Page:Scientific Papers of Josiah Willard Gibbs.djvu/49

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THERMODYNAMICS OF FLUIDS.

13

which is then reduced to

{\frac {d\epsilon }{\epsilon }}={\frac {1}{c}}d\eta -{\frac {a}{c}}{\frac {dv}{v}}

,

and by integration to

log\epsilon ={\frac {\eta }{c}}-{\frac {a}{c}}\,logv.

^[1]

(d)

The constant of integration becomes 0, if we call the entropy 0 for the state of which the volume and energy are both unity.

Any other equations which subsist between $v,p,t,\epsilon$ and $\eta$ may be derived from the three independent equations (a)), (b) and (d). If we eliminate $\epsilon$ from (b) and (d), we have

\eta =a\,log\,v+c\,log\,t+c\,log\,c

.

(e)

Eliminating $v$ from (a) and (e), we have

\eta =(a+c)log\,t-a\,log\,p+c\,log\,c+a\,log\,a

.

(f)

Eliminating $t$ from (a) and (e), we have

\eta =(a+c)log\,v+c\,log\,p+a\,log\,{\frac {c}{a}}

.

(g)

If $v$ is constant, equation (e) becomes

\eta =c\,log\,t+Const.

,

i.e, the isometrics in the entropy-temperature diagram are logarithmic curves identical with one another in form,—a change in the value of $v$ having only the effect of moving the curve parallel to the axis of $\eta$ . If $p$ is constant, equation (f) becomes

\eta =(a+c)log\,t+Const.

,

so that the isopiestics in this diagram have similar properties. This identity in form diminishes greatly the labour of drawing any considerable number of these curves. For if a card or thin board be cut in the form of one of them, it may be used as a pattern or ruler to draw all of the same system.

The isodynamics are straight in this diagram (eq. b).

To find the form of the isothermals and isentropics in the volume-pressure diagram, we may make $t$ and $\eta$ constant in equations (a) and (g) respectively, which will then reduce to the well-known equations of these curves:—

pv=Const.

,

and

p^{c}v^{a+c}=Const.

,

↑ If we use the letter

\epsilon

to denote the base of the Naperian system of logarithms, equation (d) may also be written

\epsilon =\epsilon ^{\frac {\eta }{c}}v^{\frac {a}{c}}

This may be regarded as the fundamental thermodynamic equation of an ideal gas. See the last note on page 2. It will be observed, that there would be no real loss of generality if we should choose, as the body to which the letters refer, such a quantity of the gas that one of the constants

a

and

c

should be equal to unity.

[1] If we use the letter $\epsilon$ to denote the base of the Naperian system of logarithms, equation (d) may also be written

$\epsilon =\epsilon ^{\frac {\eta }{c}}v^{\frac {a}{c}}$
This may be regarded as the fundamental thermodynamic equation of an ideal gas. See the last note on page 2. It will be observed, that there would be no real loss of generality if we should choose, as the body to which the letters refer, such a quantity of the gas that one of the constants $a$ and $c$ should be equal to unity.

[1]