For a polytropic process PV^n = constant, Cm (molar heat capacity) of an ideal gas is given by

To find the molar heat capacity (Cm) of an ideal gas for a polytropic process described by the equation PV^n = constant, we can follow these steps:

1. Start with the polytropic equation PV^n = constant, where P is the pressure, V is the volume, and n is the polytropic index.

2. Take the natural logarithm of both sides of the equation to simplify it: ln(PV^n) = ln(constant).

3. Apply the logarithmic properties to simplify the equation further: ln(P) + n*ln(V) = ln(constant).

4. Differentiate both sides of the equation with respect to V to find the derivative of ln(P) with respect to V: d(ln(P))/dV + n*d(ln(V))/dV = 0.

5. Use the ideal gas law, PV = nRT, where R is the ideal gas constant and T is the temperature, to substitute for P: d(ln(nRT))/dV + n*d(ln(V))/dV = 0.

6. Simplify the equation by rearranging terms: n*d(ln(V))/dV = -d(ln(nRT))/dV.

7. Recognize that d(ln(V))/dV is the reciprocal of V, and d(ln(nRT))/dV is the reciprocal of Cm: n/V = -1/Cm.

8. Rearrange the equation to solve for Cm: Cm = -V/n.

Therefore, the molar heat capacity (Cm) of an ideal gas for a polytropic process described by the equation PV^n = constant is given by Cm = -V/n.