Page:Scientific Papers of Josiah Willard Gibbs.djvu/447

will then vary directly as $[{\text{Q}}]$. But if the effect is obtained entirely in heat, $t''$ will have a perfectly definite value.

It is easy to show that these results are in complete accordance with Helmholtz's differential equation. We have only to differentiate the value which we have found for the electromotive force. For this purpose equation (5) is most suitable. It will be convenient to write ${\text{E}}$ for the electromotive force ${\text{V}}'-{\text{V}}''$, and for the differences $\Delta \epsilon ,\Delta \eta$ to write the fuller forms $\epsilon ''-\epsilon ',\eta ''-\eta '$, where the single and double accents distinguish the values before and after the passage of the current. We may also set $p(v'-v'')$ for ${\text{W}}_{\text{P}}$, where $p$ is the pressure (supposed uniform) to which the cell is subjected, and $v''-v'$ is the increase of volume due to the passage of the current. If we also omit the accent on the $t$, which is no longer required, the equation will read

${\text{E}}=\epsilon ''-\epsilon '-t(\eta ''-\eta ')+p(v''-v').$

(8)

If we suppose the temperature to vary, the pressure remaining constant, we have

$d{\text{E}}=d\epsilon ''.d\epsilon '-td\eta ''+td\eta '-(\eta ''-\eta ')dt+pdv''-pdv'.$

(9)

Now, the increase of energy de is equal to the heat required to increase the temperature of the cell by dt diminished by the work done by the cell in expanding. Since $d\eta '$ is the heat imparted divided by the temperature, the heat imparted is $td\eta '$, and the work is obviously $pdv'$. Hence

$d\epsilon '=td\eta '-pdv',$

and in like manner

$d\epsilon ''=td\eta ''-pdv''.$

If we substitute these values, the equation becomes

$d{\text{E}}=(\eta '-\eta '')dt.$

(10)

We have already seen that $\eta '-\eta ''$ represents the integral $\int {\frac {d[{\text{Q}}]}{t}}$ of equations (2) and (4), which by equation (2) is equal to the reversible heat evolved, Q, divided by the temperature of the cell, which we now call $t$. Substitution of this value gives

${\frac {d{\text{E}}}{dt}}=-{\frac {\text{Q}}{t}},$

(11)

which is Helmholtz's equation.

These results of the second law of thermodynamics are of course not to be applied to any real cells, except so far as they approach the condition of reversible action. They give, however, in many cases limits on one side of which the actual values must lie. Thus, if we set $\leqq$ for $=$ in equations (2), (4), (5), (6), and $\geqq$ for $=$ in (8), the formula will there hold true without the limitation of reversibility.