Page:Scientific Papers of Josiah Willard Gibbs.djvu/411

These expressions for energy and entropy will undoubtedly apply to mixtures of different gases, whatever their chemical relations may be (with such limitations and with such a degree of approximation as belong to other laws of the gaseous state), when no chemical action can take place under the conditions considered. If we assume that they will apply to such cases as we are now considering, although chemical action is possible, and suppose the equilibrium of the mixture with respect to chemical change to be determined by the condition that its entropy has the greatest value consistent with its energy and its volume, we may easily obtain an equation between $m_{1},m_{2}$, etc., $t$ and $v$.^[1]

The condition that the energy does not vary, gives

$(m_{1}c_{1}+m_{2}c_{2}+{\text{etc.}})dt+(c_{1}t+{\text{E}}_{1})dm_{1}+(c_{2}t+{\text{E}}_{2})dm_{2}+{\text{etc.}}=0.$

(1)

The condition that the entropy is a maximum implies that its variation vanishes, when the energy and volume are constant.

This gives

${\frac {m_{1}c_{1}+m_{2}c_{2}+{\text{etc.}}}{t}}dt+\left({\text{H}}_{1}-a_{1}+c_{1}\log _{\text{N}}t-a_{1}\log _{\text{N}}{\frac {m_{1}}{v}}\right)dm_{1}$ $+\left({\text{H}}_{2}-a_{2}+c_{2}\log _{\text{N}}t-a_{2}\log _{\text{N}}{\frac {m_{2}}{v}}\right)dm_{2}+{\text{etc.}}=0.$

(2)

Eliminating $dt$, we have

$\left({\text{H}}_{1}-a_{1}-c_{1}-{\frac {{\text{E}}_{1}}{t}}+c_{1}\log _{\text{N}}t-a_{1}\log _{\text{N}}{\frac {m_{1}}{v}}\right)dm_{1}$ $+\left({\text{H}}_{2}-a_{2}-c_{2}-{\frac {{\text{E}}_{2}}{t}}+c_{2}\log _{\text{N}}t-a_{2}\log _{\text{N}}{\frac {m_{2}}{v}}\right)dm_{2}+{\text{etc.}}=0.$

(3)

If the case is like that of the peroxide of nitrogen, this equation will have two terms, of which the second may refer to the denser component of the gas-mixture. We shall then have $a_{1}=2a_{2},$, and $dm_{1}=-dm_{2}$, and the equation will reduce to the form

$\log {\frac {m_{2}v}{m_{1}^{2}}}=-{\text{A}}-{\text{B}}\log t+{\frac {\text{C}}{t}},$

(4)

where common logarithms have been substituted for Naperian, and ${\text{A, B}}$ and ${\text{C}}$ are constants. If in place of the quantities of the components we introduce the partial pressures, $p_{1},p_{2}$, due to these components and measured in millimeters of mercury, by means of the relations

$m_{1}={\frac {p_{1}v}{a_{1}t}}$$m_{2}={\frac {p_{2}v}{a_{2}t}},$

1. For certain a priori considerations which give a degree of probability to these assumptions, the reader is referred to the paper already cited.