Page:Scientific Papers of Josiah Willard Gibbs.djvu/193

The values of $c,a,E,$ and $H$ will then be constant and $m$ will denote the total quantity of gas. As the equation will thus be reduced to the form of (260), it is evident that an ideal gas-mixture, as defined by (273) or (279), when the proportion of its components remains unchanged, will have all the properties which we have assumed for an ideal gas of invariable composition. The relations between the specific heats of the gas mixture at constant volume and at constant pressure and the specific heats of its components are expressed by the equations

$c=\sum _{1}{\frac {m_{1}c_{1}}{m}},$

(280)

and $c+a=\sum _{1}{\frac {m_{1}(c_{1}+a_{1})}{m}}\cdot$

(281)

We have already seen that the values of $t,v,m_{1},\mu _{1}$ in a gas-mixture are such as are possible for the component $G_{1}$ (to which $m_{1}$ and $\mu _{1}$ relate) existing separately. If we denote by $p_{1},\eta _{1},\psi _{1},\epsilon _{1},\chi _{1},\zeta _{1}$ the connected values of the several quantities which the letters indicate determined for the gas $G_{1}$ as thus existing separately, and extend this notation to the other components, we shall have by (273), (274), and (279)

$p=\sum _{1}p_{1}$$\eta =\sum _{1}\eta _{1}$$\psi =\sum _{1}\psi _{1};$

(282)

whence by (87), (89), and (91)

$\epsilon =\sum _{1}\epsilon _{1}$$\chi =\sum _{1}\chi _{1}$$\zeta =\sum _{1}\zeta _{1}.$

(283)

The quantities $p,\eta ,\psi ,\epsilon ,\chi ,\zeta$ relating to the gas-mixture may therefore be regarded as consisting of parts which may be attributed to the several components in such a manner that between the parts of these quantities which are assigned to any component, the quantity of that component, the potential for that component, the temperature and the volume, the same relations shall subsist as if that component existed separately. It is in this sense that we should understand the law of Dalton, that every gas is as a vacuum to every other gas.

It is to be remarked that these relations are consistent and possible for a mixture of gases which are not ideal gases, and indeed without any limitation in regard to the thermodynamic properties of the individual gases. They are all consequences of the law that the pressure in a mixture of different gases is equal to the sum of the pressures of the different gases as existing each by itself at the same temperature and with the same value of its potential. For let $p_{1},\eta _{1},\psi _{1},\epsilon _{1},\chi _{1},\zeta _{1};p_{2},$ etc.; etc. be defined as relating to the different gases existing each by itself with the same volume, temperature, and potential as in the gas-mixture; if

$p=\sum _{1}p_{1},$

then $\left({\frac {dp}{d\mu _{1}}}\right)_{t,\mu _{2},...\mu _{n}}=\left({\frac {dp_{1}}{d\mu _{1}}}\right)_{t};$