One mole of an ideal gas with heat capacity `C_V` goes through a process in which its entropy `S` depends on `T` as `S = alpha//T`, where `alpha` is a constant. The gas temperature varies from `T_1` to `T_2` Find :
(a) the molar heat capacity of the gas as function of its temperature ,
(b) the amount of heat transferred to the gas ,
( c) the work performed by the gas.

Explanation

(a) `C = T (dS)/(dT) = - (alpha)/(T)`
(b) `Q = int_(T_1)^(T_2) CdT = alpha 1n (T_1)/(T_2)`
( c) `W = Delta Q - Delta U = alpha 1n (T_1)/(T_2) + C_V (T_1 - T_2)`
Since for an ideal gas `C_V` is constant
and `Delta U = C_v(T_2 - T_1)`
(`U` does not depend on `V`).