that is,

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

(525)

Since equation (503) gives

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

(526)

the preceding equation may be written

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

(527)

Again, equation (526) gives

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

(528)

Eliminating ${\displaystyle d\Omega }$ from these equations, we get

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

(529)

If we set

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

(530)

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

(531)

we have

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

(532)

The corresponding thermodynamic equations are

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

(533)

${\displaystyle \psi =\epsilon -T\eta ,}$

(534)

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

(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 (${\displaystyle m_{1}}$, ${\displaystyle m_{2}}$, 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 ${\displaystyle m_{1}}$, ${\displaystyle m_{2}}$, etc., are proportional to the numbers of the different kinds of molecules or atoms. The quantities ${\displaystyle \mu _{1}}$, ${\displaystyle \mu _{2}}$, 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.