For the case of an ideal gas find the equation of the process (in the variables `T, V`) in which the molar heat capacity varies as :
(a) `C = C_V + alpha T` ,
(b) `C = C_V + beta V `,
( c) `C = C_v + ap` ,
where `alpha, beta` and `a` are constants.

Explanation

Heat capacity is given by `C = C_V + (RT)/(V) (dV)/(dT)`
(a) Given `C = C_V + alpha T`
So, `C_V + alpha T = C_V + (RT)/(V)(dV)/(dT)`or, `(alpha)/(R) dT = (dV)/(V)`
Integrating both sides, we get `(alpha)/(R)T = 1n C_0 = 1n VC_0 , C_0` is a constant.
Or, `V. C_0 = e^(alpha T//R)` or `V . e^(alpha T//R) = (1)/(C_0) = constant`
(b) `C = C_V + beta V`
and `C = C_V + (RT)/(V) (dV)/(dT)` so, `C_V (RT)/(V) (dV)/(dT) = C_V + beta V`
or, `(RT)/(V)(dV)/(dT) = beta V` or, `(dV)/(V^2) = (beta)/(R) (dT)/(T)` or, `V^-2 = (dT)/(T)`
Integrating both sides, we get `(R)/(beta) (V^-1)/(beta - 1) = 1n T + 1n C_0 = 1n T.C_0`
So, `1n T.C_0 = -(R)/(betaV) T. C_0 = e^(-R//beta V) ` or, `T e^(-R//beta V) = (1)/(C_0) = constant`
( c) `C = C_V + ap` and `C = C_V + (RT)/(V) (dV)/(dT)`
So, `C_V + ap = C_V + (RT)/(V) (dV)/(dT)` so, `ap = (RT)/(V) (dV)/(dT)`
or, `a (RT)/(V) = (RT)/(V) (dV)/(dT)` (as `p = (RT)/(V)` for one mole of gas)
or, `(dV)/(dT) =a` ro, `dV = adT`or `dT = (dV)/(a)`
So, `T = (V)/(a) + constant` or `V - aT = constant`.