«Earlier Derivation of the Solubility-Product Principle.»—A very simple relation was derived by Nernst[291] for the combined conditions of chemical and of physical equilibrium, where difficultly soluble electrolytes (precipitates) were concerned. We shall develop the relation first for a simple salt, such as silver acetate.
When water is added to solid silver acetate, the salt will dissolve. If an excess of the acetate is used, equilibrium will result between the solid salt and its solution, when the solution is saturated at the temperature used. As the salt dissolves, it is more or less ionized, and in the saturated solution we have a [p140] condition of chemical equilibrium between the salt and its ions:
CH_{ 3}COOAg ⇄ CH_{ 3}COO^{ −} + Ag^{ +}. (1)
If the law of chemical equilibrium is applied to this reversible action, we have (p. 98)
[CH_{ 3}COO^{ −}] × [Ag^{ +}] / [CH_{ 3}COOAg] = K_{ Ionization}. I
The nonionized silver acetate is present in two phases, in the solid phase and also in solution:
Applying the law of physical equilibrium to this system, we have further (p. 121)
[CH_{ 3}COOAg] / [CH_{ 3}COOAg]_{ solid} = K. II
The concentration of a pure solid, as we have seen, may be considered a constant at a given temperature. Consequently, if we consider the question of the size of the solid particles as a minor factor and negligible, we shall conclude, that, for saturated solutions of silver acetate, the concentration of the solid silver acetate being a constant, the concentration of the nonionized or molecular silver acetate, the first term of our constant ratio II, must also have some definite, constant value at a given temperature. We may call this concentration the "molecular solubility" of silver acetate and may put
for a ‹saturated› solution of silver acetate in water at the given temperature. Now, since the concentration of the nonionized silver acetate, [CH_{ 3}COOAg], in the saturated solution also forms the second term of equation I, representing the condition of chemical equilibrium between the acetate and its ions, we obtain, by combining I and III,
[CH_{ 3}COO^{ −}] × [Ag^{ +}] = K_{ Ionization} × K_{ mol. sol.} = K_{ Solubility-Product}. IV
Further, if the assumption is made that the presence of foreign electrolytes, in not too concentrated solutions, does not affect either the molecular solubility, K_{ mol. sol.}, of silver acetate or its tendency to ionize—as expressed in K_{ Ionization}—then, the [p141] relation, which has been developed, would hold for saturated aqueous solutions of silver acetate in the presence of foreign electrolytes, as well as for a saturated, pure, aqueous solution. A single, simple equation would thus express the conditions for simultaneous chemical and physical equilibrium between a difficultly soluble ionogen, of the type of silver acetate, and its saturated solutions, at a given temperature, in the presence or the absence of foreign electrolytes.