POTENTIAL TEMPERATURE AND ENTROPY BAUER 499 



We have in general: 



h ~ s a = S (log b - log d a ) = c v (log d b , - log a .) . . (6) 



or the entropy change in passing from a to & by any process what- 

 soever — revefsible or irreversible — can be measured ideally by the 

 temperature changes incurred in allowing the body to expand under 

 standard pressure between the initial and final adiabats. 



For other substances. — If the substance acted upon be not a perfect 

 gas we have: 



r» r b ' dh r b ' do r b do 



J.*-*- 5 --J, 't'L^T'J^T' • (7) 



Here c p is not a constant as in the case of a perfect gas, but varies 

 with temperature and may even be discontinuous, hence it is impos- 

 sible, in general, to carry out the integration of the right-hand 

 member. This we know, however, that c p is invariably positive, 

 i.e., heat must always be supplied to a substance to raise its tem- 

 perature under a constant pressure. Since 



ds - cJL (8) 



u 



it follows that the sign of ds is the same as that of dd, so that when- 

 ever the entropy increases, the potential temperature does likewise. 

 This, while true for cases treated, is not so, in general, as previously 

 explained. 



In the foregoing paragraphs the law of potential temperature has 

 been deduced from that of entropy; however, an independent deduc- 

 tion can readily be made if desired. 



For example, we may build up the law of potential temperature 

 in precisely the same manner as in the case of the entropy law by 

 taking typical examples of natural processes and showing that 

 nature unaided invariably tends to increase the potential tempera- 

 ture. 



Thus take the well-known case of the sudden expansion of a 

 perfect gas without the performance of external work. It is very 

 easy to show on the pv diagram, since the adiabat is a steeper 

 curve than the isotherm, that the potential temperature in the final 

 stage is greater than in the initial stage. 



So again with the case of heat conduction. Suppose we have 

 the same mass of the same perfect gas enclosed in each of two 

 vessels a and b of the same size and enclosed in a non-conducting 

 vessel. The temperature of a is greater than b. Connect now a 



