One mole of an ideal gas whose adiabatic axponent equal `gamma` undergoes a process in which the gas pressure relates to the temperature as `p = aT^alpha`, where `a` and `alpha` are consists. Find :
(a) the work performed by the gas if its temperature gets an increment `Delta T` ,
(b) the molar heat capacity of the gas in the process , at what value of `alpha` will the heat capacity be negative ?

Explanation

Given `p = a T^alpha` (for one mole of gas)
So, `pT^-alpha = a` or, `p ((p V)/(R))^-alpha = a`,
or, `p^(1 - alpha) V^-alpha = aR^-alpha` or, `pV^(alpha//(alpha - 1)) = constant`
Here ploytropic exponent `n = (alpha)/(alpha - 1)`
(a) In the polytropic process for one mole of gas :
`A = (R Delta T)/(1 - n) = (R Delta T)/((1 - (alpha)/(alpha - 1)) ) = R Delta T (1 - alpha)`
(b) Molar heat capacity is given by
`C = (R)/(gamma - 1) - (R)/(n - 1)=(R)/(gamma - 1) -(R)/((alpha)/(alpha - 1) -1) =(R)/(gamma - 1) + R (1 - alpha)`.