253 
See. 20 -o- Eq. 116-124 
of x, and T as a function of x, the latter as an auxiliary quantity be- 
cause T will appear in the integrand above. P will be obtained by in- 
tegrating (dP/dx), and T by integrating (aT/ax)g. 
From pure thermodynamics we obtain the following relations: 
(¢) = oe g) (116) 
dv Ss C., at Vv 
aP = ap ’ (117) 
ei y@), 
Eel 7 
Cy \at/y/\av/p 5 
V 32 
Cera aT arr) av (119) 
ee ou it (a k ; 
in which C_, is the heat capacity of the M grams of burnt gases at the 
temperature T treated as ideal gases, C, the same quantity for the real 
gases at Tandv. ¥ is c,,/C, for the real gas. 
The density ( equals M/v so 
aC = —M/v- . (120) 
dv 
Also, from the definition of x, 
dx = -(x/v) dv + (xh'/njaT , (121) 
(= = "kK + ce =P k= xh a) 
dv Ss Vv h \dv 3 v. Bee any 
= -x hCy + vh'T (dP/aT), (122 ) 
hv CG 
These equations may now be combined to give the results desired, i.e. 
ar zs (ele fav = Joyaig ML ; 
ax S ay g dx /g x [he (ar /aP), + vh T] , (123 ) 
an expression which can be integrated to give T as a function of x, for 
Siven To 5 Xo. 
Also aP ate dv =o ¥ hy C dF dv (124) 
ax J, avj, \ax Js x [h C+ vhit aP/dT),, 
which can be integrated to give P as a function of x if Po and xp are known. 
