One mole of an ideal gas goes through a process in which the entropy of the gas changes with temperature `T` as `S = aT + C_V 1n T`, where `a` is a positive constant. `C_V` is the molar heat capacity of this gas at constant volume. Find the volume dependence of the gas temperature in this process if `T = T_0` at `V = V_0`.

Explanation

Along the process line : `S = aT + C_V 1n T`
or the specific heat is : `C = T (dS)/(dT) = aT + C_V`
On the other hand : `dQ = CdT = C_V dT + pdV` for an ideal gas.
Thus, `pdV = (RT)/(V) dV = aT dT`
or `(R)/(a) (dV)/(V) = dT` or, `(R)/(a) 1n V + constant = T`
Using `T = T_0` when `V = V_0` we get, `T = T_0 + (R)/(a) 1n (V)/(V_0)`.