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

The process equation for the thermodynamic process is given as
`T=T_0e^(alphaV)`…(3.78)
For a general thermodynamic process we know the molar heat capacity is given by
`C=C_V+(RPdV)/(PdV+VdP)` ..(3.79)
From gas law we can relate pressure and volume of gas as
`P=(nRT)/V`
From equation-(3.74)
`P=(nRT_0)/V e^(alphaV)`..(3.80)
Differentiating this equation , we get
`PdV+VdP=nRT_0 alpha e^(alphaV)dV`...(3.81)
From equation -(3.79), (3.80) and (3.81), we get
`C=C_V+(R((nRT_0)/V e^(alphaV))dV)/(nRT_0alphae^(alphaV)dV)`
or `C=C_V+R/(alphaV)`..(3.82)
Equation-(3.82) gives the molar specific heat of the gas undergoing the given process and students should note that this molar specific heat is given a function of volume of gas thus this process is a nonpolytropic process.