Reversible Vs Irreversible Process Comparator

Physics Thermodynamics • Heat Engines and Second Law

Written by STEM Calculators Team Published January 4, 2026 Updated February 24, 2026

5. Reversible vs. Irreversible Process Comparator

Compare two processes between the same endpoints for an ideal gas: reversible isothermal expansion (maximum work) vs. irreversible free expansion (model). For isothermal ideal gas, $W_{\mathrm{rev}} = nRT \ln\!\left(\frac{V_2}{V_1}\right)$ , $\Delta S_{\mathrm{sys}} = nR \ln\!\left(\frac{V_2}{V_1}\right)$ . Reversible: $\Delta S_{\mathrm{univ}}=0$ . Free expansion: $W\approx 0$ , $\Delta S_{\mathrm{univ}}>0$ .

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

Endpoints + settings

Amount $n$ (mol): Temperature $T$ (K): Initial volume $V_1$ (L): Final volume $V_2$ (L): Pressure input: Initial pressure $P_1$ (atm): Irreversible PV “path” model: Shade rev work area: In-plot labels:

Graph + animation controls

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

Each plot supports drag-to-pan, wheel-to-zoom, and double-click (Reset view). Graphs are stacked vertically to avoid being too small.

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Steps

Enter values and click Solve.

PV paths (stacked)

Reversible PV path (isothermal, max work)

Hover: (V,P)=…

Irreversible PV “path” (model, not quasi-static)

Hover: (V,P)=…

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Reversible vs. Irreversible Processes (same endpoints)

Two processes can start and end at the same thermodynamic states but differ in the path. Path matters because quantities like work and heat depend on the path, while state functions like internal energy $U$ and entropy $S$ depend only on endpoints.

Key ideas

Reversible (ideal limit): quasi-static, no dissipation, can be reversed by an infinitesimal change. It produces maximum work for a given expansion between fixed endpoints.
Irreversible: finite gradients (pressure/temperature), friction, turbulence, free expansion, etc. It generates entropy: $\Delta S_{\text{gen}} = \Delta S_{\text{univ}} > 0$ .
Second law accounting: $\Delta S_{\text{univ}} = \Delta S_{\text{sys}} + \Delta S_{\text{surr}} \ge 0$ . Equality corresponds to the reversible idealization.

Isothermal ideal-gas expansion example

Consider an ideal gas expanding isothermally from $V_1$ to $V_2$ at temperature $T$. The entropy change of the gas depends only on endpoints:

\[ \Delta S_{\text{sys}} = nR\ln\!\left(\frac{V_2}{V_1}\right). \]

Reversible isothermal path (maximum work)

For a reversible isothermal expansion, the gas pressure satisfies $P(V)=nRT/V$, so the work done by the gas is:

\[ W_{\text{rev}} = \int_{V_1}^{V_2} P\,dV = \int_{V_1}^{V_2} \frac{nRT}{V}\,dV = nRT\ln\!\left(\frac{V_2}{V_1}\right). \] \[ \Delta U = 0 \ \text{(ideal gas, isothermal)} \ \Rightarrow\ Q_{\text{rev}} = W_{\text{rev}}. \] \[ \Delta S_{\text{surr}} = -\frac{Q_{\text{rev}}}{T} = -nR\ln\!\left(\frac{V_2}{V_1}\right) \ \Rightarrow\ \Delta S_{\text{univ}} = 0. \]

Free expansion (irreversible)

In a free expansion into vacuum (idealized insulated container), the gas does no boundary work: $$W=0$$ , and receives no heat: $$Q=0$$ . Nevertheless, the gas entropy increases by the same endpoint formula:

\[ W_{\text{free}}=0,\quad Q_{\text{free}}=0,\quad \Delta S_{\text{sys}} = nR\ln\!\left(\frac{V_2}{V_1}\right), \] \[ \Delta S_{\text{surr}}=0 \ \Rightarrow\ \Delta S_{\text{univ}} = nR\ln\!\left(\frac{V_2}{V_1}\right) > 0. \]

Lost work (university note)

A common way to quantify irreversibility is “lost work”, comparing the reversible maximum work to what actually occurs:

\[ W_{\text{lost}} = W_{\text{rev}} - W_{\text{actual}} \ge 0. \] If an environment temperature $T_0$ is used, exergy destruction is often written as \[ X_{\text{dest}} = T_0\,\Delta S_{\text{gen}} = T_0\,\Delta S_{\text{univ}}. \]

What the PV plots mean

The reversible isothermal PV curve is a true quasi-static path, so the area under $P(V)$ matches the work.
Free expansion is not quasi-static; it has no well-defined intermediate PV states. The calculator shows a dashed “jump” to emphasize that the path is not a real equilibrium curve.