Polytropic Process

A polytropic process is one in which the equation of the pressure versus volume curve is of the form PV^N = Constant. Where, N is a constant throughout the process and may adopt any numerical value.

The thermodynamics process discussed earlier, may in some instances, be regarded as particular cases of the polytropic process, where each process has a corresponding value for the exponent N = Constant.

For isochoric process, N → ± ∞

For the isobaric process, N = 0

For the isothermal process, N = 1

And for the adiabatic process, N = γ = C_P/ C_V.

The values of N for the last three above mentioned processes are apparent, for the isochoric process, if we extract the n^th root on both sides of the equation PV^N = Constant, we get P^1/N x V = Constant.

For V = Constant, P^1/N = 1 (or) 1/ N = 0 then, N → ±∞.

Thus, for a polytropic process, in the general case,

PV^N = Constant.

Work done in Polytropic process:

We have calculated it in adiabatic process. Here also, work done.

\(W=\frac{nR}{(N-1)}\left[ {{T}_{i}}-{{T}_{f}} \right]\) (Or) \(W=\frac{-nR}{\left( N-1 \right)}\left[ {{T}_{i}}-{{T}_{f}} \right]\).

\(W=-\frac{nR\Delta T}{(N-1)}\).

For one mole of a gas, n = 1

∴ \(Work\,\,done\,(W)=-\frac{R\Delta T}{(N-1)}\).