Page:Elementary Principles in Statistical Mechanics (1902).djvu/224

that is,

$d\Omega ={\frac {\Omega +\mu _{1}{\overline {\nu }}_{1}\ldots \mu _{h}{\overline {\nu }}_{h}-{\overline {\epsilon }}}{\Theta }}\,d\Theta -\Sigma {\overline {\nu }}_{1}\,d\mu _{1}-\Sigma {\overline {A}}_{1}\,da_{1}.$

(525)

Since equation (503) gives

${\frac {\Omega +\mu _{1}{\overline {\nu }}_{1}\ldots +\mu _{h}{\overline {\nu }}_{h}-{\overline {\epsilon }}}{\Theta }}={\overline {\mathrm {H} }},$

(526)

the preceding equation may be written

$d\Omega ={\overline {\mathrm {H} }}\,d\Theta -\Sigma {\overline {\nu }}_{1}\,d\mu _{1}-\Sigma {\overline {A}}_{1}\,da_{1}.$

(527)

Again, equation (526) gives

$d\Omega +\Sigma \mu _{1}\,d{\overline {\nu }}_{1}+\Sigma {\overline {\nu }}_{1}\,d\mu _{1}-d{\overline {\epsilon }}=\Theta \,d{\overline {\mathrm {H} }}+{\overline {\mathrm {H} }}\,d\Theta .$

(528)

Eliminating $d\Omega$ from these equations, we get

$d{\overline {\epsilon }}=-\Theta \,d{\overline {\mathrm {H} }}+\Sigma \mu _{1}\,d{\overline {\nu }}_{1}-\Sigma {\overline {A}}_{1}\,da_{1}.$

(529)

If we set

$\Psi ={\overline {\epsilon }}+\Theta {\overline {\mathrm {H} }},$

(530)

$d\Psi =d\epsilon +\Theta \,d{\overline {\mathrm {H} }}+{\overline {\mathrm {H} }}\,d\Theta ,$

(531)

we have

$d\Psi ={\overline {\mathrm {H} }}\,d\Theta +\Sigma \mu _{1}\,d{\overline {\nu }}_{1}-\Sigma {\overline {A}}_{1}\,da_{1}.$

(532)

The corresponding thermodynamic equations are

$d\epsilon =T\,d\eta +\Sigma \mu _{1}\,dm_{1}-\Sigma A_{1}\,da_{1},$

(533)

$\psi =\epsilon -T\eta ,$

(534)

$d\psi =-\eta \,dT+\Sigma \mu _{1}\,dm_{1}-\Sigma A_{1}\,da_{1}.$

(535)

These are derived from the thermodynamic equations (114) and (117) by the addition of the terms necessary to take account of variation in the quantities ($m_{1}$, $m_{2}$, etc.) of the several substances of which a body is composed. The correspondence of the equations is most perfect when the component substances are measured in such units that $m_{1}$, $m_{2}$, etc., are proportional to the numbers of the different kinds of molecules or atoms. The quantities $\mu _{1}$, $\mu _{2}$, etc., in these thermodynamic equations may be defined as differential coefficients by either of the equations in which they occur.^[1]

1. Compare Transactions Connecticut Academy, Vol. III, pages 116 ff.