One mole of an ideal monatomic gas undergoes the process `p=alphaT^(1//2)`, where `alpha` is a constant.
(a) Find the work done by the gas if its temperature increases by 50K.
(b) Also, find the molar specific heat of the gas.

Explanation

`P = alpha T^(1//2)`
where `alpha` is constant
`n = 1 mol`, monatomic
`W = int PdV = int (alpha T^(1//2)) dV`
From Eq. (i),
`T = (alpha T^(1//2))/(nR) implies T^(1//2) = (alpha V)/(nR)`
Differntiating both sides,
`(1)/(2 sqrtT) dT = (alpha dV)/(nR) implies dV = (nR)/(alpha 2 sqrtT) dT`
`W = int alpha T^(1//2) (nR)/(2T^(1//2)) dT`
`= (n alpha R)/(2) int_(T_(1))^(T_(2)) dT = (R )/(2) xx 50 = 25R = 207.7 J`
b. `Q = Delta U + W`
`C (T_(2) - T_(1)) = (R )/((gamma -1)) (T_(2) - T_(1)) + (R )/(2) (T_(2) - T_(1))`
`C = (R )/((gamma - 1)) + (R )/(2) = ((gamma + 1)/(gamma - 1)) (R )/(2)`