Page:Scientific Papers of Josiah Willard Gibbs.djvu/173

$-{\frac {A}{m_{1}}},$ the value of $A$ will be positive, and will be independent of $m_{1}.$ Then for small values of ${\frac {m_{2}}{m_{1}}}$ we have by (210), approximately,

$m_{2}\left({\frac {d\mu _{2}}{dm_{2}}}\right)_{t,p,m_{1}}=A,$

(213)

i.e., $\left({\frac {d\mu _{2}}{d\log {m_{2}}}}\right)_{t,p,m_{1}}=A.$

(214)

If we write the integral of this equation in the form

$\mu _{2}=A\log {\frac {Bm_{2}}{m_{1}}},$

(215)

$B$ like $A$ will have a positive value depending only upon the temperature and pressure. As this equation is to be applied only to cases in which the value of $m_{2}$ is very small compared with $m_{1},$ we may regard ${\frac {m_{1}}{v}}$ as constant, when temperature and pressure are constant, and write

$\mu _{2}=A\log {\frac {Cm_{2}}{v}},$

(216)

$C$ denoting a positive quantity, dependent only upon the temperature and pressure.

We have so far considered the composition of the body as varying only in regard to the proportion of two components. But the argument will be in no respect invalidated, if we suppose the composition of the body to be capable of other variations. In this case, the quantities $A$ and $C$ will be functions not only of the temperature and pressure but also of the quantities which express the composition of the substance of which together with $S_{2}$ the body is composed. If the quantities of any of the components besides $S_{2}$ are very small (relatively to the quantities of others), it seems reasonable to assume that the value of $\mu _{2},$ and therefore the values of $A$ and $C,$ will be nearly the same as if these components were absent.

Hence, if the independently variable components of any body are $S_{a},...S_{g}$ and $S_{h},...S_{k}$ the quantities of the latter being very small as compared with the quantities of the former, and are incapable of negative values, we may express approximately the values of the potentials for $S_{h},...S_{k}$ by equations (subject of course to the uncertainties of the assumptions which have been made) of the form

$\mu _{h}=A_{h}\log {\frac {C_{h}m_{h}}{v}},$

(217)

$....................................$

$\mu _{k}=A_{k}\log {\frac {C_{k}m_{k}}{v}},$

(218)

in which $A_{h},C_{h},...A_{k},C_{k}$ denote functions of the temperature, the pressure, and the ratios of the quantities $m_{a},...m_{g}.$