Polytropic Process Solver

Physics Thermodynamics • First Law of Thermodynamics

Written by STEM Calculators Team Published January 2, 2026 Updated February 24, 2026

7. Polytropic Process Solver

Polytropic process: $PV^n=\text{const}$ . For an ideal gas with constant heat capacities, this tool computes $$P_2$$ , $$T_1,T_2$$ , $$W$$ , $\Delta U$ , and $$Q$$ . Special limits: $n\to 1$ (isothermal) and $n=\gamma$ (adiabatic for ideal gas).

Inputs support: pi, e, sqrt(), sin, cos, exp, log. Use * for multiplication.

Inputs

Amount $n_{\text{mol}}$ (mol): Volume unit: Pressure unit: Initial pressure $P_1$: Initial volume $V_1$: Final volume $V_2$: Polytropic index $n$: $n$ slider: Heat capacity ratio $\gamma$ (for $\Delta U$ + adiabatic overlay): Custom $\gamma$: Overlay adiabatic $PV^\gamma=\text{const}$: Overlay isothermal $n=1$: Shade work area $\int P\,dV$:

Tip: move the $n$ slider to see the PV curve change.

Sign convention: $W>0$ for expansion work done by the system. Then $\Delta U = Q - W\Rightarrow Q=\Delta U + W$ .

Graph + progress controls

Progress $\tau\in[0,1]$: Play speed:

PV graph supports drag-to-pan, wheel-to-zoom, double-click or “Reset view” to restore.

Ready

Steps

Enter values and click Solve.

PV graph (polytropic curve + overlays)

Polytropic: $PV^n=\text{const}$ (with optional isothermal/adiabatic comparison)

Hover: (V,P)=… Marker: τ=1

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Theory: Polytropic Process

A polytropic process is defined by $PV^n=\text{const}$ , where $n$ is the polytropic index. Many real processes can be approximated by some effective $n$ between common limits.

1) Core relation

\[ PV^n = K \quad (\text{constant along the path}) \] \[ P_2 = P_1\left(\frac{V_1}{V_2}\right)^n \]

2) Work for $n\neq 1$

Boundary work (quasi-static) is $W=\int_{V_1}^{V_2}P\,dV$. Using $P=K/V^n$:

\[ W=\int_{V_1}^{V_2}\frac{K}{V^n}\,dV = \frac{K}{1-n}\left(V_2^{1-n}-V_1^{1-n}\right) = \frac{P_2V_2 - P_1V_1}{1-n} \quad (n\neq 1). \]

3) Isothermal limit $n\to 1$

As $n\to 1$, the polytropic relation approaches $PV=\text{const}$ (isothermal for an ideal gas), and the work becomes the logarithmic form:

\[ W = n_{\text{mol}}RT\ln\!\left(\frac{V_2}{V_1}\right)\quad (n=1). \]

4) Temperature change for an ideal gas

Combine $PV^n=\text{const}$ with $PV=n_{\text{mol}}RT$: $T\propto V^{1-n}$. Therefore,

\[ \frac{T_2}{T_1}=\left(\frac{V_2}{V_1}\right)^{1-n}. \]

5) $\Delta U$ and $Q$ (constant heat capacities)

For an ideal gas with approximately constant heat capacities, $\Delta U=n_{\text{mol}}C_v(T_2-T_1)$. With $\gamma=C_p/C_v$, $C_v=\frac{R}{\gamma-1}$ and $C_p=\frac{\gamma R}{\gamma-1}$. Using the first law $\Delta U=Q-W$, we get $Q=\Delta U+W$.

\[ C_v=\frac{R}{\gamma-1},\quad \Delta U=n_{\text{mol}}C_v(T_2-T_1),\quad Q=\Delta U+W. \]

6) Special comparison: adiabatic $n=\gamma$

For a reversible adiabatic ideal-gas process, $PV^\gamma=\text{const}$. The calculator shows this as an “adiabatic limit” overlay so you can compare shapes.

7) PV curve & work area

On a $P$-$V$ plot, the work is the area under the curve between $V_1$ and $V_2$, with sign determined by the direction (expansion $V_2>V_1$ tends to give $W>0$).

Website tip: include an “adiabatic limit” explanation next to the $n$ slider to link the polytropic family to common reversible processes.

Steps

PV graph (polytropic curve + overlays)

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