Polytropic Process

Many expansion and compression processes of gases are polytropic, such that they follow the equation

$$P{V}^n=C\text{or}P=C{V}^{-n}$$

In general, they follow the equation

$$P_1{V}_1^n=P_2{V}_2^n$$

Recall that the amount of work done in a thermodynamic process is determined by the process path:

$$\begin{array}{rl}{W}_{1-2}& ={\int }_{V_1}^{V_2}P(V)dV\\ & ={\int }_{V_1}^{V_2}C{V}^{n}dV\end{array}$$

Changing the exponent $n$ allows us to describe many different processes. The value of $n$ for a given process is typically found by curve-fitting experimental data.

$n\ne 1$

If $n\ne 1$, we have:

$$W_{1-2}=\frac{C{V}^{-n+1}}{-n+1}{|}_{V_1}^{V_2}=\frac{C{V}_2^{1-n}-C{V}_1^{1-n}}{1-n}$$

Since $C=P_1{V}_1^n=P_2{V}_2^n$, we have:

$$\boxed{W_{1-2}=\frac{P_2{V}_2-P_1{V}_1}{1-n}}$$

If we are working with an ideal gas, this can be simplified to a temperature-dependent form:

$$W_o=\frac{mR(T_2-T_1)}{1-n}$$

$n=1$

If $n=1$, we have:

$$W_{1-2}={\int }_{V_1}^{V_2}\frac{C}{V}dV=C\ln V{|}_{V_1}^{V_2}=C\ln \left(\frac{V_2}{V_1}\right)$$

Thus, we have:

$$\begin{array}{r}{W}_{1-2}=P_1{V}_1\ln \left(\frac{V_2}{V_1}\right)\\ =\boxed{P_2{V}_2\ln \left(\frac{V_2}{V_1}\right)}\end{array}$$

This can again be simplified for an ideal gas with $PV=mRT$. In this case, this would be an isothermal process, since $R$ and $m$ are constant in the system, so $T$ must also be constant to have $P_1{V}_1=P_2{V}_2$.

$n=0$

If $n=0$, we have:

$$W_{1-2}=P(V_2-V_1)$$

which is an isobaric process.

$n=\infty$

If $n=\infty$, we have:

$$W_{1-2}=0$$

which is isochoric process.

$n=\gamma$

For a ratio of specific heat, $\gamma =\frac{c_p}{c_v}$, which is approximately $1.4$ for air.