Page:Scientific Papers of Josiah Willard Gibbs.djvu/133

Or, when the $r$ bodies considered have not the same independently variable components, if we still denote by $n$ the number of independently variable components of the $r$ bodies taken as a whole, the number of independent variations of phase of which the system is capable will still be $n+2-r$. In this case, it will be necessary to consider the potentials for more than $n$ component substances. Let the number of these potentials be $n+h$. We shall have by (97), as before, $r$ relations between the variations of the temperature, of the pressure, and of these $n+h$ potentials, and we shall also have by (43) and (51) $h$ relations between these potentials, of the same form as the relations which subsist between the units of the different component substances.

Hence, if $r=n+2$, no variation in the phases (remaining coexistent) is possible. It does not seem probable that $r$ can ever exceed $n+2$. An example of $n=1$ and $r=3$ is seen in the coexistent solid, liquid, and gaseous forms of any substance of invariable composition. It seems not improbable that in the case of sulphur and some other simple substances there is more than one triad of coexistent phases; but it is entirely improbable that there are four coexistent phases of any simple substance. An example of $n=2$ and $r=4$ is seen in a solution of a salt in water in contact with vapor of water and two different kinds of crystals of the salt.

Concerning $n+1$ Coexistent Phases.

We will now seek the differential equation which expresses the relation between the variations of the temperature and the pressure in a system of $n+1$ coexistent phases ($n$ denoting, as before, the number of independently variable components in the system taken as a whole). In this case we have $n+1$ equations of the general form of (97) (one for each of the coexistent phases), in which we may distinguish the quantities $\eta ,v,m_{1},m_{2},$ etc., relating to the different phases by accents. But $t$ and $p$ will each have the same value throughout, and the same is true of $\mu _{1},\mu _{2},$, etc., so far as each of these occurs in the different equations. If the total number of these potentials is $n+h$, there will be $h$ independent relations between them, corresponding to the $h$ independent relations between the units of the component substances to which the potentials relate, by means of which we may eliminate the variations of $h$ of the potentials from the equations of the form of (97) in which they occur.

Let one of these equations be

$v'dp=\eta 'dt+m'_{a}d\mu _{a}+m'_{b}d\mu _{b}+{\text{ etc.,}}$

(124)

and by the proposed elimination let it become

$v'dp=\eta 'dt+A'_{1}d\mu _{a}+A'_{2}d\mu _{b}...+A'_{n}d\mu _{n}.$

(125)