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

(a) By the first law of thermodynamics
`dQ = dU + dA = vC_VdT + pd V`
Molar specfic heat according to definition
`C = (dQ)/(v dT) = (C_V dT + pdV)/(vdT)`
=`(v C_V dT + (vRT)/(V) dV)/(v dT) = C_V + (RT)/(V) (dV)/(dT)`,
We have `T = T_0 e^(alpha V)`
After differentiating, we get `dT = alpha T_0 e^(alpha V). dV`
So, `(dV)/(DT) = (1)/(alpha T_0 e^(alpha V))`,
Hence `C= = C_V + (RT)/(V).(1)/(alpha T_0 e^(alpha V)) = C_V + (RT_0 e^(alpha V))/(alpha VT_0 e^(alpha v)) = C_v + (R)/(alpha V)`
(b) Process is `p = p_0 e^(alpha V)`
`P = (RT)/(V) = p_0 e^(alpha V)`
or, `T = (p_0)/(R) e^(alpha v). V`
So, `C = C_V + (RT)/(V) (dV)/(dT) = C_V + p_0 e^(alpha V).(R)/(p_0 e^(alpha V)(1 + alpha V)) = C_V + (R)/(1 + alpha V)`.