Page:Scientific Papers of Josiah Willard Gibbs.djvu/172

136 ideas, let us suppose that $$S_{1}$$ denotes water, and $$S_{2}$$ a salt (either anhydrous or any particular hydrate). The addition of the salt to water, previously in a state capable of equilibrium with vapor or with ice, will destroy the possibility of such equilibrium at the same temperature and pressure. The liquid will dissolve the ice, or condense the vapor, which is brought in contact with it under such circumstances, which shows that $$\mu_{1}$$ (the potential for water in the liquid mass) is diminished by the addition of the salt, when the temperature and pressure are maintained constant. Now there seems to be no a priori reason for supposing that the ratio of this diminution of the potential for water to the quantity of the salt which is added vanishes with this quantity. We should rather expect that, for small quantities of the salt, an effect of this kind would be proportional to its cause, i.e., that the differential coefficient in (211) would have a finite negative value for an infinitesimal value of $$m_{2}.$$ That this is the case with respect to numerous watery solutions of salts is distinctly indicated by the experiments of Wüllner on the tension of the vapor yielded by such solutions, and of Rüdorff on the temperature at which ice is formed in them; and unless we have experimental evidence that cases are numerous in which the contrary is true, it seems not unreasonable to assume, as a general law, that when $$m_{2}$$ has the value zero and is incapable of negative values, the differential coefficient in (211) will have a finite negative value, and that equation (212) will therefore hold true. But this case must be carefully distinguished from that in which $$m_{2}$$ is capable of negative values, which also may be illustrated by a solution of a salt in water. For this purpose let $$S_{1}$$ denote a hydrate of the salt which can be crystallized, and let $$S_{2}$$ denote water, and let us consider a liquid consisting entirely of $$S_{1}$$ and of such temperature and pressure as to be in equilibrium with crystals of $$S_{1}.$$ In such a liquid, an increase or a diminution of the quantity of water would alike cause crystals of $$S_{1}$$ to dissolve, which requires that the differential coefficient in (211) shall vanish at the particular phase of the liquid for which $$m_{2} = 0.$$

Let us return to the case in which $$m_{2}$$ is incapable of negative values, and examine, without other restriction in regard to the substances denoted by $$S_{1}$$ and $$S_{2},$$ the relation between $$\mu_{2}$$ and $$\frac{m_{2}}{m_{1}}$$ for any constant temperature and pressure and for such small values of $$\frac{m_{2}}{m_{1}}$$ that the differential coefficient in (211) may be regarded as having the same constant value as when $$m_{2} = 0,$$ the values of $$t, p,$$ and $$m_{1}$$ being unchanged. If we denote this value of the differential coefficient by