257-260] General Equation of Calorimetry 215 



(ii) the energy of n other momentoids (e.g. momentoids of rotation) also 

 governed by the principal temperature ; and 



(iii) the energy of various degrees of freedom, governed by subsidiary 

 temperatures r x , r 2 ... . 



General Equation of Calorimetry. 



259. We now examine the Calorimetry of such a gas, without at 

 present making the simplifying assumption that the gas obeys Boyle's 

 Law. 



The general equation of calorimetry is, as before (equation (391)), 



d<& = Rv b Tdv + NdE ........................ (511), 



where vj, is the molecular density at an imaginary boundary, and E is the 

 mean value of the energy of the molecules. The value of the mean energy 

 is, from equations (457), 



E = ^R{(n + 3)T+r 1 + r,+ ...} ............... (512). 



Instead of equation (511), the equation of calorimetry can now be written 

 d<& = Rv b Tdv + kNR {(n + 3)dT + dr 1 + dr 2 + ...} ...... (513), 



and on division by T 



(514). 



CiT (l T 



On the right-hand side the terms -, -^ are not perfect differentials, 



so that there is no theorem corresponding to that of 187 (the Second Law 

 of Thermodynamics) when the energy of subsidiary temperatures is taken 

 into account. 



Specific Heat at Constant Volume. 



260. To find C v , the specific heat at constant volume, we suppose that a 

 quantity d<& of heat is absorbed by the gas, while the volume remains constant. 

 In this case equation (513) assumes the form 



d(^ = ^NR{(n + ^dT+dr 1 + dr z +...} ............ (515), 



in which, since r a = yf^ (T) etc., we have 



Putting d<& = C v JNmdT we obtain 



...)} ............ (516). 



It therefore appears that C v is, in general, a function of the temperature, 

 but that it may be regarded as constant at temperatures at which the gas 

 is not incandescent. The value of C v , we notice, is in no way affected by 

 deviations from Boyle's law. 



