What is the magnitude of work performed by the gas to increase its internal energy by ? — Thermodynamics and Thermochemistry Chemistry Question

💡 Solution & Explanation

Internal energy $U = n C_v T = \frac{PV}{\gamma-1}$. Equating to the given relation: $\frac{PV}{\gamma-1} = aV^\alpha \implies P = (\gamma-1)aV^{\alpha-1}$. Work $W = \int P dV = \int (\gamma-1)aV^{\alpha-1} dV = \frac{\gamma-1}{\alpha}a [V_2^\alpha - V_1^\alpha]$. Since $\Delta U = aV_2^\alpha - aV_1^\alpha$, $W = \frac{\gamma-1}{\alpha}\Delta U$.