170 Physical Properties [CH. vn 



199. From equations (402) and (404) we obtain by subtraction 



Since m is proportional to the molecular weight of the particular gas we are 

 discussing, this equation expresses Carnot's Law : 



The difference of the two specific heats of a gas is inversely proportional to 

 the molecular weight of the gas. 



This law can be expressed in a different form. The specific heats referred 

 to unit volume instead of to unit mass are of course C p p, C v p and equation 

 (405) may be written 



the last transformation depending on equation (399). Hence : 



At a given temperature and pressure the difference of the two specific 

 heats per unit mme is the same for all gases. 



200. It is found by experiment that, at any rate for a large number of 

 gases, Cp and C v are approximately independent of the temperature. This, 

 as is shewn by a reference to the formulae (402) and (404), must mean that 



rl f? 



-TJJ-, is a constant, and therefore that the mean energy of a molecule of the 



CL-L 



gas stands in a constant ratio to the translational energy. Let us denote this 

 ratio by (1 + {3), so that /3 is the ratio of internal to translational energy. 

 Then 



= (1+/3)RT .............................. (407), 



r) T? 



so that = \R (1 + /3). 



al 



dE 



Substituting this value for -^ in equations (402) and (404), we obtain 



CLJL 



If we denote the ratio C P /C V by 7, we obtain by division, 



_0 y= 1 + 

 7 ~ 



