Page:Scientific Papers of Josiah Willard Gibbs.djvu/199

The foregoing considerations appear sufficient to justify the definition of an ideal gas-mixture which we have chosen. It is of course immaterial whether we regard the definition as expressed by equation (273), or by (279), or by any other fundamental equation which can be derived from these.

The fundamental equations for an ideal gas-mixture corresponding to (255), (265), and (271) may easily be derived from these equations by using inversely the substitutions given on page 156. They are

$\sum _{1}(c_{1}m_{1})\log {\frac {\epsilon -\sum _{1}(E_{1}m_{1})}{\sum _{1}(c_{1}m_{1})}}=\eta +\sum _{1}(a_{1}m_{1}\log {\frac {m_{1}}{v}}-H_{1}m_{1}),$

(291)

$\sum _{1}(c_{1}m_{1}+a_{1}m_{1})\log {\frac {\chi -\sum _{1}(E_{1}m_{1})}{\sum _{1}(c_{1}m_{1}+a_{1}m_{1})}}=\eta +\sum _{1}\left(a_{1}m_{1}\log {\frac {pm_{1}}{\sum _{1}(a_{1}m_{1})}}-H_{1}m_{1}\right),$

(292)

$\zeta =\sum _{1}\left(E_{1}m_{1}+m_{1}t(c_{1}+a_{1}-H_{1})\right)-\sum _{1}(c_{1}m_{1}+a_{1}m_{1})t\log t+\sum _{1}\left(a_{1}m_{1}t\log {\frac {pm_{1}}{\sum _{1}(a_{1}m_{1})}}\right)\cdot$

(293)

The components to which the fundamental equations (273), (279), (291), (292), (293) refer, may themselves be gas-mixtures. We may for example apply the fundamental equations of a binary gas-mixture to a mixture of hydrogen and air, or to any ternary gas-mixture in which the proportion of two of the components is fixed. In fact, the form of equation (279) which applies to a gas-mixture of any particular number of components may easily be reduced, when the proportions of some of these components are fixed, to the form which applies to a gas-mixture of a smaller number of components. The necessary substitutions will be analogous to those given on page 156. But the components must be entirely different from one another with respect to the gases of which they are formed by mixture. We cannot, for example, apply equation (279) to a gas-mixture in which the components are oxygen and air. It would indeed be easy to form a fundamental equation for such a gas-mixture with reference to the designated gases as components. Such an equation might be derived from (279) by the proper substitutions, But the result would be an equation of more complexity than (279). A chemical compound, however, with respect to Dalton's law, and with respect to all the equations which have been given, is to be regarded as entirely different from its components. Thus, a mixture of hydrogen, oxygen, and vapor of water is to be regarded as a ternary gas-mixture, having the three components mentioned. This is certainly true when the quantities of the compound gas and of its components are all independently variable in the gas-mixture, without change of temperature