Page:Scientific Papers of Josiah Willard Gibbs.djvu/217

represents the excess of the heat evolved over the work done by external forces when a mass of the gas is compressed at constant temperature until a unit of NO₂ has been converted into N₂O₄. This quantity will be constant if $B=0$, i.e., if the specific heats at constant volume of NO₂ and N₂O₄ are the same. This assumption would be more simple from a theoretical stand-point and perhaps safer than the assumption that $B'=0$ If $B=0,B'=a_{2}$. If we wish to embody this assumption in the equation between $D,p$, and $t$, we may substitute

$6.5228+\log _{10}(t_{c}+273)-{\frac {2977.4}{t_{c}+273}}$

for the second member of equation (336). The relative densities calculated by the equation thus modified from the temperatures and pressures of the experiments under discussion will not differ from those calculated from the unmodified equation by more than .002 in any case, or by more than .001 in the first series of experiments.

It is to be noticed that if we admit the validity of the volumetrical relation expressed by equation (333), which is evidently equivalent to an equation between $p,t,v$, and $m$ (this letter denoting the quantity of the gas without reference to its molecular condition), or if we admit the validity of the equation only between certain limits of temperature and for densities less than a certain limit of density, and also admit that between the given limits of temperature the specific heat of the gas at constant volume may be regarded as a constant quantity when the gas is sufficiently rarefied to be regarded as consisting wholly of NO₂,—or, to speak without reference to the molecular state of the gas, when it is rarefied until its relative density $D$ approximates to its limiting value $D_{1}$,—we must also admit the validity (within the same limits of temperature and density) of all the calorimetrical relations which belong to ideal gas-mixtures with convertible components. The premises are evidently equivalent to this,—that we may imagine an ideal gas with convertible components such that between certain limits of temperature and above a certain limit of density the relation between $p,t$, and $v$ shall be the same for a unit of this ideal gas as for a unit of peroxide of nitrogen, and for a very great value of $v$ (within the given limits of temperature) the thermal capacity at constant volume of the ideal and actual gases shall be the same. Let us regard $t$ and $v$ as independent variables; we may let these letters and $p$ refer alike to the ideal and real gases, but we must distinguish the entropy $\eta '$ of the ideal gas from the entropy $\eta$ of the real gas. Now by (88)

${\frac {d\eta }{dv}}={\frac {dp}{dt}},$

(337)

therefore${\frac {d}{dv}}{\frac {d\eta }{dt}}={\frac {d}{dt}}{\frac {d\eta }{dv}}={\frac {d}{dt}}{\frac {dp}{dt}}={\frac {d^{2}p}{dt^{2}}}\cdot$

(338)