Page:Scientific Papers of Josiah Willard Gibbs.djvu/412

where $a_{1}$ denotes a constant, we have

${\begin{aligned}\log {\frac {p_{2}}{p_{1}^{2}}}&=-({\text{A}}+\log 2a_{1})-(1+{\text{B}})\log t+{\frac {\text{C}}{t}}\\&=-{\text{A}}'-{\text{B}}'\log t+{\frac {\text{C}}{t}},\\\end{aligned}}$

(5)

where ${\text{A}}'$ and ${\text{B}}'$ are new constants. Now if we denote by $p$ the total pressure of the gas-mixture (in millimeters of mercury), by ${\text{D}}$ its density (relative to air of the same temperature and pressure), and by ${\text{D}}_{1}$ the theoretical density of the rarer component, we shall have

$p:p+p_{2}::{\text{D}}_{1}:{\text{D}}.$

This appears from the consideration that $p+p_{2}$ represents what the pressure would become, if without change of temperature or volume all the matter in the gas-mixture could take the form of the rarer component. Hence,

$p_{2}=p{\frac {{\text{D}}-{\text{D}}_{1}}{{\text{D}}_{1}}},$ $p_{1}=p-p_{2}=p{\frac {2{\text{D}}_{1}-{\text{D}}}{{\text{D}}_{1}}},$

By substitution in (5) we obtain

$\log {\frac {{\text{D}}_{1}({\text{D}}-{\text{D}}_{1})}{(2{\text{D}}_{1}-D)^{2}}}=-{\text{A}}'-{\text{B}}'\log t+{\frac {\text{C}}{t}}+\log p.$

(6)

By this formula, when the values of the constants are determined, we may calculate the density of the gas-mixture from its temperature and pressure. The value of ${\text{D}}_{1}$ may be obtained from the molecular formula of the rarer component. If we compare equations (3), (4) and (5), we see that

${\text{B}}'={\text{B}}+1,$⁠${\text{B}}={\frac {c_{1}-c_{2}}{a_{2}}}\cdot$

Now $c_{1}-c_{2}$ is the difference of the specific heats at constant volume of NO₂ and N₂O₄. The general rule that the specific heat of a gas at constant volume and per unit of weight is independent of its condensation, would make $c_{1}=c_{2},{\text{B}}=0$, and ${\text{B}}'=1$. It may easily be shown, with respect to any of the substances considered in this paper,^[1] that unless the numerical value of ${\text{B}}'$ greatly exceeds unity, the term ${\text{B}}'\log t$ may be neglected without serious error, if its omission is compensated in the values given to ${\text{A}}'$ and ${\text{C}}$. We may therefore cancel this term, and then determine the remaining constants by comparison of the formula with the results of experiment.

1. For the case of peroxide of nitrogen, see pp. 180, 181 in the paper cited above.