An ideal gas with the adiabatic exponent `gamma` undergoes a process in which its internal energy relates to the volume as `u = aV^alpha`. Where `a` and `alpha` are constants. Find :
(a) the work performed by the gas and the amount of heat to be transferred to this gas to increase its internal energy by `Delta U` ,
(b) the molar heat capacity of the gas in this process.

Explanation

Given `U = aV^alpha`
or, `v C_V T = a V^alpha`, or, `vC_V (pV)/(v R) = a V^alpha`
or, `a V^alpha.(R)/C_V.(1)/(pV) = 1`, or, `V^(alpha - 1).p^-1 = (C_V)/(Ra)`
or `pV^(1 - alpha) = (Ra)/(C_V) = constant = a(gamma - 1)[as C_V = (R)/(gamma - 1)]`
So polytropric index `n = 1 - alpha`.
(a) Work done by the gas is given by
`A = (-v R Delta T)/(n -1) `and `Delta U = (v R Delta T)/(gamma - 1)`
Hence `A = (-Delta U(gamma - 1))/(n -1) = (Delta U (gamma-1))/(alpha)(as n = 1 - alpha)`
By the first law of thermodynamics, `Q = Delta U + A`
=`Delta U + (Delta U(gamma - 1))/(alpha) = DeltaU[1 + (gamma - 1)/(alpha)]`
(b) Molar heat capacity is given by
`C = (R)/(gamma - 1)-(R)/(n - 1)=(R)/(gamma - 1) -(R)/(1 - alpha - 1)`
=`(R)/(gamma - 1) +(R)/(alpha) (as n = 1 - alpha)`.