199-202] Specific Heats 171 



Adiabatic Motion. 



201. Let us suppose the pressure, volume and temperature to change in 

 such a way that no heat enters or leaves the gas. Then since d(& = 0, we 

 have from equation (400), 



or, on substituting for E from equation (407), and dividing by RNT, 



^ dT dv 



A 

 =0. 



T v 



Hence upon integration 



TV* <i+0) = constant (409), 



or again, since T is proportional to pv, 



p v 3(1+0) = constant, 

 or pv"* = constant ........................ (410). 



This is the general relation between pressure and volume in a motion of 

 the gas in which no heat enters or leaves the gas a type of motion which is 

 known as "adiabatic." 



Since /3 cannot be negative, we see from equation (409) that in adiabatic 

 motion an increase in v is accompanied by a decrease in T, and vice vei'sd. 



Calculation of <y. 



202. Let us now examine more closely the values of the two specific 

 heats, abandoning the simplifying assumption that the mean internal energy 

 of a molecule bears a constant ratio to the mean energy of translation. 



As in equation (160) the energy of a molecule may be regarded as the 

 sum of the potential energy V and of the energies of n momentoids. The 

 mean energy of each momentoid is by equation (265) equal to ^RT. The 

 total mean energy of a molecule is accordingly given by 



E = V + friRT ........................... (411). 



This gives, upon differentiation with respect to T, 



(412) - 



and the substitution of this in equations (402) and (404) leads to 



1 dV R 



