Page:Scientific Papers of Josiah Willard Gibbs.djvu/303

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EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES.

267

same phases but without any surface of discontinuity. (A mass thus existing without any surface of discontinuity must of course be entirely surrounded by matter of the same phase.)^[1]

The form in which the values of $\left({\frac {d\sigma }{dt}}\right)_{p}$ and $\left({\frac {d\sigma }{dp}}\right)_{t}$ are given in equation (580) is adapted to give a clear idea of the relations of these quantities to the particular state of the system for which they are to be determined, but not to show how they vary with the state of the system. For this purpose it will be convenient to have the values of these differential coefficients expressed with reference to ordinary components. Let these be specified as usual by $_{1}$ and $_{2}$ .

If we eliminate $d\mu _{1}$ and $d\mu _{2}$ from the equations

-d\sigma =\eta _{S}dt+\Gamma _{1}d\mu _{1}+\Gamma _{2}d\mu _{2},

dp=\eta _{V'}dt+\gamma _{1}'d\mu _{1}+\gamma _{2}'d\mu _{2}

dp=\eta _{V''}dt+\gamma _{1}''d\mu _{1}+\gamma _{2}''d\mu _{2}

we obtain

d\sigma ={\frac {B}{A}}dt+{\frac {C}{A}}dp,

(581)

↑ If we set

V=-{\frac {\Gamma _{'}}{\gamma '}}-{\frac {\Gamma _{''}}{\gamma ''}},

(a)

H_{S}=\eta _{S}-{\frac {\Gamma _{'}}{\gamma '}}\eta _{V'}-{\frac {\Gamma _{''}}{\gamma ''}}\eta _{V''},

(b)

and in like manner

E_{S}=\epsilon _{S}-{\frac {\Gamma _{'}}{\gamma '}}\epsilon _{V'}-{\frac {\Gamma _{''}}{\gamma ''}}\epsilon _{V''},

(c)

we may easily obtain, by means of equations (93) and (507),

E_{S}=tH_{S}+\sigma -pV.

(d)

Now equation (580) may be written

d\sigma =-H_{S}dt+Vdp.

(e)

Differentiating (d), and comparing the result with (e), we obtain

dE_{S}=tdH_{S}-pdV.

(f)

The quantities

E_{S}

and

H_{S}

might be called the superficial densities of energy and entropy quite as properly as those which we denote by

\epsilon _{S}

and

\eta _{S}

. In fact, when the composition of both of the homogeneous masses is invariable, the quantities

E_{S}

and

H_{S}

are much more simple in their definition than

\epsilon _{S}

and

\eta _{S}

, and would probably be more naturally suggested by the terms superficial density of energy and of entropy. It would also be natural in this case to regard the quantities of the homogeneous masses as determined by the total quantities of matter, and not by the surface of tension or any other dividing surface. But such a nomenclature and method could not readily be extended so as to treat cases of more than two components with entire generality.
In the treatment of surfaces of discontinuity in this paper, the definitions and nomenclature which have been adopted will be strictly adhered to. The object of this note is to suggest to the reader how a different method might be used in some cases with advantage, and to show the precise relations between the quantities which are used in this paper and others which might be confounded with them, and which may be made more prominent when the subject is treated differently.

[1] If we set

$V=-{\frac {\Gamma _{'}}{\gamma '}}-{\frac {\Gamma _{''}}{\gamma ''}},$ (a)

$H_{S}=\eta _{S}-{\frac {\Gamma _{'}}{\gamma '}}\eta _{V'}-{\frac {\Gamma _{''}}{\gamma ''}}\eta _{V''},$ (b)
and in like manner

$E_{S}=\epsilon _{S}-{\frac {\Gamma _{'}}{\gamma '}}\epsilon _{V'}-{\frac {\Gamma _{''}}{\gamma ''}}\epsilon _{V''},$ (c)
we may easily obtain, by means of equations (93) and (507),

$E_{S}=tH_{S}+\sigma -pV.$ (d)
Now equation (580) may be written

$d\sigma =-H_{S}dt+Vdp.$ (e)
Differentiating (d), and comparing the result with (e), we obtain

$dE_{S}=tdH_{S}-pdV.$ (f)
The quantities $E_{S}$ and $H_{S}$ might be called the superficial densities of energy and entropy quite as properly as those which we denote by $\epsilon _{S}$ and $\eta _{S}$ . In fact, when the composition of both of the homogeneous masses is invariable, the quantities $E_{S}$ and $H_{S}$ are much more simple in their definition than $\epsilon _{S}$ and $\eta _{S}$ , and would probably be more naturally suggested by the terms superficial density of energy and of entropy. It would also be natural in this case to regard the quantities of the homogeneous masses as determined by the total quantities of matter, and not by the surface of tension or any other dividing surface. But such a nomenclature and method could not readily be extended so as to treat cases of more than two components with entire generality.
In the treatment of surfaces of discontinuity in this paper, the definitions and nomenclature which have been adopted will be strictly adhered to. The object of this note is to suggest to the reader how a different method might be used in some cases with advantage, and to show the precise relations between the quantities which are used in this paper and others which might be confounded with them, and which may be made more prominent when the subject is treated differently.

[1]