Page:Scientific Papers of Josiah Willard Gibbs.djvu/399

constant volume, $a$ denotes the constant value of $pv\div mt,{\text{E}}$ and ${\text{H}}$ depend upon the zeros of energy and entropy. The two last equations may be abbreviated by the use of different constants. The properties of fundamental equations mentioned above may easily be verified in each case by differentiation.

The law of Dalton respecting a mixture of different gases affords a point of departure for the discussion of such mixtures and the establishment of their fundamental equations. It is found convenient to give the law the following form:—

The pressure in a mixture of different gases is equal to the sum of the pressures of the different gases as existing each by itself at the same temperature and with the same value of its potential.

A mixture of ideal gases which satisfies this law is called an ideal gas-mixture. Its fundamental equation in $p,t,\mu _{1},\mu _{2}$, etc. is evidently of the form

$p=\textstyle \sum _{1}\displaystyle \left(a_{1}e^{\frac {{\text{H}}_{1}-c_{1}-a_{1}}{a_{1}}}t^{\frac {c_{1}+a_{1}}{a_{1}}}e^{\frac {\mu _{1}-{\text{E}}_{1}}{a_{1}t}}\right),$

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where $\textstyle \sum _{1}$ denotes summation with respect to the different components of the mixture. From this may be deduced other fundamental equations for ideal gas-mixtures. That in $\psi ,t,v,m_{1},m_{2}$, etc. is

$\psi =\textstyle \sum _{1}\displaystyle \left({\text{E}}_{1}m_{1}+m_{1}t\left(c_{1}-{\text{H}}_{1}-c_{1}\log t+a_{1}\log {\frac {m_{1}}{v}}\right)\right)\cdot$

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Phases of dissipated energy of ideal gas-mixtures.—When the proximate components of a gas-mixture are so related that some of them can be formed out of others, although not necessarily in the gas-mixture itself at the temperatures considered, there are certain phases of the gas-mixture which deserve especial attention. These are the phases of dissipated energy, i.e., those phases in which the energy of the mass has the least value consistent with its entropy and volume. An atmosphere of such a phase could not furnish a source of mechanical power to any machine or chemical engine working within it, as other phases of the same matter might do. Nor can such phases be affected by any catalytic agent. A perfect catalytic agent would reduce any other phase of the gas-mixture to a phase of dissipated energy. The condition which will make the energy a minimum is that the potentials for the proximate components shall satisfy an equation similar to that which expresses the relation between the units of weight of these components. For example, if the components were hydrogen, oxygen and water, since one gram of hydrogen with eight grams of oxygen are chemically equivalent to nine grams of water, the potentials for these substances in a phase of dissipated energy must satisfy the relation

$\mu _{\text{H}}+8\mu _{\text{O}}=9\mu _{\text{W}}.$