Find the molar heat capacity of an ideak gas in a polytropic process `p V^n = const` if the adiabatic exponent of the gas is equal to `gamma`. At what values of the polytropic constant `n` will the heat capacity of the gas be negative ?

Explanation

In the polytropic process work done by the gas
`A = (vR[T_i - T_f])/(n - 1)`
(where `T_i` and `T_f` are initial and final temperature of the gas like in adiabatic process)
and `Delta U = (vR)/(gamma - 1)(T_f - T_i)`
By the first law of thermodynamics `Q = Delta U + A`
=`(vR)/(gamma - 1)(T_f - T_i)+(vR)/(n -1) (T_i - T_f)`
=`(T_f - T_i) vR[(1)/(gamma - 1) -(1)/(n -1)] = (vR[n - gamma])/((n - 1)(gamma -1)) Delta T`
According to definition of molar heat capacity when number of moles `v = 1` and `Delta T = 1` then `Q` = Molar heat capacity.
Here, `C_n = (R(n - gamma))/((n - 1)(gamma - 1)) lt 0 "for" 1 lt n lt gamma`.