Page:Scientific Papers of Josiah Willard Gibbs.djvu/128

and $b$ parts of a salt which is in contact with vapor of water and crystals of the salt, it is necessary that the value of $\zeta$ for the quantity $a+b$ of the solution should be equal to the sum of the values of $\zeta$ for the quantities a of the vapor and 6 of the salt. Similar propositions will hold true in more complicated cases. The reader will easily deduce these conditions from the particular conditions of equilibrium given on page 74.

In like manner we may extend the definition of $\chi$ to any mass or combination of masses in which the pressure is everywhere the same, using $\epsilon$ for the energy and $v$ for the volume of the whole and setting as before

$\chi =\epsilon +pv.$

(118)

If we denote by $Q$ the heat received by the combined masses from external sources in any process in which the pressure is not varied, and distinguish the initial and final states of the system by accents we have

$\chi ''-\chi '=\epsilon ''-\epsilon '+p(v''-v')=Q.$

(119)

This function may therefore be called the heat function for constant pressure (just as the energy might be called the heat function for constant volume), the diminution of the function representing in all cases in which the pressure is not varied the heat given out by the system. In all cases of chemical action in which no heat is allowed to escape the value of $\chi$ remains unchanged.

Potentials.

In the definition of the potentials $\mu _{1},\mu _{2},$ etc., the energy of a homogeneous mass was considered as a function of its entropy, its volume, and the quantities of the various substances composing it. Then the potential for one of these substances was defined as the differential coefficient of the energy taken with respect to the variable expressing the quantity of that substance. Now, as the manner in which we consider the given mass as composed of various substances is in some degree arbitrary, so that the energy may be considered as a function of various different sets of variables expressing quantities of component substances, it might seem that the above definition does not fix the value of the potential of any substance in the given mass, until we have fixed the manner in which the mass is to be considered as composed. For example, if we have a solution obtained by dissolving in water a certain salt containing water of crystallization, we may consider the liquid as composed of $m_{S}$ weight-units of the hydrate and $m_{W}$ of water, or as composed of $m_{s}$ of the anhydrous salt and $m_{w}$ of water. It will be observed that the values of $m_{S}$ and