ABHIJIT PURKAYASTHA AND JOHNWALKLEY
2138
Scaled Particle Theory for Nonelectrolyte Solutions of Dilute Solid Solutes by Abhijit Purkayastha and John Walkley* Department of Chemistry, Simon Fraser University, Burnaby 3, British Columbia, Canada
(Received October
4, 1971)
Pihblication costs assisted by the Department of Chemistry, S i m o n Fraser University
The entropy of solution of a solid solute dissolved in a nonpolar solvent is examined in terms of a modified Yosim cycle. I t is seen to give good agreement with experiment even for those systems wherein the solute-solvent molar volume ratio is large. The scaled particle theory, devcloped by Reiss, et uZ.,l as a statistical mechanical theory for fluids, has been used by srveral workers to predict the thermodynamic properties of dissolved g a s e ~ . ~Common ,~ to most theories attempting to predict these properties is the assumption that the dissolution process consists of two stages: (i) the creation of a cavity in the solvent to accommodate the solute molecule and (ii) the introduction of the solute molccule into the cavity. For the solubility properties of a solid solute the two-step "hole making-hole filling" process is not suitable. One of the more interesting thermodynamic properties of such systems is the entropy of solution or, in usual notation, the (sz - szs) term.4 One route to the prediction of this property using scaled particle theory lies in the following imaginary six-step process similar to one discussed by Yosim.6 Step I . The pure solid solute at the experimental temperature is converted to a hypothetical supercooled liquid at that temperature. Scaled particle theory is not applicable to the solid phase nor does it correctly describe the melting process of solids.6 The entropy change for step I must therefore be obtained from experimental heat capacity data Aszl = nZ(Asf298) = nz(Asf,,,,*
- ACps - P la Tmpt/298)
(the usual notation is used throughout). Step I I . The liquid solvent and the liquid solute are separately "discharged" at constant volume to hardsphere fluids. AS''
=
- (niAS''charge(1)
'hAS''ohrtrge(2))
Step III. The hard-sphere fluids are vaporized at constant temperature to a hard-sphere gas. Yosim6 obtained from scaled particle theory the entropy of vaporization of a hard-sphere fluid as
diameter of the substance in the gas phase, and Vg is the molar volume of the substance in the gas phase, assumed to obey ideal gas laws. Step IV. The ideal hard-sphere gases produced in step I11 are mixed at constant pressure. If we assume ideal mixing, then AsIv = -nlR In XI - nzR In xz
where x1 and xz are the mole fraction compositions of solvent and solute. Step V . The hard-sphere gas mixture is compressed to the volume of the solution, Vsp. Using the Lebomitzlb equation of state for a mixture of hard spheres the entropy charge associated with this step can be written Asv = -R(nl
+ nz) X + 6a13a23+ x1xz2(6a13a23 -
zl2xZ(6al5az - 3al4az2
+ 6~1~2')+ - ~ c V ~ P [ X+~ ' U ~ ~ x1r2(61.1aZ2+ a1%) + xZ2aZ3][~Ps,Z98 -
3UizU24
3
-I
Here Vp is the molar volume of the liquid, c = nN0/6, where No is Avogadro's number, aiis the hard-sphere T h e Journal of Physical Chemistry, Vol. 76,No. 15, 1979
2
~
~
2
~
1
c(zla13
+
~ 2 ~ 2 1-2 3 )
1
Step V I . The compressed mixture of hard spheres is now "recharged," the resulting solution being equivalent to the solvent saturated with the solute at the temperature and pressure of the experiment. Asv1 = (nl inz)ASV'chargdi,z) We define an excess entropy term AsE = AsM -
niR In xi i = 1,2
where AsM is the total entropy of mixing. Collecting (1) (a) H. Reiss, H. L. Frisch, and J. L. Lebowitz, J . Chem. Phys., L. Lebowitz, Phys. Rea. A, 133, 895 (1964). (2) R. Pierotti, J. Phys. Chem., 67, 1840 (1963). (3) E. Wilhelm and R. Battino, J . Chem. Thermodyn., 3, 379 (1971). (4) J. H . Hildebrand and R. L. Scott, "Solubility of Nonelectrolytes," Dover Publications, New York, N. Y., 1964. (5) S. J. Yosim, J . Chem. Phys., 43, 286 (1965). (6) S. J. Yosim and B. B. Owens, ibid., 39, 2222 (1963). 31, 369 (1959); (b) J.
3cRn,a,3(2Vp - CUi3) 2(Vp - CUi3)
~
SCALED PARTICLE THEORY FOR KONELECTROLYTE SOLUTIONS all entropy changes in steps I through VI and making the assumption that xz + 0 (ie., very dilute solution) and the consequent assumption that for such dilute solutions the sum of the entropy changes associated with the “charging” and “discharging” steps mag be ignored,2 then we can formulate the excess entropy term. Making the substitution Vsp = n1V1 nzVz, we can then write the partial molal entropy of solution for component 2 of the mixture (the dilute solute) as
+
ASzE = AS1
- R In
(Vzp
+
- caz3)
+ + +
3cRuz3(2Vw~ - c a z 3 ) / ( Vz ~~ 1 1 , ~ ~ ) ~ - 6cul3V1P}/(V1~ - ca13)z l/zR{ R ( V Z-- C U ~ ~ ) / [ X I ~xzVz ~ -~(51~1~ xzaz3)1 R In [ ( X I ~ I Z Z ~ Z ) - ~ ( 5 1 ~ 1xzaz3)1 ~
+
+ +
ASzE(exptl)= R ( b In xz/b In T ) - R In x2 (for these very dilute solutions, since R ( b In az/b In 5 2 ) ‘v 1). In Table I we compare experimental excess entropy of mixing data for iodine and for stannic iodide in a range of solvents with those values calculated from the expression derived above.’!* The agreement is seen to be very good. Table I : Excess Partial Molal Entropy of Solution
System
a
See ref 7.
See ref 8.
term associated with the first of the proposed steps, AS,‘. This term should be of constant valuc irrespective of the solvent, and in Table I1 we give values for this term obtained by subtracting the sum of the entropy terms ASP from the experimental excess entropy of solution. It is seen that a reasonable constancy is found for the AS2 term for the three solvents for which experimental data exist. Similar agreement is observed for the perfluoro compound, Sn(C6FJ4, for which the solute-solvent volume disparity is as large as 5 : 1.
+
4.34 4.50 5.79 4.71 1.35 2.42
5 . 04a 4.42a 5 . 24a 4.86“ 1.67b 3.05b
+
Table I1 : Comparison of Asfzss Values
+
The above equation gives the excess entropy of component 2 at constant pressure and may be compared to the experimental term
Iz-CClr Iz-C-CsHie 12-CClzFCClFz Iz-i-C~His Sn14-CClr Sn14-CC1~FCClF~
2139
System
sfm(calcd), cal deg-1 mol-1
Sdexptl), cal deg-1 mol-1
Sn(CsH6)4-c-CsHl~ Sn(CsHj)4-CC14 Sn(CeH;)4-CClzFCClFz Sn (C~HF,)~-CC~~ Sn(CeH:)4-c-C& Sn(CsH:)4-toluene
9.8 7.8 8.2 7.0 7.2 9.0
32. 87a 2 7 . 365 33.27“ 2 8 . 38b 33.41b 30.22b
a
See ref 9.
b
- R In zs(expt1)
16.7ga 13. 566 19.145 10,66b 15.51b 10 62b
See ref 10.
If step I is removed, the process can obviously be adapted to the calculation of the (Sz - &g) term for gases dissolved in nonpolar solvents. Battino3 has recently extended the Pierotti2 theory to the calculation of (h - @). Use of our proposed process for SFa gas gives values of -11.6 ( C - C ~ Hsolvent) I~ and -11.32 solvent), which compare well to the values obtained by Battino of -11.5 and -11.0; experimental values are -15.90 and -14.5 eu, respectively. Values obtained from the two theories agree, the result being independent of the supposed process of solution. I t would seem that as construed above scaled particle theory can be used to calculate the entropy of solution of dilute nonelectrolyte systems and that solute-solvent size disparity does not throw any particular strain upon the use of the theory.
Systems which show a considerable disparity in solute-solvent molar volumes are of great interest. Even regular solution theory must concede that there (7) J. H . Hildebrand and R. L. Scott, “Regular Solutions,” Prenticeis an entropy term for such systems for which only apHall, Englewood Cliffs, E.J., 1962. proximate theoretical expressions have been d e r i ~ e d . ~ (8) E . B. Smith and J. Walkley, Trans. Faraday Sac., 5 6 , 1276 (1960). For a solute such as tetraphenyltin(1V) the excess (9) M. Vitoria and J. Walkleg, ibid., 6 5 , 57, 62 (1969). entropy of mixing is large; see Table II.9110 Unfor(10) A . Purkayastha, Ph.D. Thesis, Simon Fraser University, tunately, for this solute we cannot evaluate the entropy Burnaby 2, British Columbia, Canada, 1971.
The Journal of Physical Chemistry, Vol. 76, KO.16,2972
