Page:Scientific Papers of Josiah Willard Gibbs.djvu/196

to vary, while the composition of the liquid and the temperature remain unchanged. If we denote the increments of pressure and of the potential for $S_{1}$ by $dp$ and $d\mu _{1}$, we shall have by (272)

$d\mu _{1}=\left({\frac {d\mu _{1}}{dp}}\right)_{t,m}^{(L)}dp=\left({\frac {dv}{dm_{1}}}\right)_{t,p,m}^{(L)}dp,$

the index (L) denoting that the expressions to which it is affixed refer to the liquid. (Expressions without such an index will refer to the gas alone or to the gas and liquid in common.) Again, since the gas is an ideal gas-mixture, the relation between $p_{1}$ and $\mu _{1}$ is the same as if the component $S_{1}$ existed by itself at the same temperature, and therefore by (268)

$d\mu _{1}=a_{1}td\log {p_{1}}.$

Therefore ⁠$a_{1}td\log {p_{1}}=\left({\frac {dv}{dm_{1}}}\right)_{t,p,m}^{(L)}dp.$

(285)

This may be integrated at once if we regard the differential coefficient in the second member as constant, which will be a very close approximation. We may obtain a result more simple, but not quite so accurate, if we write the equation in the form

$dp_{1}=\gamma _{1}\left({\frac {dv}{dm_{1}}}\right)_{t,p,m}^{(L)}dp,$

(286)

where $\gamma _{1}$ denotes the density of the component $S_{1}$ in the gas, and integrate regarding this quantity also as constant. This will give

$p_{1}-p'_{1}=\gamma _{1}\left({\frac {dv}{dm_{1}}}\right)_{t,p,m}^{(L)}(p-p'),$

(287)

where $p'_{1}$ and $p'$ denote the values of $p_{1}$ and $p$ when the insoluble component of the gas is entirely wanting. It will be observed that $p-p'$ is nearly equal to the pressure of the insoluble component, in the phase of the gas-mixture to which $p$ relates. $S_{1}$ is not necessarily the only common component of the gas and liquid. If there are others, we may find the increase of the part of the pressure in the gas-mixture belonging to any one of them by equations differing from the last only in the subscript numerals.

Let us next consider the effect of a gas which is absorbed to some extent, and which must therefore in strictness be regarded as a component of the liquid. We may commence by considering in general the equilibrium of a gas-mixture of two components $S_{1}$ and $S_{2}$ with a liquid formed of the same components. Using a notation like the previous, we shall have by (98) for constant temperature,

$dp=\gamma _{1}d\mu _{1}+\gamma _{2}d\mu _{2},$

and ⁠$dp=\gamma _{1}^{(L)}d\mu _{1}+\gamma _{2}^{(L)}d\mu _{2},$

whence ⁠$(\gamma _{1}^{(L)}-\gamma _{1})d\mu _{1}=(\gamma _{2}^{(L)}-\gamma _{2})d\mu _{2}$