with van der Waals' and Clausius' Characteristics. 415 



In the case of saturation we obtain from Planck's memoir 

 (loc. cit.) 



j/ = A(V + l — rcos<£), v' = i(r+ 1 + rcos </>) 

 _ 3^/B(r+ 1)1 r sin ^ 

 T ~ 2(^ sin 2 <j> + 2r+l) 



6 y/'^r* sin 2 0-1) 

 — (r -I- 1)*(V- sin* </> + 2r + l)r sin <f> 

 X/t _ 3i*»sin»ft + 4r- , -l 

 RT = (r + l)r sin <£ tan <£ ' 



wherc , cos<£cot 2 </> 



log cot l(f>— COS0 



our previous mode of calculationfthen gives for the saturation 

 specific heats 



g-fc , fc-$' 2( > 2 sin 2 ft + 2?'+l) 

 S aT1 R - ?» 2 sin 2 </> 



O- 2 sin 2 0-l) 2 -3(r+l)±(2r"sin 2 (/)-l)rcos(/) 

 X (r 2 sin 2 ^-l)(2r+2-r 2 sin 2 0)-2(r+l) ' 



and the work of vaporization w is JHTtttcos^. 



Calculation for different values of (f> gives the following 

 table (p. 416), whence it appears that 



(i.) S — it is always positive, is infinite for t = and r = \, 

 and has a minimum value 15*333 E for t=*83235: 



(ii.) s'— h is always negative, is — oc for t = and t=1, 

 and has a maximum value — 1T355B for t = -80571>: 



(iii.) $ — &' is always positive, is x for r = and t=1, and 

 has a minimum value 26*741 R for t=*81962: 



(iv.) the latent heat X increases continuously from to 

 co as r falls from 1 to 0: 



(v.) the work of vaporization has a maximum value 

 ■68567 RT when t = *76610, being for t=0 and t = 1. 



With this characteristic too inversion may occur in the 

 sign of S', but now when *< 1*088 or, on the former assump- 

 tion that fe/R = N + i for an N-atomic gas, only if the gas 

 has at least eleven atoms in its molecule. 



The contrast between these conditions and the corresponding 

 ones for v. d. Waals ; characteristic, especially in regard 

 to 3-fe and X, is very marked and is very clearly shown 

 by the curves that have been plotted. This contrast subsists 

 further in the neighbourhood of the critical point : for, if 

 as before we write :' 2 1 — r, we find tt=1 — 7.; 2 , X RT 

 = (21/ V 2> = 14*849*, (*-fe}/R— 1-35 = 7*425/*= (fe -s')/ R 

 4- 1*35, and the work of vaporization is one-seventh of the 

 latent heat. 



