) -t nHzO(s, stable) AS," refers to the process, ref. 13 unless otherwise noted. nHzO(s, metastable) A(g). S,, configurational A.nH2O(s) entropy, is 0.4 e.u. for the hydrates in the first group and is assumed to be zero for the hydrates in the second and third groups. Free volume calculated from eq. 6. W. P. Banks, B. 0. Heaton, and F. F. Blankenship, J. Phys. Chem., 58,962 (1957).
-
+
+
by the methods employed for structure I. In this S, for propane hydrate is made nearly case, AS," equal to that of krypton-quinol clathrate. The value of G, - G, which is finally calculated is 190 cal./mole, and this and other values are included in Table 11. Previously obtained values of G, - G, are 196 cal./mole6 and 88-128 cal./mole.16 In addi-
+
(15) AH,,bi a : J. Timmermans, "Les Constantes Physiques des Composes Organiques Cristallises," Masson et Cie, 1953, p. 414; d , and da: H. M. Powell, J . Chem. SOC.,61 (1948),and W.C. Child, Jr., unpublished. (16) R. M.Barrer and D. J. Ruzicka, Trans. Faraday Soc., 58, 2239 (1962).
Volume 68, Number 7 July, 196.4
1838
,J. O'M. BOCKRIS, S.YOSHIKAWA, A N D S.R. RICHARDS
tion, there are available three recent and reliable analyses of structure I1 hydrates in equilibrium with stable ice and gaseous guest. The values of y found are 0.999 f 0.012, 1.004 f 0.007, and 1.000 f 0.012, corresponding to SFaI17 CBrC11'2, l 3 and (Y3rzF2'* hydrates, respectively. (The uncertainties given are standard deviations:) Within the experimental errors the values center on unity. -4valuc of G, - G, = 220 cal./molc (;y = 0.999) is not inconsistent with these results, but as stated earlier, no meaningful value of G, - G, can be deduced from the analyses. As before, AS,' S,for a number of structure I1 hydrates has been calculated, listed in Table 111, and plotted in Fig. 1. It should be noted that within each
group of clathrates (quinol. structures I and I1 ice) S,. on log Vr is close to that the dependence of AS,' predicted by cy. 4 in the square-well approximation.
+
+
Acknowledgment. Financial aid from Carleton College, and thc hospitality of the Chemistry Departnierit of the T'niversity C'ollcge of Wales in Aberystwyth, are gratefully acknowlcdged. The author is indebted to Dr. Mansel 1lavic.s for helpful and stimulating discussions and to Dr. I). S.Glew for his comments on an early version of this article. (17) L. D. Sortland, 1f.S~. Thesis, rniversity of Alberta, 1962. (18) D. N.Glew. private rornmunication.
Diffusion of Unlike Ions into Liquid Sodium Chloride
by J. O'M. Bockris, S. Yoshikawa, and S. R. Richards The Elcctrochcmistry Laboratory, C'nic,et d i / of Pennuylvanin. Phdadelphia. Pennsylvania (Rrceired January S I , 1964)
The diffusion coefficients (D)of dilute amounts of Li (
