What is the relation between and of gas in this process? — Thermodynamics and Thermochemistry Chemistry Question

💡 Solution & Explanation

Polytropic heat capacity is $C = C_v + \frac{R}{1-x}$. Substituting $C = -C_v$ gives $-C_v = C_v + \frac{R}{1-x} \implies -2C_v = \frac{R}{1-x}$. Since $C_v = \frac{R}{\gamma-1}$, we get $\frac{-2R}{\gamma-1} = \frac{R}{1-x} \implies 1-x = \frac{1-\gamma}{2} \implies x-1 = \frac{\gamma-1}{2}$. The polytropic relation $TV^{x-1} = \text{const}$ becomes $TV^{(\gamma-1)/2} = \text{constant}$.