Page:Scientific Papers of Josiah Willard Gibbs.djvu/171

On the Values of the Potentials when the Quantity of one of the Components is very small.

If we apply equation (97) to a homogeneous mass having two independently variable components $S_{1}$ and $S_{2}$ and make $t,p,$ and $m_{1}$ constant, we obtain

$m_{1}\left({\frac {d\mu _{1}}{dm_{1}}}\right)_{t,p,m_{1}}+m_{2}\left({\frac {d\mu _{2}}{dm_{2}}}\right)_{t,p,m_{1}}=0.$

(210)

Therefore, for $m_{2}=0,$ either

$\left({\frac {d\mu _{1}}{dm_{1}}}\right)_{t,p,m_{1}}=0,$

(211)

ro $\left({\frac {d\mu _{2}}{dm_{2}}}\right)_{t,p,m_{1}}=\infty .$

(212)

Now, whatever may be the composition of the mass considered, we may always so choose the substance $S_{1}$ that the mass shall consist solely of that substance, and in respect to any other variable component $S_{2},$ we shall have $m_{2}=0.$ But equation (212) cannot hold true in general as thus applied. For it may easily be shown (as has been done with regard to the potential on pages 92, 93) that the value of a differential coefficient like that in (212) for any given mass, when the substance $S_{2}$ (to which ra 2 and // 2 relate) is determined, is independent of the particular substance which we may regard as the other component of the mass; so that, if equation (212) holds true when the substance denoted by $S_{1}$ has been so chosen that $m_{2}=0,$ it must hold true without such a restriction, which cannot generally be the case.

In fact, it is easy to prove directly that equation (211) will hold true of any phase which is stable in regard to continuous changes and in which $m_{2}=0,$ if $m_{2}$ is capable of negative as well as positive values. For by (171), in any phase having that kind of stability, $\mu _{1}$ is an increasing function of $m_{1}$ when $t,p$ and $m_{2}$ are regarded as constant. Hence, $\mu _{1}$ will have its greatest value when the mass consists wholly of $S_{1}$ i.e., when $m_{2}=0.$ Therefore, if $m_{2}$ is capable of negative as well as positive values, equation (211) must hold true for $m_{2}=0.$ (This appears also from the geometrical representation of potentials in the $m$–$\zeta$ curve. See page 119.)

But if $m_{2}$ is capable only of positive values, we can only conclude from the preceding considerations that the value of the differential coefficient in (211) cannot be positive. Nor, if we consider the physical significance of this case, viz., that an increase of $m_{2}$ denotes an addition to the mass in question of a substance not before contained in it, does any reason appear for supposing that this differential coefficient has generally the value zero. To fix our