166
YUNG-CHIWu
liminary communication.12 The latter were evaluated on the assumption that log K A varied linearly with l / e over the whole range of composition. The rate constant seems to decrease slightly in either case in contradiction to the expected increased rate a t low di-
AND
HAROLD L. FRIEDMAN
electric constant.ll The major reason, however, for the rapid decrease in the importance of the free anion reaction is not change in reactivity, but a rapid decrease in free anion concentration as the medium becomeslesspolar.
Heats of Dilution of Some Electrolyte Solutions in D,O and Comparison
of Thermodynamic Excess Functions in D,O and H,O
by Yung-Chi Wu and Harold L. Friedman* I B X Watson Research Center, Yorktown Heights, ,Vew York 10698 (Received J u l y 6 , 1966)
Heats of dilution of LiC1, NaC1, and NaI solutions in DzO have been measured in the range 1 to 3 M to supplement earlier results of Birnthaler and Lange. The solvent isotope effect on the excess enthalpies is much larger than that on the excess free energies found by Frank and Berwin. The excess entropies in DzO have been calculated from these data and compared with those in water with the result that, within the experimental error, we have Sex(in HzO) - Sex(in DzO) = k,m, where m is the aquamolality (moles of salt per 55.51 moles of solvent). We find Tk, = 54 for LiC1, 35 for NaC1, and 35 for NaI, all in units of cal. mole-'M-'. These results, as well as the solvent-isotope effects on the entropies of solvation, seem to be consistent with the model of aqueous solutions developed by Frank and Evans, Gurney, and NBmethy and Scheraga.
1. Introduction At 25", the liquids DzO and HzO have the same dielectric constant e within 0.4% and the same d In t/dT within 0.870.l Therefore, calculations based on the primitive model,2 namely, the Born equation for the solvation energy and the Debye-Hiickel theory for the themiodynaniic excess functions, lead to closely similar results for solutions in the two solvents. On the other hand, DzO has more structure than HzO at the same t e m p e r a t ~ r e ,forms ~ , ~ stronger hydrogen bonds, and has significantly different zero-point energy, even in certain librational mode^.^,^ Therefore, the comparison of thermodynamic properties of ionic solutions in the two solvents may be expected to provide clues as to the degree to which these properties are sensitive to structural, hydrogen-bonding, and The Journal of Physical Chemistry
zero-point energy effects.6 I n the case of solutions of a nonelectrolyte, argon, Ben-Waim4 has in this way recently confirmed the view put forward earlier by
* Department of Chemistry, State University of New York, Stony Brook, N. Y. 11790. (1) A. A. Maryott and E. R. Smith, National Bureau of Standards Circular 514, U. S. Government Printing Office, Washington, D. C., 1951. (2) In the primitive model for electrolyte solutions the solvent is represented as a structureless fluid whose properties are the macroscopic properties of the real solvent. The ions are represented as hard spheres with charges at the center. In some applications the sphere is assumed to have the dielectric constant of the solvent, in others that of a vacuum. (3) G. Nemethy and H. A. Scheraga, J . Chem. Phys., 36, 3382 (1962); 41, 680 (1964). (4) A. Ben-Naim, ibid., 42, 1512 (1965). (5) J. D. Bernal and G. Tamm, Nature, 135, 229 (1935). (6) It is not meant to be implied that these are three independent effects.
167
HEATSOF DILUTION OF ELECTROLYTE SOLUTIONS
Eley’ and Frank and Evans8 that the structural effects on the solvation entropy are large indeed. I n the case of electrolyte solutions, however, Swain and Baderg found that the isotope effect on ionic solvation varies from ion to ion in a way that is parallel to effects on the infrared spectrum of water in the libration region (15 p ) and they interpret the spectral changes in terms of the Bernal-Fowler model of the solutions,’” emphasizing the differences from the Frank and Evans model. The present study was stimulated by the work of Frank and Kerwin,ll who determined the osmotic and activity coefficients of a number of alkali halides in D 2 0 and compared the results with those in H2O. The solvent isotope effect that they determined in this way is very small, suggesting that nonprimitive model effects of the type referred to above, although important, occur in both the entropy of dilution and the enthalpy of dilution in such a way as nearly to cancel in the free energy of dilution, ie., in the osmotic and activity coefficients. The objective of the work reported here is to see whether or not the enthalpies of dilution are in accord with this speculation. Birnthaler and Langel2,l3measured the enthalpy of dilution of a number of ionic solutions in both H2O and DzO, but there are some inconsistencies in their results, as noted in detail below and, in order to arrive a t reliable values, further experiments are required. Our own calorimeter was designed for a different kind of measurement and is just barely suitable for the present purpose, but in view of the interest in the results, it seemed useful to resolve the question posed above to the extent that the instrument allows. Our experiments lead to the apparent molal heat content as a junction of molality in D20, +~2(m). The notation $ ~ ( mwill ) be used for the corresponding quantity in HzO. In either case m is the aquamolality, the number of moles of solute per 55.51 moles of solvent. Solutions of the same aquamolality in the two solvents are a t very nearly the same volume concentration.11 Similarly, M is employed as an abbreviation for the units of aquamolality. Thus “2 M” is to be read “2 aquamolal.”
to be a trace of dissolved glass. The salts were reagent grade materials, oven-dried before use. I n series 1 experiments, samples of ionic solutions in D20, in the concentration range 1 < m < 3 M were mixed with a thousandfold excess of HzO or dilute aqueous solution, and the heat effect was observed. The results are summarized in Table I. Table I: Summary of Series 1 Mewurementsa*b
Experiment A 1. 206.3 mg. of 1.356 M 285.6 g. of HzO
The calorimeter has been described e1se~here.I~ The preparation and breaking of the sample bulbs has been improved, leading to a reduction of the uncertainty due to the heat of bulb breaking to A10 pdeg. in the temperature rise. The D content of the D 2 0 was stated by the supplier15 to be 99.8%. This material was used as received although it had a specsc conductivity of lO-j/ohm cm. The impurity seemed
599 f 29
LiCl in DzO 2. 3.
197.1 mg. of 1.783 M LiCl in DzO 186.7 mg. of 2.736 M
Solution from 1 Solution from 2
262
23
-213 f 17
LiCl in DzO Experiment B 1. 222.5 mg. of 2.962 M 286.1 g. of HzO NaI in DzO 2. 238.1 mg. of 2.100 M Solution from 1 NaI in DzO 3. 212.6 mg. of 1.576 M Solution from 2
353 rJ, 10 422 f 15 640 i20
NaI in DzO Experiment C 1. 231.1 mg. of 1.541 M 287.4 g. of HzO N a I in DzO 2. 266.2 mg. of 2.062 M Solution from 1 NaI in DzO 3. 294.9 mg. of 3.263 M Solution from 2 N a I in D20
602 f 20 453 i 15 368 =k 10
M is aquamolality: moles per 55.51 moles of solvent. AH is corrected for the heat of bulb breaking. This is calories per mole of salt in the “mix” column.
I n reducing these data we need data for the LM of the process HzO(1)
+ D20(1) = 2HDOO)
(1)
and for the AH” of the process A+(D20)
2. Experimental Section
AH, cal./molec
With
Mix
+ B-(D20)
= A+(H20)
+ B-(H20)
(2)
(7) D. D.Eley, Trans. Faraday SOC.,35, 1281 (1939). (8) H.S. Frank and M. W. Evans, J . Chem. Phys., 13, 607 (1946). (9) G. C. Swain and R. F. W. Bader, Tetrahedron, 10, 182 (1960). (10) J. D.Bernal and R. H. Fowler, J . Chem. Phys., 1, 515 (1933). (11) R. E. Kerwin, Ph.D. Thesis, University of Pittsburgh, 1964. (12) W. Birnthaler and E. Lange, 2. Eleldrochem., 43, 643 (1937). (13) W.Birnthaler and E. Lange, ibicl., 44, 679 (1938). (14) H.L. Friedman and Y.-C. Wu, Rev. Sci. Instr., 36,1236 (1966). (15) Isomet Corporation, Palisades Park, N. J.
Volume 70,Number 1 January 1966
168
YUNG-CHI Wu
1mole of AB (std. in HzO)
+
55.51
moles of DzO(std. in HzO)
(A’ - k)
+
T +
55.51 moles of D ~ O
AH\
Table II : Series 2 Measurements AH,
Mix
With
oal./molea
1. 0.4384 mole of LiCl (4 2. 216.9 mg. of 2.374 M LiCl in D2O 3. 308.5 mg. of 2.915 M NaCl in D20 4. 238.4 mg. of 2.431 M NaI in DtO
323.77 g. of DzO
-8456 f 24
Solution from 1
-380 i 16
Solution from 2
384 f 10
Solution from 3
463 i 15
This is calories per mole of salt in the “mix” column.
I
+
1 mole of AB in 55.51 moles of D ~ O ml
HAROLD L. FRIEDMAN
moles of HzO (pure)
+Ll(ms)
1 mole of AB in
AND
3. Results It is convenient to present our results for 4~2(m)
K
1 mole of AB (std. in DzO)
+ %moles
of pure DzO
+
(Ag -?3
55.51 - -
moles of pure HzO
Figure 1. Ent,halpy cycle for the reduction of series 1 data. The quantity me measure is AH. The notation std. refers to the usual hypothetical 1 M standard state.
for the appropriate salt AB. Doehlemann and Lange16 found AHl = 32 cal./mole of DzO reacted
A 6 ~ ( m= ) 6~1(m) - 6 ~ ~ ( m ) (3) and together with our treatment of the data of Birnthaler and Lange12J3for the same quantities. LiCZ. Our results for 4 ~ 2are shown in Figure 2, together with the data of Birnthaler and Lange.l3vZ1 The indicated limits of error in our series 1 data come about equally from the uncertainty in the heat of bulb breaking and from our estimate of 1 cal./mole as the uncertainty in AH1. We consider that all of our data and those of Birnthaler and Lange are in satisfactory agreement, especially since the earlier determinations of Birnthaler and Lange depend on combining measured heats of dilution with an unspecified extrapolation procedure. Three sets of 4 ~ data 1 are shown in Figure 2. The results of Lange and Durr22 are, after proper23extrapolation to m = 0, in good agreement with the recent data of Wu and while the data of Birnthaler and Langel3l2lare quite different. In the absence of experimental details of their work we can only conclude that their 4 ~ results 1 are in error.24
independent of the H/D composition of the reacting solutions. Lange and Martin1‘J8 studied reaction 2 a t a final concentration level in the range 0.1 < m < 0.3 M . We have made small corrections to their results, to allow for the difference in heats of dilution in HzO and DzO, in order to arrive a t AH,”. (These values are tabulated in section 4.) Finally, we need +L(m) in the dilute solution obtained by mixing in our calorimeter. This solution contains a small per(16) E. Doehlemann and E. Lange, 2.Elektrochem., 41, 639 (1935). centage of HDO in the solvent as a result of the mixing (17) E. Lange and W. Martin, ibitZ., 42, 662 (1936). process. However, in this range of m, 4 ~ ~ ( m=) (18) E. Lange and W. Martin, 2. physik. Chem., A180, 233 (1937). +L(m) = $ L I ( ~ > within the accuracy required here, (19) V. B. Parker, Thermal Properties of Aqueous Uni-Univalent and we have used the tabulations of Parkerlgfor + ~ ~ ( m ) . Electrolytes, U. S. National Bureau of Standards, NSRDS-NBS 2, Washington, D. C., 1965. The way in which the various data are combined to (20) Our calorimeter14accommodates up to four sample bulbs, and m ) our series 1 measurements is best obtain + ~ ~ (from in these experiments only the tirst bulb of DzO solution is mixed with pure (Le., natural H/D ratio) HeO. The reduction of the data seen by reference to Figure 1.20 for the remaining bulbs then requires an additional step, omitted In series 2 experiments, various samples were mixed, 1 the more dilute from Figure 1 for simplicity, which involves the 6 ~ of solution before each mixing step. in succession, with a single filling of DzO in the calorim(21) These data were obtained from Figure 2 of ref. 13 with the aid eter. These measurements are summarized in Table of a microscope. No indication is given of the accuracy of these 11. Because these final concentrations are so low, data. the corrections to infinite dilution are easily made with (22) E. Lange and F. D-, 2. physik. Chem., AlZ1, 361 (1926). an accuracy of i2 cal./mole in each case. (23) Y. C. Wu, Ph.D. Thesis, University of Chicago, 1957. The Journal of Physical Chemistry
HEATS OF DILUTION OF ELECTROLYTE SOLUTIONS
169
600
0
-200
400 W
2 0
z
W
\
-I
-I 4
2 -400
0
\
-I
a
0
200
-600
0
I 2 AQUAMOLALITY
3
Figure 2. LiCl data: 0, Birnthaler and Lange; A, this paper, series 1; 0, this paper, series 2. The line through the A(PL points fits the equation A(PL = 47m.
The A+L values in Figure 2 were obtained from the experimental values of 4 ~ 2and the smoothed curve through the data of Wu and Youngzafor cpL1 and fitted to the relation 4rp~= khm
(4)
as shown. In order to verify the value of AHz' used in calculating our +L2 in the series 1 measurements, we made an independent determination of the heat of soliition of LiCl in DzO (Table 11). After correction for qk2in the final solution, the result is 4Ho = -8472 f 24 cal./mole, in satisfactory agreement with one of the values (-8445 f 20, -8577 20) that we derive from the two measurements by Lange and Ma~%in.~'J* NaCZ. From the heat effect of the series 2 experiment with NaCl (Table 11),after making a small correction for the change in cpLz of the dilute solution, we obtain +~~(2.915) = -392 f 12 cal./mole. With values of +Ll(m), taken from Parker,19 we find A&,(2.915) = 99 f 12 cal./mole = [35 f 41 X 2.915 cal./mole. Birnthaler and Langel2 measured + ~ ~ ( m c)pLZ(m1) in a considerable number of dilution experiments, in six of which both m2 and ml were less than 3 M. The
*
YUNC-CHIWu AND HAROLD L. FRIEDMAN
170
values apparently reflect large errors in the determinations. Nul. Our results for # ~ ~ (are m )shown in Figure 3. The #Ll(m) curve in this figure is based on the tabulation of Parker,l9 and A+L values in Figure 3 are obtained by difference between our data and the #Ll(m) curve. There seem to be no earlier determinations of +L2 for NaI. 4.
Excess Functions For the rest it is convenient to change notation,
noting that for any particular solution of a single electrolyte
&, = He"
solute.25, 26 Corresponding to He" is G"", the total excess Gibbs free energy per mole of s o l ~ t e .For ~ ~ a~ ~solution ~ of a single symmetrical electrolyte it is given in terms of the familiar partial molal excess functions, the osmotic coefficient +, and the mean activity coefficient yf,by the equation
+ In y*J
(6)
The corresponding entropy quantity is
Sex= [H"
+ E,m
- Ge"]/T
(7)
In m
f B,m
+ .. ,
(8)
but for symmetrical electrolytes25bE, = 0. Furthermore, for HzO and D2O the Debye-Huckel limiting law coefficient S, is very nearly the same. Therefore, Aye" Yex(m,HzO) - Ye"(m, D20) is of the form
AY""
=
aB,m
+ (higher-order terms in m)
(9)
This is the theoretical justification for representing our results as we did in section 3, namely, A#L = khm. The accuracy of the available data offers no basis for including higher terms in m. Frank and K e d found in their studied1 that
At$(m) = -km
(10)
where k is characteristic of the salt and explained, in the same way as we have done, that no lower-order terms are expected. They also point out that it follows from the GibbsDuhem equation that A h y& = -2km
Then, in view of eq. 6 , we have The Journal of Phyeical Chemktry
-2RTkm
(12)
k,m and, in view of eq. 7
TAS"" =
[kh
- kg]m
(13)
= Tk,m
For the electrolytes studied here, the values of kh, k,, and k, are collected in Table 111. From these coefficients and the excess functions for HzO solutions,27 we have prepared the comparison of the excess functions in HzO and D20 given in Figure 4.
Table III: Solvent Isotope Effects on Excess Functions and on Solvation Energetics a t 25" Salt
(11)
LiCl
NrtCl Cal. mole -1 M--
-1
Tk,
-7 47 f 8 54 & 8
AGz"
-111 zk 5ob
-116"
AH$"
-423f20 -312 f70
-545zk30 -429 f 30
kh
TA&"
Any of these three excess functions, call it Ye", has an m-dependence of the form
Yex= S,m'"
=
(5)
where He" is the total excess enthalpy per mole of
G" = 2RT[1 - #
AG""
34 =k 5 35 zk 5
NaI
-4 31 f 6 35 f 6
0 f loob - 8 3 0 h 20 -830 f 120
a From Frank and Kerwin.11 The uncertainties are negligible on the scale of our results. Estimated from solubilities and
activity coefficients.
5. Discussion The results in Figure 4 c o d r m the suspicion, mentioned in the Introduction, that the effect in changing solvent from H20 to D2O is much greater in He" and TS"" than in G"". I n a general way the observations may be interpreted in terms of molecular interactions on the basis of the model developed by Frank and Evanss from considerations of ionic entropies and applied by Frank28 to interpret the part of Sex ~
~~
(26) (a) H. L. Friedman, I. C h m . Phys., 32, 1361 (1960); (b) "Ionic Solution Theory," Interscience Publishers, New York, N. Y., 1962, Chapter 16. (26) The same notation was formerlya6used for exceas functions per kilogram of solvent, which is more convenient for treating solutions of more than one electrolyte. (27) Gex was calculated from activity and osmotic coefficients tabulated by Robinson and Stokes, "Electrolyte Solutions," Butterworth and Co. Ltd., London, 1966. Hex for NaCl was taken from the tabulation of Parker.ls (28) H. S. Frank, "Chemical Physics of Ionic Solutions," a selection of invited papers presented at an International Symposium on Electrolvte Solutions monsored bv the Electrochemical Societv. B. E. Conway and R. G. Banadas,-Ed., John Wiley and Sons, In;.; New York, N. Y., 1965.
171
HEATSOF DILUTION OF ELECTROLYTE SOLUTIONS
leased to the bulk solvent and so undergoes a change of state. solvent (in cosphere) +solvent (normal) (14) Statistical-mechanical calculations indicate that, in the range from 1 to 3 M in aqueous solutions of 1-1 electrolytes a t 2 5 O , the average mutual volume of the cospheres per mole of electrolyte is proportional to This has the implication that the part of Yex that is characteristic of the electrolyte species ought to be proportional to molality in this range.3z This is actually observed for the alkali halides in water for G“, Hex, and Sexas may be seen by examining the coefficients
[ Y e X *-~Y e X ~ a ~ l ]= / mK,(AB)
- K,(NaCI)
(15)33
Some examples are shown in Table IV. APUAMOLALITY
-
Figure 4. Total excess functions a t 25”: , in HzO; , in DzO. The e x c m free energies in HzO derive from tables in Robinson and Stokes” with the use of eq. 6. The excess enthalpies in HzO are derived as described in this paper.
-----
Table N : Coefficients of Species-Characteristic Terms of Excess Functions for Aqueous Solutions a t 25” -Gal. mole-’ va--
Solute
&(AB) - K,(NaCl)” T [K,( AB) - K.( NaCl)]
O.04T[KB(AI3) - K.(NaC1)1
that is characteristic of the ionic species. Essentially the same model has been applied by Gurneyz9to discuss the part of In yk that is characteristic of the solute species. First, considering an ion at infinite dilution in the solvent, we define the cosphere of the ion as a sphere, centered on the ion, that encloses all of the solvent that has properties essentially different from those of the bulk solvent. On the other hand, the solvent outside the cosphere is identical with the pure solvent with respect to properties such as p(P,T,E), density as a function of pressure, temperature, and electric field, and e(P,T,E),the dielectric constant as a function of these same variables. The interaction of the ion with the solvent in this region may be calculated on a macroscopic basis while within the cosphere the existing molecular structure must be taken into account. The evidence for this model is well k n ~ although ~ n it ~ is not often explicitly accounted for in theoretical estimations of ionic solvation energetics.so Now, as the concentration of an electrolyte solution is increased from zero, the overlap of the cospheres of the ions becomes important and contributes to the part of Yex that is characteristic of the electrolyte species. A simple approximation to the effect of overlap of the cospheres is that a volume of solvent equal to the mutual volume of the cospheres is re-
T[L,(AB)
- k.(NaCl)lb
LiCl
NaI
96 f 2 227 f 23 9 19 f 13
64 f 3 -133 f 24 -5
O f 1 1
These coefficients are obtained by fitting the HzO solution data in Figure 4 to eq. 15 in the range from 1to 3 m. The limits of error tabulated here indicate the largest deviations of the coefficients from the value a t 2 M. These coefficients are calculated from Tk,values in Table 11.
For a given electrolyte and molality, the mutual volume of the cospheres is expected to be practically the same in DzO as in HZO. Then the solvent isotope (29) R. W.Gurney, “Ionic Processes in Solution,” Dover Publications, Inc., New York, N. Y., 1953. (30) For recent calculations which appear to demonstrate that the solvation energetics can be accounted for without explicit allowance for the structures in the cospheres, see E. Glueckauf, Trans. Faaraday SOC.,60, 677 (1964),and R. M.Noyes, J . Am. Chem. SOC.,86, 971 (1964). (31) calculations. The same calcula~ H. ~ L. Friedman, ~ ~ unpublished ~ tions indicate that at much lower concentrations the dependence of the mutual volume on concentration is far from proportional and that in the proportional range the overlap of cospheres of and - pairs is as important as that of - pairs. (32) This corresponds to the Akerlof-Thomas rule that In ?+(AB) In y*(KCl) is proportional to m. See G . Akerlof and H. C. Thomas, J. Am. Chem. Soc., 56, 593 (1934). (33) This equation defines the difference in K , for AB from K, for NaC1. The absolute value of K yis the part of Yes that is characteristic of the electrolyte species. In principle, it may be calculated by subtracting from Ye= the primitive model contribution for a 1-1 electrolyte of some standard ion size, but the absolute values are not employed here.
+
Volume YO, Number 1
++
January 1966
172
effect on K,(AB) must be attributed to the solvent isotope effect on AY in reaction 14. For the entropy this difference may be estimated in the following way: we assume that the water within the cospheress4 has the same average molar entropy as the liquid solvent in a hypothetical nonhydrogenbonded state, and that the entropy change in reaction 14 is proportional to the degree of hydrogen-bondedness of the solvent as measured by the parameter XHB of This is the fraction of the N6methy and S ~ h e r a g a . ~ potential hydrogen bonds formed in the equilibrium state of the real solvent. They find that X H B = 0.444 in water at 25" and 4% larger than this in DzO. Therefore, it is reasonable that the entropy decrease of water released from the cospheres when they overlap should be about 453, more in DzO than in Hz0.35136The resulting quantity is compared with the appropriate experimental quantity in the last two rows of Table IV. We conclude that the isotope effect on Sex,that we have determined experimentally, is quite consistent with the model discussed here. It is also clear that there is plenty of room for extending and improving the data as well as for refining the calculations based on the model.36z When one tries to deduce the isotope effect on the ion-solvation entropies from the same model by the method we have used, he encounters the problem that in the solvation process, as opposed to the overlapping of cospheres, all of the solvent in the cosphere is necessarily displaced. It is apparent that the average molar entropy of m-ater in the first solvation layer about an ion such as Li+ is much less than that of pure water in the hypothetical xHB = 0 state. Therefore, in such a calculation one ought to treat the inner and outer parts of the cosphere differently, but this requires the introduction of some parameters to represent the effects in the inner cosphere. The simpler treatment, in which the latter effects are neglected, is summarized in Table V. In this table S h y d is the standard increase in entropy on removing a mole of salt from water relative to that for KaCl as zero. Subtracting the NaCl term serves to isolate the specific effects from those occurring outside the cosphere in the same approximate way as in the excess function calculation. The comparison with experiment in the last three rows of the table suggests that the model discussed here is, according to this very approximate treatment, consistent with these data as well. The same physical interpretation of
YUNG-CHIWu
AND
HAROLD L. FRIEDMAN
Table V: Species-Characteristic Terms of Hydration Entropies, 25' Solute
-Cd. LiCl
Shydo
8
0. 04Shyd ASz'(AB) A&"(AB)
0.3 0.4 0.21
- ASz'(NaCl)b - ASzo(NaC1)"
mole-1 deg.-I-----, KCl NaI
-9 -0.4 -0.14 -0.08
-9 -0.4 -1.3 -0.38
a Standard entropies are taken from W. Latimer, "Oxidation Potentials," Prentice-Hall, New York, N. Y., 1952. For example, the entry for LiCl is calculated from Latimer's tables as 8 = S'(Li+, g) - So(Li+,aq) S'(Na+, aq) - So(Na+,g). Calculated from Table I11 for LiCl and N a I and from Frank and Kerwinllfor KC1. For example, the entry for LiCl is 0.4 = [-312 - (-429)]/298. Calculated from the ionic entropies of transfer tabulated by Swain and Bader.9 These also depend on the enthalpy data of Lange and Martin,1'?18 but are calculated from the enthalpies in a nonthermodynamic way with the use of certain infrared data.
+
part of these data was made earlier by Creyson3' and by Frank and Kerwin.ll
Acknowledgment. We wish to thank Professor Henry
S. Frank for suggesting this problem and for several helpful discussions. (34) It is only necessary to make this assumption for the part of the water subject to displacement when the cospheres overlap as opposed to the part, if any, that is bound to the ion so that it is not displaced when the ion collides with another in the solution. (35) More specifically, we assume that as ZEB changes from 1 (perfect ice crystal) to 0, the entropy changes linearly. If the entropy of vaporization from the hypothetical ZHB = 0 state, which has free volume V O ,to~ the standard gas state, in which the volume per mole is VG,is assumed to be given by R In (VG/VO), then from the known standard entropy36 of vaporization of water ( Z H B = 0.444) and the fact that ZFIB is 4%larger for water than for DzO, we deduce that the standard entropy of vaporization of DsO at 25' ought to be 0.51 gibbs/mole larger than for HzO. The experimental difference*E is 0.90 gibbs/mole. This tends t o support our approximation about the dependence of entropy on ZHB. Of course the exact dependence is implicit in the NQmethy-Scheragacalculations. The same sort of check fails when applied to the crystal-liquid transition. At 3.82' the entropy of fusion36 of HzO is 5.39 gibbs/mole, while that of DzO is 5.42 gibbs/mole. If ZEB = 1 in the crystals before melting, the NBmethy-Scheraga theory with linear dependence of S on ZHB leads one to expect a difference of about 0.5 gibbs/ mole in the opposite direction. (36) K. E;. Kelley, United States Department of the Interior, Bureau of Mines, Bulletin 477, U. S. Government Printing O5ce, Washington, D. C., 1950. (36a) NOTEADDED IN PROOF.Recent measurements we have made of +LZ for CsCl indicate that for this salt The is also positive but it is in poor quantitative agreement with the value given by the expression in the third row of Table IV. (37) J. Greyson, J . Phys. Chem., 66, 2218 (1962).
