The statistical mechanics of regular solutions

Abstract

In a previous paper the methods of statistical mechanics were used to obtain the laws of ideally dilute solutions and of perfect solutions. The methods there used will be extended in the present paper to apply to solutions of a more general type which may be called “regular solutions,” a term due to Hildebrand. These various kinds of solutions may be defined as follows. A solution will be ideally dilute if there are no long range forces between the solute molecules and if the ratio of solute to solvent molecules is so small that of all relevant possible configurations the number of them, in which two solute molecules are within range of each other’s field of force, is negligible. A solution will be perfect if, starting with any given configuration, the interchange of position of any molecule of any one species with any molecule of any other species does not alter the total potential energy of the system. We shall apply the epithet “regular” to solutions whose properties correspond, with fair accuracy, to those of a simplified model which we shall now describe. For the sake of simplicity we shall consider a mixture of two species “A” and “B.” The extension to mixtures of more than two species will be obvious. In our model of a regular solution we postulate first the absence of long-range (electrostatic) forces between the molecules. Our second assumption is that the “A” and “B” molecules may be treated as spheres of at least approximately the same size. Thirdly, we assume that each molecule whether of “A” or “B” is directly surrounded by the same number r of other molecules. If the molecules are close- packed, r will have the value 12, but for our present purpose, there is no need to assign any specified value to r, provided its value is the same for the “A” molecules as for the “B” molecules. Fourthly, we assume that the liquids “A” and “B” mix in all proportions without volume change. Our fifth assumption is that, for varying configurations (all of the same volume) of the system, the total potential energy may be regarded as the sum of contributions of each pair of molecules in direct contact. This assumption is slightly less drastic than assuming that the field of a given molecule does not extend beyond the further side of the next molecule. It is equivalent to ignoring differences between the field of an “A” molecule and that of a “B” molecule at distances exceeding one molecular diameter. Obviously our first assumption is included in our fifth.