Many compression and expansion processes are approximated by a polytropic relation

$$pv^n=\text{constant},$$

where the exponent $n$ is chosen to model the process.

For an ideal gas with $n\ne1$, the specific boundary work is

$$w_b=\frac{p_2v_2-p_1v_1}{1-n}=\frac{R(T_2-T_1)}{1-n}.$$

Special values recover familiar idealizations: $n=0$ gives constant pressure, $n=1$ gives an isothermal ideal-gas process, and $n=k=c_p/c_v$ corresponds to a reversible adiabatic ideal-gas process with constant specific heats.

For $n=1$, integration instead gives

$$w_b=RT\ln\frac{v_2}{v_1}.$$

A polytropic exponent is a process model, not an intrinsic material property. Fitting $n$ to compressor or expansion data can summarize combined heat-transfer and irreversibility effects, but it does not identify their physical causes.