Page:Scientific Papers of Josiah Willard Gibbs.djvu/209

only a part of the phases are actually capable of existing, we might still suppose the particular phases which alone can exist to be determined by some other principle than that of the free convertibility of the components (as if, perhaps, the case were analogous to one of constraint in mechanics), it may easily be shown that such a hypothesis is entirely untenable, when the quantities of the proximate components may be varied independently by suitable variations of the temperature and pressure, and of the quantities of the ultimate components, and it is admitted that the relations between the energy, entropy, volume, temperature, pressure, and the quantities of the several proximate components in the gas-mixture are the same as for an ordinary ideal gas-mixture, in which the components are not convertible. Let us denote the quantities of the $n'$ proximate components of a gas-mixture $A$ by $m_{1},m_{2}$, etc., and the quantities of its $n$ ultimate components by ${\mathsf {m}}_{1},{\mathsf {m}}_{2}$, etc. ($n$ denoting a number less than $n'$), and let us suppose that for this gas-mixture the quantities $\epsilon ,\eta ,v,t,p,m_{1},m_{2}$, etc. satisfy the relations characteristic of an ideal gas-mixture, while the phase of the gas-mixture is entirely determined by the values of ${\mathsf {m}}_{1},{\mathsf {m}}_{2}$, etc., with two of the quantities $\epsilon ,\eta ,v,t,p$. We may evidently imagine such an ideal gas-mixture $B$ having $n'$ components (not convertible), that every phase of $A$ shall correspond with one of $B$ in the values of $\epsilon ,\eta ,v,t,p,m_{1},m_{2}$, etc. Now let us give to the quantities ${\mathsf {m}}_{1},{\mathsf {m}}_{2}$, etc. in the gas-mixture $A$ any fixed values, and for the body thus defined let us imagine the $v-\eta -\epsilon$ surface (see page 116) constructed; likewise for the ideal gas-mixture $B$ let us imagine the $v-\eta -\epsilon$ surface constructed for every set of values of $m_{1},m_{2}$, etc. which is consistent with the given values of ${\mathsf {m}}_{1},{\mathsf {m}}_{2}$, etc., i.e., for every body of which the ultimate composition would be expressed by the given values of ${\mathsf {m}}_{1},{\mathsf {m}}_{2}$, etc. It follows immediately from our supposition, that every point in the $v-\eta -\epsilon$ surface relating to $A$ must coincide with some point of one of the $v-\eta -\epsilon$ surfaces relating to $B$ not only in respect to position but also in respect to its tangent plane (which represents temperature and pressure); therefore the $v-\eta -\epsilon$ surface relating to $A$ must be tangent to the various $v-\eta -\epsilon$ surfaces relating to $B$, and therefore must be an envelop of these surfaces. From this it follows that the points which represent phases common to both gas-mixtures must represent the phases of dissipated energy of the gas-mixture $B$.

The properties of an ideal gas-mixture which are assumed in regard to the gas-mixture of convertible components in the above demonstration are expressed by equations (277) and (278) with the equation

$\epsilon =\sum _{1}(c_{1}m_{1}t+m_{1}E_{1}).$

(324)

It is usual to assume in regard to gas-mixtures having convertible components that the convertibility of the components does not affect