Question

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.

Answer 1

The work performed by the gas is Q = = ΔU[1+ γ−1 / α] and the molar heat capacity is C = R / γ−1 + R / α

Explanation:

We are given that:

U = aVα

or, vCVT = aV^α, or vCV pV / vR = aV^α

or, aVα. R / CV.1 / pV = 1

or, V^α−1.p^−1 = CV / Ra

pV^1 − α = Ra / CV = constant = a(γ−1) [as CV = R / γ−1]

So polytropric index n = 1−α.

(a) Work done by the gas is given by

A = −vRΔT / n−1 and ΔU = vRΔT / γ−1

Hence A = −ΔU(γ−1) / n−1 = ΔU(γ−1) / α (as n = 1 − α )

By the first law of thermodynamics, Q = ΔU + A

Q = ΔU + ΔU(γ−1) / α = ΔU[1+ γ−1 / α]

(b) Molar heat capacity is given by

C = R / γ − 1 − R / n − 1 = R / γ − 1 − R / 1 − α − 1

C = R / γ−1 + R / α (as n = 1 − α)