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T o test the thermodynamic feasibility of their proposal, assume that CaF2 exists as a continuous series of nearly stoichiometric solids in which the chemical potential of the component CaFz varies as a function of the defect content. Assume further that one of these solids with a solubility product of 3.4 X (solid A) is placed into a solution in equilibrium with solid B (ion activity product = 6.5 X According to the thesis of Stearns and Berndt, the solution in equilibrium with solid B would tend to convert the defect structure of solid A into that of solid B with an increase in Gibbs free energy of 0.4 kcal/mol of solid A. Since this process would not be accompanied by a significant change in the composition of the solution or in the chemical potential of CaFz in solid B, the net result would be a spontaneous increase in the Gibbs energy of the system, a thermodynamic impossibility. Actually, CaF2 from the solution should precipitate on solid A because this would be accompanied by a decrease in Gibbs energy; the consequent decrease in the solution concentration would lead to dissolution of solid B, with a further decrease in Gibbs energy, so that at final equilibrium only solid A would be present and the ion activity product in the solution would be 3.4 X 10-12. Thus, in general, only the solid with defect composition corresponding to the lowest solubility product (i-e., lowest chemical potential of the component CaF2) would be the true equilibrium solid within any series of defect solids with nearly the same composition but different Gibbs energies. I t is shown next that this conclusion is compatible with the Gibbs-Duhem equation. Of various possible equilibrium models for the defect structure of CaF2, Stearns and Berndt used one in which Ca2+ and F- can be defined as the components. This choice, since it does not take into account neutrality of charge, cannot lead to significant variations in stoichiometry, and it is contrary to their eq 1-4 which are applicable only when the liquid and solid phases are a t the same potential. However, the GibbsDuhem equation applied to their model gives
mcaz+dpcaz+= -mF- dpFwhere mca2+and mF- closely approximate the stoichiometric coefficients 1and 2, respectively, and p is a chemical potential. Thus, as long as the composition of the solid is nearly stoichiometric, a change in the chemical potential of Ca2+ would have a compensatory change in the chemical potential of F-, and the chemical potential of CaF2 in the solid would remain essentially constant. More suitable, electrically neutral models of the defect solid represented by other choices of components (e.g., Ca and F,; CaF2 and Ca or F2; and Ca(OH)2, HF, and H20) can be shown to yield the same result. Stearns and Berndt chose to ignore the 100-fold difference in the solubility products they calculated from the data of Aum6ras2 as compared with those derived from their own data. Instead, they noted an apparent similarity in that both sets of data seem to define similarly shaped curves, but with less than twofold variations in the apparent solubility products. It seems tenuous, however, to attribute significance to a small variation and at the same time to disregard a 100-fold difference. I t is more reasonable to attribute the variation in their apparent solubility product to inadequate descriptions of the equilibria within the solution phase, to difficulties in calculating the activities of Ca2+and F- ions at such high ionic strengths, and to experimental error. These considerations do not preclude either of the two possibilities: (i) that solids prepared under equilibrium conditions can vary in composition (although these variations tend to be small except at elevated temperatures), and (ii) that
the Gibbs energies of stoichiometric solids prepared under irreversible conditions may exceed their equilibrium values. The proposal of Stearns and Berndt, however, relates to the Gibbs energy of a nearly stoichiometric solid under equilibrium conditions. This proposal, as shown here, is contrary to theory and is based on inadequate experimental evidence; it should not be accepted as a satisfactory basis for invalidating the useful and well-established solubility product concept when it is applied to a solid with sensibly constant composition. References and Notes (1) R. I. Stearns and A. F. Berndt, J. Phys. Chem., 80, 1060 (1976) (2) M. Aumeras, J. Chim. Phys. Phys., Chim. Biol., 24, 548 (1927).
American Dental Association Health Foundation Research Unit National Bureau of Standards Washington, D.C. 20234
Walter E. Brown
Received August 2, 1976
Reply to Comments on Solubility Product Variation
Sir: Since the appearance of our recent article1 a number of investigators have written commentaries on this work. Three letters are reproduced here and others have been sent directly to us. Some of the commentators are disturbed by the conclusion and find it difficult to accept. Our paper challenges a concept universally accepted for over a century and, therefore, this is to be expected. In trying to show that our conclusion is invalid, some of these commentators focus the thrust of their criticism on the experimental results and the calculations by which they were derived. In so doing they miss the main point. Our paper is a theoretical paper and not an experimental one. Our theory is based on the logical and systematic application of the basic laws of thermodynamics. It is not the result of an experimental investigation. We presented a theoretical, deductive proof that the classical solubility product is not constant a t constant temperature and pressure but must vary with activities in the liquid phase. We do not believe that the validity of the deductive proof has been seriously questioned. Experimental data were included in the paper because a theoretical argument is more easily accepted if experimental data are available to support it. If these data were not there, then one could say, “Yes, theoretically the classical solubility product does vary but the variation is far too small to be measured experimentally and is therefore of no consequence”. Without experimental data we could not answer this. We agree that the Debye-Huckel limiting law has its shortcomings and perhaps we should have considered the formation of complex ions other than the CaAc+. Many of the other criticisms of the experimental techniques and calculations have a degree of validity. In this connection it is interesting to note that Professor Richard W. Ramette of Carlton College has recalculated our results using the Davies equation for the approximation of activities and found that the maximum persists. His results support our theory and are satisfying in that his values of the solubility product calculated from our The Journal of Physical Chemistry, Vol. 80, No. 24, 1976
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data are closer to the accepted literature value. Moreover, he has obtained independent experimental data with CaFz that further support our theory. In an attempt to overcome Pitzer's objections surrounding the validity of the Debye-Huckel limiting law we have conducted some experiments in which the total ionic strength was quite low (
