2707
Comrnunicatlons to the Editor
COMMUNICATIONS TO THE EDITOR
Alleged Solubility Product Variability at Constant Pressure and Temperature
Sir: In a recent paper, Stearns and Berndtl ascribe apparent variations in solubility product with solution composition to the effects of nonstoichiometric composition of the solid phase which was calcium fluoride. At high temperatures, departures from stoichiometric composition occur for compounds of multivalent elements, but it seems most unlikely that any actual departure of a compound such as CaF2 from stoichiometric composition a t room temperature could cause effects of the magnitude noted. On the other hand, the treatment by Stearns and Berndt of the activity coefficients in solution could easily be in error by the variation noted. Stearns and Berndt assumed that the activity coefficient for a given solute was a function purely of ionic strength and independent of the specific ions actually present. While that is an acceptable basis for rough estimates, it has been known, at least since the papers of Br@nsted,2that activity coefficients actually depend on interactions among the particular ions present, and Guggenheim and Turegon3 presented a method of representing these specific ion interactions. Kim and the writer4 recently developed a refined set of equations appropriate for mixed electrolytes yielding excellent agreement with data for mixtures as complex as sea water as well as for several mixtures of three solutes and a large number of mixtures of two solutes. The solutions used by Stearns and Berndtl involving acetate buffers, could, in principle, be treated by the equations of Pitzer and Kim,4 but the calculations would be complex and not all of the parameters are readily available. The effects of specific ion interaction on the activity coefficient for CaF2 can be illustrated, however, in simpler systems where the important parameters have been tabu1ated.j For example, consider solutions of constant ionic strength I with NaCl(m1) and either CaClZ(m2) or NaF(m3). If we neglect the solubility of CaF2, I = ml 3mz or I = ml m3. Equation 15 of Pitzer and Kim4 then yields In y(CaF2) in terms of the interaction parameters /I(1) and /I(1) for NaCl, CaClZ, and NaF, all of which are kn0wn.j The parameter for Na+-Caz+ interaction4 is known to be zero; those for F--Cl- and for CaZf-F- interactions are not known and will be neglected, but they could not change our general conclusion. Table I gives the ratio of the product y ~ ~ z + yfor ~ -a*solution with CaCl2 or NaF added to that for the solution of NaCl alone of the same ionic strength. This ratio would be unity for the treatment used by Stearns and Berndt. An ionic strength of 0.6 was chosen to correspond to their conditions. It is noted that the ratio calculated from the equations recognizing specific ion effects differs from unity by nearly a factor of 2 for 0.1 M excess Ca2+. In the paper of Stearns and Berndt the maximum departure from constancy of their solubility product was also about a factor of 2 for a similar 0.1 M excess of Ca2+.Hence it is clear that the entire effect of Stearns and Berndt is no larger than variations which might be expected
+
+
TABLE I: Calculated Values of the Ratio of the Product yCsZ+YF-’ for Solutions with Excess CaClz or NaF to that for NaCl a t Constant Ionic Strength of 0.6 M
m(CaC12)
Ratio
m (NaF)
Ratio
0.01
0.953 0.908 0.791 0.632
0.01
0.976 0.953 0.887 0.784
0.02 0.05 0.1
0.02 0.05 0.1
in view of the approximations in their calculations of activity coefficients. Thus there is no need to assume any variation in solubility product of CaFz with solution composition a t constant pressure and temperature. References and Notes (1) R. I. Stearns and A. F. Berndt, J. Phys. Chem., 80, 1060 (1976). (2) J. N. Bronsted, Kgl. Dan. Vidensk. Selsk., Mat. Fys. Medd., 4, (4)(1921); J. Am. Chem. SOC.,44, 677 (1922); 45, 2898 (1923). (3) E. A. Guggenheim and J. C . Turgeon, Trans. Faraday SOC., 51, 747 (1955). (4) K. S. Pitzer and J. J. Kim, J. Am. Chem. SOC.,96, 5701 (1974). ( 5 ) K. S. Pitzer and G. Mayorga, J. Phys. Chem., 77, 2300 (1973).
Department of Chemistry University of California Berkeley, California 94720 Received June 14, 1976
Kenneth S. Pitzer
On the Variable So’ubility Product of Calcium Fluoride Publication costs assisted by the University of Missouri-Rolla
Sir: Recently, Stearns and Berndt presented a theoretical development and experimental evidence for the variability of the solubility product of calcium fluoride at constant temperature and pressure.l Basing their theoretical development on the statement (referred to “any standard text on physical chemistry”), “When two phases are in equilibrium then the chemical potentials of each and every chemical species are the same in both phases,” they equate the chemical potentials of individual ions in the solid and aqueous phases, and proceed to a thermodynamic proof that the chemical potential of the solid must vary as the concentrations of the ions in solution are varied. The quoted statement above may a t first glance appear to be a paraphrasing of a statement by Moore,2 “For any component i in the system, the value of the chemical potential wL1must be the same in every phase, when the system is in equilibrium at constant T and P.” However, shortly preceding this statement, Moore specifically defines components as, “. . . those constituents the concentrations of which may be independently varied in the various phases.” Clearly, the concentrations of cations and anions in a phase The Journal of Physlcal Chemistry, Vol. 80, No. 24, 1976
