Page:Scientific Papers of Josiah Willard Gibbs.djvu/307

Rh We must notice the difference between this formula and (587). In (593) the quantity of heat $$Q$$ is determined by the condition that the temperature and pressures shall remain constant. In (587) these conditions are equivalent and insufficient to determine the quantity of heat. The additional condition by which $$Q$$ is determined may be most simply expressed by saying that the total volume must remain constant. Again, the differential coefficient in (593) is defined by considering $$p$$ as constant; in the differential coefficient in (587) $$p$$ cannot be considered as constant, and no condition is necessary to give the expression a definite value. Yet, notwithstanding the difference of the two cases, it is quite possible to give a single demonstration which shall be applicable to both. This may be done by considering a cycle of operations after the method employed by Sir William Thomson, who first pointed out these relations. The diminution of volume (per unit of surface formed) will be and the work done (per unit of surface formed) by the external bodies which maintain the pressure constant will be  Compare equation (592).

The values of $$Q$$ and $$W$$ may also be expressed in terms of quantities relating to the ordinary components. By substitution in (593) and (595) of the values of the differential coefficients which are given by (581), we obtain where $$A, B$$, and $$C$$ represent the expressions indicated by (582)–(584). It will be observed that the values of $$Q$$ and $$W$$ are in general infinite for the surface of discontinuity between coexistent phases which differ infinitesimally in composition, and change sign with the quantity $$A$$. When the phases are absolutely identical in composition, it is not in general possible to counteract the effect of extension of the surface of discontinuity by any supply of heat. For the matter at the surface will not in general have the same composition as the homogeneous masses, and the matter required for the increased surface cannot be obtained from these masses without altering their phase. The infinite values of $$Q$$ and $$W$$ are explained by the fact that when the phases are nearly identical in composition, the extension of the surface of