Polytropic Process Work of a Closed System with Ideal Gas

Symbols

• \(W\) — work done by the gas
• \(m\) — mass of the gas
• \( R \) — gas constant
• \(V_1\) — initial volume
• \(V_2\) — final volume
• \(T_1\) — initial thermodynamic temperature
• \(T_2\) — final thermodynamic temperature
• \(n\) — polytropic index

Note: \( mRT_1 = p_1V_1 = p_2V_2 = mRT_2 \Rightarrow T_1 = T_2 = T \rightarrow \text{Isothermal Process} \) for \( n = 1.0 \)

From polytropic process work of a closed system, we have:

\( \displaystyle W = \int_{V_1}^{V_2} p \, dV = \begin{cases} p_1 V_1 \ln\left(\frac{V_2}{V_1}\right) & \text{if } n = 1.0 \\ \frac{p_2 V_2 - p_1 V_1}{1 - n} & \text{if } n \neq 1.0 \end{cases} \)

For ideal gases, we have: \(pV = mRT \)

Substituting, we get:

\( \begin{aligned} \displaystyle W &= \begin{cases} mRT_1 \ln\left(\frac{V_2}{V_1}\right) & \text{if } n = 1.0 \\ \frac{mRT_2 - mRT_1}{1 - n} & \text{if } n \neq 1.0 \end{cases} \\ &= \begin{cases} mRT \ln\left(\frac{V_2}{V_1}\right) & \text{if } n = 1.0 \\ \frac{mR(T_2 - T_1)}{1 - n} & \text{if } n \neq 1.0 \end{cases} \end{aligned} \)

We can derive other forms by combining the polytropic and ideal gas assumptions

For polytropic processes, we have: \( \displaystyle p_1V_1^n = p_2V_2^n \Rightarrow \frac{p_2}{p_1} = (\frac{V_2}{V_1}) ^ n \ \)

To relate temperature to volume:

\( \displaystyle \frac{T_2}{T_1} = \frac{p_2 V_2 / (mR)}{p_1 V_1 / (mR)} = \frac{p_2}{p_1} \cdot \frac{V_2}{V_1} = \left(\frac{V_1}{V_2}\right)^{n} \frac{V_2}{V_1} = \left(\frac{V_1}{V_2}\right)^{n-1} \)

Then for \( n \neq 1 \),
\( \displaystyle W = \frac{mR(T_2 - T_1)}{1 - n} = \frac{mRT_1(\frac{T_2}{T_1} - 1)}{1 - n} = \frac{mRT_1[(\frac{V_1}{V_2})^{n-1} - 1]}{1 - n} \)