An ideal gas has an adiabatic exponent `gamma`. In some process its molar heat capacity varies as `C = alpha//T`,where `alpha` is a constant Find :
(a) the work performed by one mole of the gas during its heating from the temperature `T_0` to the temperature `eta` times higher ,
(b) the equation of the process in the variables `p, V`.

Explanation

(a) By the first law of thermodynamics `A = Q - Delta U`
or, =`CdT - C_V dT = (C - C_V)dT` (for one mole)
Given `C = (alpha)/(T)`
So, `A = int_(T_0)^(eta T_0)((alpha)/(T) - C_V) dT= alpha 1n (eta T_0)/(T_0) - C_V (eta T_0 - T_0)`
=`alpha 1n eta = C_V T_0 (eta - 1) = alpha 1n eta + (RT)/(gamma - 1) (eta -1)`
(b) `C = + (dQ)/(dT) = (RT)/(V) (dV)/(dT) + C_V`
Given `C = (alpha)/(T)`, so `C_V + (RT)/(V) (dV)/(dT) = (alpha)/(T)`
or, `(R)/(gamma - 1)(1)/(RT) + (dV)/(V) = (alpha)/(RT^2) dT`
or, `(dV)/(V) = (alpha)/(RT^2) dT - (1)/(gamma - 1).(dT)/(T)`
or, `(gamma - 1)(dV)/(V) = (alpha(gamma-1))/(RT^2) dT - (dT)/(T)` ltbrltgt Intergrating both sides, we get
or `(gamma - 1)1n V = -(alpha(gamma - 1))/(RT) -1n T + 1n K`
or `1n V^(gamma - 1) (T)/(K) = (-alpha(gamma - 1))/(RT)`
`1n V^(gamma - 1).(pV)/(RK) = (-alpha(gamma - 1))/(pV)`
or, `(pV^gamma)/(RK) = e^(alpha(gamma - 1)//pV)`
or, `pV^gamma e^(alpha (gamma - 1)//(pV)) = RK = constant`.