Page:Scientific Papers of Josiah Willard Gibbs.djvu/410

determining the proportions of the component gases necessary for the equilibrium of such a mixture under any given conditions, these substances afford an appropriate test for such a law.

In a former paper^[1] by the present writer, equations were proposed to express the relation between the temperature, the pressure or the volume, and the quantities of the components in such a gas-mixture as we are considering a gas-mixtwe of convertible components in the language of that paper. Applied to the vapor of the peroxide of nitrogen, these equations led to a formula giving the density in terms of the temperature and pressure, which was shown to agree very closely with the experiments of Deville and Troost, and much less closely, but apparently within the limits of possible error, with the experiments of Playfair and Wanklyn. Since the publication of that paper, new determinations of the density have been published in different quarters, which render it possible to compare the equation with the results of experiment throughout a wider range of temperature and pressure. In the present paper, all experimental determinations of the density of this vapor which have come to the knowledge of the writer are cited, and compared with the values demanded by the formula, and a similar comparison of theory and experiment is made with respect to each of the other substances which have been mentioned.

The considerations from which these formulae were deduced may be briefly stated as follows. It will be observed that they are based rather upon an extension of generally acknowledged principles to a new class of cases than upon the introduction of any new principle.

The energy of a gas-mixture may be represented by an expression of the form

$m_{1}(c_{1}t+{\text{E}}_{1})+m_{2}(c_{2}t+{\text{E}}_{2})+{\text{etc.,}}$

with as many terms as there are different kinds of gas in the mixture, $m_{1},m_{2}$, etc. denoting the quantities (by weight) of the several component gases, $c_{1},c_{2}$, etc., their several specific heats at constant volume, ${\text{E}}_{1},{\text{E}}_{2}$, etc., other constants, and $t$ the absolute temperature. In like manner the entropy of the gas-mixture is expressed by

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

where $v$ denotes the volume, and ${\text{H}}_{1},a_{1},{\text{H}}_{2},a_{2}$, etc. denote constants relating to the component gases, $a_{1},a_{2}$, etc. being inversely proportional to their several densities. The logarithms are Naperian.

1. "On the Equilibrium of Heterogeneous Substances," this volume, page 55. The equations referred to are (313), (317), (319), and (320), on pages 171 and 172. The applicability of these equations to such cases as we are now considering is discussed under the heading "Gas-mixtures with Convertible Components," page 172.