Boltzmann's Minimum Function. (16) We have seen that, the law of distribution being 2 X1X2 = n D' nS Let us consider a varied system in which with these same values of x12 x12, &c. the law of distribution is Ce 2T 1+q, where 1+q is any positive function of x1, x2, &c., and g must 9 satisfy the following conditions, namely :— (1) In order that the number of systems may be the same in the varied as in the normal state, (2) In order that the values of x1, x2, &c., may not be altered, SS. · · &c., 157 These conditions being satisfied the system will, on free interchange of energy, pass out of the varied state into the normal state. And it can be now shown that the function B = = ·SS... ƒ (x1. . . x) {log ƒ (≈1 . . . *„)−1}dæ1... dæ diminishes in the process. (x1 For let Bo be the value of B in the normal state, when ns ƒ (x1 . . . x) = Ce, B its value when n8 -n8 2T1+q (log C— nS ...E -1+log1+q)dx1. . . dx„ because =off. =C ... e 1 + q log 1+qdx1 . . . dx E 0 Now since 1+g is positive, 1+qlog 1+q-q is necessarily positive, unless q=0, and is then zero; B-B is therefore positive. And given T and the coefficients a1, b12, &c., B has its least possible value when q=0, or ƒ (x1 And this least possible or minimum value differs by a con (B−B)=log(1+q), and therefore B-B, dimin ishes as q approaches zero. The Second Law of Thermodynamics. (17) In stationary motion the minimum function has the value It is a function of T and the parameters a1, b12, &c., or any parameters v1, v2, &c., on which a1, b12, &c., depend. The second law may from one point of view be regarded as the law of the variation of B when T and the parameters vary very slowly, so that stationary motion is always attained. On this assumption the proofs of the second law depend. We have seen that ST and is independent of the para d meters, or 5=0 for each v. dv It may, therefore, be the case variation of any parameter v. ds ds that work has to be done on av. It will include the work done against all external dv forces. The energy imparted during any small variation of T and the parameters will be denoted by Q. Then the second law requires that shall be a complete differential. (18) It will be sufficient to prove the law for any parameter v on which a1, b12, &c. depend. So far as this proof is concerned, there may be many such. We have in this case JQ=ƏT- av dv XI. On the Relative Strengths or "Avidities" of Weak Acids. By JOHN SHIELDS, D.Sc., Ph.D.* WHE HEN the sodium salt of an acid A is mixed with the equivalent quantity of another acid B, the base will in general be distributed between the two acids, and the ratio of distribution will depend on the relative strengths of the competing acids A and B. If represent that fraction of the neutral sodium salt which is decomposed on the addition of the acid B, then 1will represent the quantity of the original salt which remains undecomposed. By applying Guldberg and Waage's law, after equilibrium has taken place, we get where c and c, denote the velocities of the opposed reactions. The ratio of distribution of the base between the two acids is therefore proportional to the square root of the ratio of the coefficients of velocity, This ratio of distribution gives us the measure of the relative strengths or, as Julius Thomsen has called it, the "avidities" of the acids for any given dilution. As regards the measurement of the ratio of distribution, Thomsen employed a calorimetric method; but Ostwald, Gladstone, Jellet, Wiedemann, and Löwenthal and Lenssen employed various other methods, which, however, are not particularly well adapted for measuring the relative strengths of very weak acids. In the present paper I propose to show how the avidities of the weakest acids may be determined from the rate at which their salt-solutions are hydrolyzed. At this stage it will be convenient to point out that water is here regarded as a weak acid, and potash and soda as the same base since they are equally strong (cf. Reicher, Annalen, ccxxviii. p. 257). Arrhenius (Zeitsch. f. physikal. Chem. v. p. 13, 1890) has shown that when two weak acids compete for the same base, the ratio of distribution is very nearly proportional to *Communicated by the Author. the square roots of the electrolytic dissociation-constants. But the dissociation-constant is arrived at from the equation in which m is the degree of electrolytic dissociation or the ratio of the molecular conductivity at any given dilution, v, to that at infinite dilution. Now, when m is very small, the above equation becomes or, when the dilution for different acids is the same, i. e. the electrolytic dissociation ratios are as the square roots of the dissociation-constants, or directly as the ratios of distribution. According to Guldberg and Waage's law (as enunciated by Julius Thomsen), the ratios of distribution are as the square roots of the velocity-constants. If we call K the velocityconstant in the hydrolysis of aqueous salt-solutions, then ✔K becomes the measure of the dissociation ratio, or the relative strengths of the two weak acids are as K: K. To apply this method to a few specific cases, we may obtain the necessary data in my former paper on hydrolysis in aqueous solutions of salts of strong bases with weak acids (Phil. Mag. [5] xxxv. p. 365, 1893). It was there pointed out that when potassium cyanide, for example, is dissolved in water it is partially decomposed into free acid and free base, and the following equilibrium takes place : KOH+HCN· from which we get the equation C2 KCN+HOH, KOH×HCN=K(KCN×HOH), where K= of the more general equation, C1 c(KOH× HCN)=c(KCN×HOH), in which c1 and c2 are the velocity-coefficients in opposite directions (Guldberg and Waage, Journ. f. pr. Chem. [2] xix. p. 69, 1879). In the present case, however, we desire to study the formation of the salts or the ratio of distribution of the base |