184-188] The Second Law 163 



The Second Law of Thermodynamics. 

 187. If equation (391) is divided throughout by T, it becomes 



(392). 



In this equation the gas remains of the same constitution throughout 

 the change in its state. Thus E is a function of T only and v b (p) of v only. 



Hence the terms -~- and v b (p) dv are each complete differentials. 



It follows that the right-hand member of equation (392) is a complete 

 differential, say d$$, and we therefore have 



(393). 



This equation contains the second law of thermodynamics. 



If we suppose that by the introduction of heat and by changes in 

 pressure and volume, the gas is caused to pass through a series of states, 

 such that finally it is brought back to the same volume and temperature 

 from which it started, we have, since ^ is the same at the beginning and 

 end of the series of changes, 



'^? = 0.. ...(394). 



I 



Thus we have the second law in a different form : 



The value of I -~- , where the integral is taken through any closed cycle of 

 thermal processes, is zero. 



188. When equation (394) is reached from the thermodynamic stand- 

 point, the temperature is not measured on the kinetic theory scale, which, as 

 we have seen, is identical with the scale of a gas thermometer in which a 

 perfect gas is used as thermometric substance, but is measured on the 

 thermodynamical scale, which is independent of the working substance. 



The result expressed by equation (394) does not in itself permit us to 

 identify the kinetic theory temperature T, with the thermodynamic tempera- 



ture . All that we have found so far is that ^ and are both of them 



\y JL 



integrating factors of d<&, but since cftSl depends only on two variables, it 

 must in any case have an infinite number of integrating factors. 



For in general, if 



dz=Ldx + Mdy .............................. (395), 



the condition that N shall be an integrating factor of dz is that 



L , M , 

 dy 



112 



