ELECTROCHEMICAL THERMODYNAMICS. 411 



will then vary directly as [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 E 

 for the electromotive force V V", and for the differences Ae, ki\ to 

 write the fuller forms e" e', J/" */, 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 W 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 



E = e" - e' - t(rf r - if) +p(v" - v'). (8) 



If we suppose the temperature to vary, the pressure remaining con- 

 stant, we have 



= de" -dff-t djj" + tdrf- (rf f - vf) dt +p dv" -p dv'. (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 drf is the heat imparted divided 

 by the temperature, the heat imparted is tdrf, and the work is 

 obviously p dv'. Hence 



de' = tdrf pdv', 

 and in like manner 



If we substitute these values, the equation becomes 



dE = (rj'-ri")dt (10) 



We have already seen that r\ r\' represents the integral -^J 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 



^=- (ID 



dt t' 



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 ^ for = in equations (2), (4), (5), (6), and ^ for = in (8), the 

 formula will there hold true without the limitation of reversibility. 



