Page:Scientific Papers of Josiah Willard Gibbs.djvu/194

and therefore, by (98), the quantity of any component gas $G_{1}$ in the gas-mixture, and in the separate gas to which $p_{1},\eta _{1},$ etc. relate, is the same and may be denoted by the same symbol $m_{1}$. Also

$\eta =v\left({\frac {dp}{dt}}\right)_{\mu _{1},...\mu _{n}}=v\sum _{1}\left({\frac {dp}{dt}}\right)_{\mu _{1}};$

whence also, by (93)–(96),

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

All the same relations will also hold true whenever the value of $\psi$ for the gas-mixture is equal to the sum of the values of this function for the several component gases existing each by itself in the same quantity as in the gas-mixture and with the temperature and volume of the gas-mixture. For if $p_{1},\eta _{1},\psi _{1},\epsilon _{1},\chi _{1},\zeta _{1};p_{2},$ etc.; etc. are defined as relating to the components existing thus by themselves, we shall have

$\psi =\sum _{1}\psi _{1},$

whence $\left({\frac {d\psi }{dm_{1}}}\right)_{t,v,m}=\left({\frac {d\psi _{1}}{dm_{1}}}\right)_{t,v}\cdot$[1]

Therefore, by (88), the potential $\mu _{1}$ has the same value in the gas-mixture and in the gas $G_{1}$ existing separately as supposed. Moreover,

$\eta =-\left({\frac {d\psi }{dt}}\right)_{v,m}=-\sum _{1}\left({\frac {d\psi }{dt}}\right)_{v,m}=\sum _{1}\eta _{1},$

and $p=-\left({\frac {d\psi }{dv}}\right)_{t,m}=-\sum _{1}\left({\frac {d\psi }{dv}}\right)_{t,m}=\sum _{1}p_{1},$

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

Whenever different bodies are combined without communication of work or heat between them and external bodies, the energy of the body formed by the combination is necessarily equal to the sum of the energies of the bodies combined. In the case of ideal gas-mixtures, when the initial temperatures of the gas-masses which are combined are the same (whether these gas-masses are entirely different gases, or gas-mixtures differing only in the proportion of their components), the condition just mentioned can only be satisfied when the temperature of the resultant gas-mixture is also the same. In such combinations, therefore, the final temperature will be the same as the initial.

If we consider a vertical column of an ideal gas-mixture which is

1. A subscript $m$ after a differential coefficient relating to a body having several independently variable components is used here and elsewhere in this paper to indicate that each of the quantities $m_{1},m_{2},$ etc., unless its differential occurs in the expression to which the suffix is applied, is to be regarded as constant in the differentiation.