Page:Scientific Papers of Josiah Willard Gibbs.djvu/446

410 Returning to equation (4), we may observe that if $$t$$ under the integral sign has a constant value, say $$t$$, the equation will reduce to Such would be the case if we should suppose that at the temperature $$t$$ the chemical processes to which the brackets relate take place reversibly with evolution or absorption of heat, and that the heat required to bring the substances from the temperature of the cell to the temperature $$t''$$, and that obtained in bringing them back again to the temperature of the cell, may be neglected as counterbalancing each other. This is the point of view of my former letter. I do not know that it is necessary to discuss the question whether any such case has a real existence. It appears to me that in supposing such a case we do not exceed the liberty usually allowed in theoretical discussions. But if this should appear doubtful, I would observe that the equation (6) must hold in all cases if we give a slightly different definition to $$t$$, viz., if $$t$$ be defined as a temperature determined so that The temperature $$t''$$, thus defined, will have an important physical meaning. For by means of perfect thermo-dynamic engines we may change a supply of heat $$[\text{Q}]$$ at the constant temperature $$t$$ into a supply distributed among the various temperatures represented by $$t$$ in the manner implied in the integral, or vice versâ''. We may, therefore, while vastly complicating the experimental operations involved, obtain a theoretical result which may be very simply stated and discussed. For we now see that after the passage of the current we may (theoretically) by reversible processes bring back the cell to its original state simply by the expenditure of the heat $$[\text{Q}]$$ supplied at the temperature $$t''$$, with perhaps a certain amount of work represented by $$[\text{W}]$$, and that the electromotive force of the cell is determined by these quantities in the manner indicated by equation (6), which may sometimes be further simplified by the vanishing of $$[\text{W}]$$ and $$\text{W}_{\text{P}}$$.

If the current causes a separation of radicles, which are afterwards united with evolution of heat, $$[\text{Q}]$$ being in this case negative, $$t''$$ represents the highest temperature at which this heat can be obtained. I do not mean the highest at which any part of the heat can be obtained—that would be quite indefinite—but the highest at which the whole can be obtained. I should add that if the effect of the union of the radicles is obtained partly in work—$$[\text{W}]$$, and partly in heat—$$[\text{Q}]$$, we may vary the proportion of work and heat; and $$t''$$