Page:Scientific Papers of Josiah Willard Gibbs.djvu/305

of the gases specified in the gas-mixture, and $p_{2},p_{3}$, etc., $\eta _{V2},\eta _{V3}$, etc. the pressures and the densities of entropy due to these several gases, we obtain

$-\left(\eta _{S(1)}-{\frac {\Gamma _{2(1)}}{\gamma _{2}}}-{\frac {\Gamma _{3(1)}}{\gamma _{3}}}\eta _{V3}-{\text{etc.}}\right)dt-{\frac {\Gamma _{2(1)}}{\gamma _{2}}}dp_{2}-{\frac {\Gamma _{3(1)}}{\gamma _{3}}}dp_{3}-{\text{etc.}}$

(585)

This equation affords values of the differential coefficients of $\sigma$ with respect to $t,p_{2},p_{3}$, etc., which may be set equal to those obtained by differentiating the equation between these variables.

Thermal and Mechanical Relations pertaining to the Extension of a Surface of Discontinuity.

The fundamental equation of a surface of discontinuity with one or two component substances, besides its statical applications, is of use to determine the heat absorbed when the surface is extended under certain conditions.

Let us first consider the case in which there is only a single component substance. We may treat the surface as plane, and place the dividing surface so that the surface density of the single component vanishes. (See page 234.) If we suppose the area of the surface to be increased by unity without change of temperature or of the quantities of liquid and vapor, the entropy of the whole will be increased by $\eta _{S(1)}$. Therefore, if we denote by $Q$ the quantity of heat which must be added to satisfy the conditions, we shall have

$Q=t\eta _{S(1)},$

(586)

and by (514),

$Q=-t{\frac {d\sigma }{dt}}=-{\frac {d\sigma }{d\log t}},$

(587)

It will be observed that the condition of constant quantities of liquid and vapor as determined by the dividing surface which we have adopted is equivalent to the condition that the total volume shall remain constant.

Again, if the surface is extended without application of heat, while the pressure in the liquid and vapor remains constant, the temperature will evidently be maintained constant by condensation of the vapor. If we denote by $M$ the mass of vapor condensed per unit of surface formed, and by $\eta _{M'}$ and $\eta _{M''}$ the entropies of the liquid and vapor per unit of mass, the condition of no addition of heat will require that

$M(\eta _{M}''-\eta _{M}')=\eta _{S(1)}.$

(588)