Page:Scientific Papers of Josiah Willard Gibbs.djvu/188

From (260), by (87) and (91), we obtain

$\zeta =Em+mt\left(c-H-c\log t+a\log {\frac {m}{v}}\right)+pv,$

and eliminating $v$ by means of (263), we obtain the fundamental equation

$\zeta =Em+mt\left(c+a-H-(c+a)\log t+a\log {\frac {p}{a}}\right).$

(265)

From this, by differentiation and comparison with (92), we may obtain the equations

$\eta =m\left(H+(c+a)\log t-a\log {\frac {p}{a}}\right),$

(266)

$v={\frac {amt}{p}},$

(267)

$\mu =E+t\left(c+a-H-(c+a)\log t-a\log {\frac {p}{a}}\right).$

(268)

The last is also a fundamental equation. It may be written in the form

$\log {\frac {p}{a}}={\frac {H-c-a}{a}}+{\frac {c+a}{a}}\log t+{\frac {\mu -E}{at}},$

(269)

or, if we denote by $e$ the base of the Naperian system of logarithms,

$p=ae^{\frac {H-c-a}{a}}t^{\frac {c+a}{a}}e^{\frac {\mu -E}{at}}.$

(270)

The fundamental equation between $\chi ,\eta ,p$ and $m$ may also be easily obtained; it is

$(c+a)\log {\frac {\chi -Em}{(c+a)m}}={\frac {\eta }{m}}-H+a\log {\frac {p}{a}},$

(271)

which can be solved with respect to $\chi$.

Any one of the fundamental equations (255), (260), (265), (270), and (271), which are entirely equivalent to one another, may be regarded as defining an ideal gas. It will be observed that most of these equations might be abbreviated by the use of different constants. In (270), for example, a single constant might be used for $ae^{\frac {H-c-a}{a}}$, and another for ${\frac {c+a}{a}}$. The equations have been given in the above form, in order that the relations between the constants occurring in the different equations might be most clearly exhibited. The sum $c+a$ is the specific heat for constant pressure, as appears if we differentiate (266) regarding $p$ and $m$ as constant.^[1]

1. We may easily obtain the equation between the temperature and pressure of a saturated vapor, if we know the fundamental equations of the substance both in the gaseous, and in the liquid or solid state. If we suppose that the density and the specific heat at constant pressure of the liquid may be regarded as constant quantities (for such