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

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