GIBBS' PAPERS I AND II 39 



Now for constant volume p generally increases if heat is sup- 

 plied, and under adiabatic conditions the temperature generally 

 rises under compression; hence generally this second derivative 

 is negative. But for water under the temperature of maximum 

 density the results are reversed and the derivative is positive. 

 Next 



, fde\ dh ^ dh ^ 

 dp = — d\—- } = — — — drj — dv = — Bdt] — Adv, 



\dV/r, 



dvdrj dv^ 



dh dh 



dt = d[^^] = -—dr) -j- -— dv = Cdr, + Bdv. 

 drf dvdt] 



(-) = 



Solve for dt] and dv; then 



/^\ _ _ AC - B-" 

 \dv)t ~ 



C Xdri/r, A 



AC - 52 



/dp\ _ AC - B' ^ /dt\ 



\dii] Jt B \dv/p 



B 



Now as C > 0, AC — B^ >0, this means that on an isothermal p 

 must decrease with increasing v. So, too, at constant pressure 

 the temperature must increase with a supply of heat. In the 

 general case where B <0, supplying heat and maintaining a con- 

 stant temperature must decrease the pressure, or at constant 

 pressure the temperature must increase with the volume. Note 

 that equating the last two expressions and inverting the deriva- 

 tives yields the Maxwell relation obtained from the function f . 

 Lecture XXIV. Discussion of van der Waals' equation.* 



* The development may not seem logical and was probably adopted 

 for pedagogic reasons. As early as Lecture XVII the py-diagram for 

 vapor, liquid, and vapor-liquid phases was introduced, leading from 

 physical reasoning to the definition of critical locus and the conception 

 of that sort of stability or instability which is represented by the super- 

 cooled vapor or superheated liquid. On this basis in Lectures XVIII- 

 XIX properties of the thermodynamic surface were discussed. In Lec- 

 ture XX the equation of van der Waals was cited as affording possible 

 conceptual though largely unrealizable isothermals through the critical 

 region, and this type of isothermal was kept to the fore, in parallel with 



