Page:Scientific Papers of Josiah Willard Gibbs.djvu/413

In the case of a mixture of Cl₂, PCl₃ and PCl₅, equation (3) will have three terms distinguished by different suffixes. To fix our ideas, we may make these suffixes ₂, ₃ and ₅, referring to Cl₂, PCl₃ and PCl₅ respectively. Since the constants $a_{2},a_{3}$, and $a_{5}$ are inversely proportional to the densities of these gases,

$a_{2}dm_{2}=a_{3}dm_{3}=-a_{5}dm_{5},$

and we may substitute ${\frac {1}{a_{2}}},{\frac {1}{a_{3}}},{\frac {-1}{a_{5}}}$ for $dm_{2},dm_{3}$ and $dm_{5}$ in equation (3), which is thus reduced to the form

$\log {\frac {m_{5}v}{m_{2}m_{3}}}=-{\text{A}}-{\text{B}}\log t+{\frac {\text{C}}{t}}\cdot$

(7)

If we eliminate $m_{2},m_{3},m_{5}$ by means of the partial pressures $p_{2},p_{3},p_{6}$, we obtain

$\log {\frac {p_{5}}{4p_{2}p_{5}}}=-{\text{A}}'-{\text{B}}'\log t+{\frac {\text{C}}{t}},$

(8)

when ${\text{A}}',{\text{B}}'$, like ${\text{A, B}}$ and ${\text{C}}$, are constants. If the chlorine and the protochloride are in such proportions as arise from the decomposition of the perchloride, $p_{2}=p_{3}$ and $4p_{2}p_{3}=(p_{2}+p_{3})^{2}$. In this case, therefore, we have

$\log {\frac {p_{5}}{(p_{2}+p_{3})^{2}}}=-{\text{A}}'-{\text{B}}'\log t+{\frac {\text{C}}{t}}\cdot$

(9)

It will be seen that this equation is of the same form as equation (5), when $p_{5}$ in (9) is regarded as corresponding to $p_{2}$ in (5), and $p_{2}+p_{3}$ in (9), which represents the pressure due to the products of decomposition, is regarded as corresponding to $p_{1}$ in (5), which has the same signification. It follows that equation (5), as well as (6), which is derived from it, may be regarded as applying to the vapor of perchloride of phosphorus, when the values of the constants are properly determined. This result might have been anticipated, but the longer course which we have taken has given us the more general equations, (7) and (8), which will apply to cases in which there is an excess of chlorine or of the protochloride.

If the gas-mixture considered, in addition to the components capable of chemical action, contains a neutral gas, the expressions for the energy and entropy of the gas-mixture should properly each contain a term relating to this neutral gas. This would make it necessary to add $c_{n}m_{n}$ to the coefficient of $dt$ in (1), and ${\frac {c_{n}m_{n}}{t}}$ to the coefficient of $dt$ in (2), the suffix _n being used to mark the quantities relating to the neutral gas. But these quantities would disappear with the elimination of $dt$, and equation (3) and all the subsequent equations would require no modification, if only $p$ and ${\text{D}}$ are estimated (in accordance with usage) with exclusion of the pressure and weight