ON ELECTROLYSIS IN ITS PHYSICAL AND CHEMICAL BEARINGS. 345 



Eeturninof to equation (4), we may observe that if t under the integral sign has 

 a constant value, say t", the equation will reduce to 



V--V' = ^'[Q] + [W] + Wp . . . . (6) 



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", tIz., if t" be defined as a temperature determined so that 



ipj = f-^ (7> 



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 

 [Ql 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 

 versa. We may therefore, while vastly complicating the experimental operations 

 involved, obtain a theoretical result which may be very simply stated and dis- 

 cussed. For we now see that after the passage of the current we may (theoreti- 

 cally) by reversible processes bring back the cell to its original state simply by the 

 expenditure of the heat [Q] supplied at the temperature t", with perhaps a certain 

 amount of work represented by [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 [W] and Wp. 



If the current causes a separation of radicles, which are afterwards united with 

 evolution of heat, [Q] being in this case negative, t" represents the highest tem- 

 perature 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 tlie effect of the 

 union of the radicles is obtained partly in work — [W], and partly in heat — [Q], 

 we may vary the proportion of work and heat ; and t" will then vary directly as 

 [Q]. But if the effect is obtained entirely in heat, ;;" will have a perfectly definite 

 value. 



It is easy to show that these results are in complete accordance with Helm- 

 holtz'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 E for the electromotive force V— V", and 

 for the differences Af, Av to write the fuller forms e" — e', ?;" — r;', 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 Wp, where ^j 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 



'E. = i"-i'-t(ri"-ri')+p{v"-v') . . . . (8) 



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



dE = de"-d€'-tdr," + tdq'-(T]"-r]')dt+pdv"-pdt)' . . (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 drj' is the heat imparted divided by the temperature, the heat imparted is 



tdij', and the work is obviously pdv'. Hence 



de' = tdr]'-pdt)', 



