An ideal gas has a molar heat capacity `C_v` at constant volume. Find the molar heat capacity of this gas as a function of its volume `V`, if the gas undergoes the following process :
(a) `T = T_0 e^(alpha v)` ,
(b) `p = p_0 e^(alpha v)`,
where `T_0, p_0`, and `alpha` are constants.

Explanation

In this case the process equation is given as
`P=P_0 e^(alphaV)`…(3.83)
Differentiating , we get
`dP=P_0 alpha e^(alpha V) dV`…(3.84)
If C is the molar specific heat of the gas in this process then from equation-(3.79), (3.83) and (3.84) we have
`C=C_V+(R(P_0e^(alphaV))dV)/((P_0e^(alphaV))dV+V(P_0alphae^(alphaV)dV))`
or `C=C_V+R/(1+alphaV)` ...(3.85)
Equation-(3.85) gives the molar heat capacity of the gas in the given process and again we can say that this is also not a polytropic process.