242 Mr Searle, The Expansion oj a Gas, etc. 



since the increase of energy is equal to the heat absorbed minus 

 the work done. Thus 



dU _ dcf> 



dv t dv t ^ 



By a thermodynamical formula, this may be written 



dU dp /Q v 



_, , ,_■. Rt a 



But, by (1), P = ^l-tf> 



and thus -=— = — (4). 



dv t v- 



Integrating (4), we obtain 



a 



U=f{t)~ -(5), 



v 



where f(t) is some function of t. This function is closely related 

 to the specific heat at constant volume, for 



dU_df(t) 



Cv ~w v -~df (6) - 



We may notice, in passing, that, according to (6), the specific 

 heat G v depends only upon the temperature and not upon the 

 volume of unit mass. 



In the experiment under consideration no external work is 

 done by the gas, and, when the vessels are impermeable to heat, 

 no heat escapes. Thus, the internal energy of unit mass, after 

 thermal equilibrium has been established, is unchanged by the 

 expansion. Hence, if the volume of one gramme increase from 

 v to v c.c, and if, at the same time, the temperature change from 

 t to t', we have, by (5), 



/(O.-|-/(0-j. 



From (6) we see that 



f(t')-f(t)=f t C v dt, 

 and hence 



G v dt = — a{ -, 



Jt \v v , 



If the range of temperature be small, we may treat the specific 

 heat V as constant over that range, and then we may write 



,_,_«(1_4) ( 7). 



