Page:Scientific Papers of Josiah Willard Gibbs.djvu/134

It will be observed that $\mu _{a}$, for example, in (124) denotes the potential in the mass considered for a substance $S_{a}$ which may or may not be identical with any of the substances $S_{1},S_{2}$, etc., to which the potentials in (125) relate. Now as the equations between the potentials by means of which the elimination is performed are similar to those which subsist between the units of the corresponding substances (compare equations (38), (43), and (51)), if we denote these units by ${\mathfrak {S}}_{a},{\mathfrak {S}}_{b}$, etc., ${\mathfrak {S}}_{1},{\mathfrak {S}}_{2}$, etc., we must also have

$m'_{a}{\mathfrak {S}}_{a}+m'_{b}{\mathfrak {S}}_{a}+{\text{ etc.}}=A'_{1}{\mathfrak {S}}_{1}+A'_{2}{\mathfrak {S}}_{2}...+A'_{n}{\mathfrak {S}}_{n}.$

(126)

But the first member of this equation denotes (in kind and quantity) the matter in the body to which equations (124) and (125) relate. As the same must be true of the second member, we may regard this same body as composed of the quantity $A'_{1}$ of the substance $S_{1}$ with the quantity $A'_{2}$ of the substance $S_{2}$, etc. We will therefore, in accordance with our general usage, write $m_{1},m_{2}$, etc., for $A'_{1},A'_{2}$, etc., in (125), which will then become

$v'dp=\eta 'dt+m'_{1}d\mu _{1}+m'_{2}d\mu _{2}...+m'_{n}d\mu _{n}.$

(127)

But we must remember that the components to which the $m'_{1},m'_{2}$, etc., of this equation relate are not necessarily independently variable, as are the components to which the similar expressions in (97) and (124) relate. The rest of the $n+1$ equations may be reduced to a similar form, viz.,

$v''dp=\eta ''dt+m''_{1}d\mu _{1}+m''_{2}d\mu _{2}...+m''_{n}d\mu _{n},\,\,{\text{etc.}}$

(128)

By elimination of $d\mu _{1},d\mu _{2},...d\mu _{n}$ from these equations we obtain

${\begin{vmatrix}v'&m'_{1}&m'_{2}&...&m'_{n}\\v''&m''_{1}&m''_{2}&...&m''_{n}\\v'''&m'''_{1}&m'''_{2}&...&m'''_{n}\\.&.&.&...&.\\.&.&.&...&.\end{vmatrix}}dp={\begin{vmatrix}\eta '&m'_{1}&m'_{2}&...&m'_{n}\\\eta ''&m''_{1}&m''_{2}&...&m''_{n}\\\eta '''&m'''_{1}&m'''_{2}&...&m'''_{n}\\.&.&.&...&.\\.&.&.&...&.\end{vmatrix}}dt.$

(129)

In this equation we may make $v',v''$, etc., equal to unity. Then $m'_{1},m'_{2},m''_{1},$, etc., will denote the separate densities of the components

in the different phases, and rf, rf f , etc., the densities of entropy.

When $n=1$,

$(m''v'-m'v'')dp=(m''\eta '-m'\eta '')$

(130)

or, if we make $m'=1$ and $m''=1$, we have the usual formula

${\frac {dp}{dt}}={\frac {\eta '-\eta ''}{v'-v''}}={\frac {Q}{t(v''-v')}},$

(131)

in which $Q$ denotes the heat absorbed by a unit of the substance in passing from one state to the other without change of temperature or pressure.