Properties of Gases and Vapours, (fee. 271 



equation represents a very accurate approximation to the facts at 

 moderate pressures, although n is not necessarily equal to s/R in all 

 cases. 



Variation of th/j Specific Heats. 



It would appear at first sight as though the modified equation were 

 more complicated than the original of Joule and Thomson, but it leads 

 as a matter of fact to far simpler relations between the thermody- 

 namical properties, and makes it possible to attack problems which 

 would be quite intractable with the more complicated forms of empiri- 

 cal equations in vogue. 



If S and s are the specific heats at constant pressure and at constant 

 volume respectively, and < is the entropy, we have the well-known 

 relations, 



(dS/dph = 8d*<j>/dO dp = - 0(^/d0% ............ (7), 



(dsjdv) 9 = O&ydO dv = + 0(d*p/d0*) v ............ (8). 



Assuming the characteristic equation (6), it is easy to prove from these 

 relations that the values of the specific heats at any temperature and 

 pressure are given by the simple formulae, 



S - S(l+c/V) = S + n(n + I)pc/6 ............ (9), 



s = s(l+ncj\)(l-c/V) ........................... (10), 



where S and s are the constant limiting values of the specific heats 

 when p = 0. The ratio of the specific heats g S/s is given by the 

 relation, 



where g" stands for the constant limiting value of the ratio, and is equal 

 to (n+ \}fn. 



Isentropic Relations. 



The isentropic relations are greatly simplified by the assumption 

 n s/R, since in this case the ratio of the co-aggregation volume c to 

 the ideal volume V, or to the difference of specific volumes of the 

 vapour and liquid v-b, is constant at constant entropy. From the 

 characteristic equation (6), we have for the isothermal elasticity E<? and 

 the isentropic elasticity E$, since E^/Eg = S/s = g y/(l - c/V), 



E 9 = -v(dpldu) e = +pv/y (12), 



E^ = -v(dpldv)t = +g'E 6 = gpvj(v-b) (13). 



The equation of the isentropics may therefore be written in the 

 forms 



p(o - b)9 = constant, or p n (v - l) n+1 = constant, 



p/6 n+l = constant, or (v - b)6 n = constant (14). 



or 



