An ideal gas whose adiabatic exponent equals `gamma` is expanded according to the law `p = alpha V`, where `alpha` is a constant. The initial volume of the gas is equal to `V_0`. As a result of expansion the volume increases `eta` times. Find :
(a) The incident of the internal energy of the gas ,
(b) the work performed by the gas ,
( c) the molar heat capacity of the gas in the process.

Explanation

Law of the process is `p = alpha V` or `pV^-1 = alpha`
so the process is polytropic of index `n = -1`
As `p = alpha V` so, `p_i = alpha V_0` and `p_f = alpha eta V_0`
(a) Increment of the inernal energy is given by
`Delta U = (vR)/(gamma - 1)[T_f - T_i]=(p_f V_f - p_i V_i)/(gamma - 1)`
(b) Work done by the gas is given by
`A = (p_i V_i - p_f V_f)/(n -1) =(alpha V_0^2 - alpha eta V_0.eta V_0)/(-1 -1)`
=`(alpha V_0^2 (1 - eta^2))/(-2) = (1)/(2) alpha V_0^2 (eta^2 - 1)`
( c) Molar heat capacity is given by
`C_n = (R(n - gamma))/((n- 1)(gamma - 1)) = (R(-1 -gamma))/((-1-1)(gamma - 1)) =(R gamma + 1)/(2 gamma - 1)`.