Joule Expansion Simulator

Physics Thermodynamics • First Law of Thermodynamics

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

8. Joule Expansion Simulator

Free expansion into vacuum in an insulated rigid container: $$Q=0$$ , $$W=0$$ ⇒ $\Delta U=0$ . For a real gas (van der Waals), $$U(T,V)$$ depends on volume, so $$T$$ can change. This tool uses the common model $U \approx nC_vT - a\,n^2/V$ (constant heat capacities) to predict $$T_2$$ .

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

Inputs

Gas preset: Amount $n_{\text{mol}}$ (mol): Initial temperature $T_1$ (K): Initial occupied volume $V_1$: Final volume $V_2$ (after expansion): Volume unit: Pressure unit (display): Model / comparison: Entropy preview: $\gamma$ (for $C_v$ estimate): Custom $\gamma$: Heat capacity mode: Manual $C_v$ (J/mol·K):

van der Waals constants

$a$ (Pa·m$^6$/mol$^2$): $b$ (m$^3$/mol):

These are used in:

P=\dfrac{nRT}{V-nb}-a\left(\dfrac{n}{V}\right)^2

and

U\approx nC_vT - a\,n^2/V

Free expansion into vacuum (Joule expansion): $$W=0$$ , $$Q=0$$ , so $\Delta U=0$ . The PV “path” is not quasistatic — the graph shows endpoints and an illustrative marker controlled by $\tau$.

Graph + animation controls

Progress $\tau\in[0,1]$: Play speed: Overlay: isothermal at $T_1$ (vdW + ideal): Shade (illustrative) $\int P\,dV$:

PV graph supports drag-to-pan, wheel-to-zoom, double-click or “Reset view” to restore. The animation shows particles flowing through a valve into a vacuum chamber (illustrative).

Ready

Steps

Enter values and click Solve.

Valve + vacuum expansion animation (illustrative)

Rigid container: left gas chamber → right vacuum chamber (valve opens with $\tau$)

Progress: τ=0 T(τ)=… , P(τ)=…

PV graph (endpoints + comparison)

Joule expansion: endpoints (vdW vs ideal) + optional isotherms

Hover: (V,P)=… Marker: τ=0

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Joule Expansion Simulator — Theory

Joule (free) expansion is an irreversible expansion of a gas into a vacuum inside an insulated rigid container. Since there is no external pressure resisting the expansion and no heat exchange with the surroundings, the first law gives $$Q=0$$ , $$W=0$$ , hence $\Delta U=0$ . For ideal gases, internal energy depends only on temperature, so $T$ does not change. For real gases, $U$ can depend on both $T$ and $V$, so $T$ may change even when $\Delta U=0$.

1) First law for free expansion

\[ \Delta U = Q - W. \] Insulated container: $Q=0$. Expansion into vacuum: $P_{\text{ext}}\approx 0\Rightarrow W=\int P_{\text{ext}}\,dV=0$. Therefore, \[ \Delta U = 0. \]

2) Ideal-gas result: $T_2=T_1$

For an ideal gas, $U=U(T)$ only. If $\Delta U=0$, then $T$ must remain constant: $\Delta U=0\Rightarrow T_2=T_1$ . Endpoint pressures follow from the ideal equation of state:

\[ P=\frac{nRT}{V}. \]

3) van der Waals model and predicted temperature change

The van der Waals equation of state is $\left(P + a\left(\frac{n}{V}\right)^2\right)(V-nb)=nRT$ , i.e.

\[ P=\frac{nRT}{V-nb}-a\left(\frac{n}{V}\right)^2. \]

A commonly used accompanying internal-energy approximation (constant heat capacities) is:

\[ U \approx nC_vT - a\,\frac{n^2}{V}. \]

For free expansion, $\Delta U=0$. Using the model above:

\[ 0=\Delta U \approx nC_v(T_2-T_1) - a n^2\left(\frac{1}{V_2}-\frac{1}{V_1}\right), \] so \[ T_2 = T_1 + \frac{a n}{C_v}\left(\frac{1}{V_2}-\frac{1}{V_1}\right). \]

If $a>0$ and $V_2>V_1$, then $\frac{1}{V_2}-\frac{1}{V_1}<0$, so the model typically predicts cooling (\(T_2

4) Entropy change (endpoint expression)

The process is irreversible, so entropy increases. In the constant-$C_v$ van der Waals model, an endpoint-state expression for entropy can be written as $S=nC_v\ln T + nR\ln(V-nb)+\text{const}$, giving:

\[ \Delta S_{\text{vdW}} \approx nC_v\ln\!\left(\frac{T_2}{T_1}\right)+nR\ln\!\left(\frac{V_2-nb}{V_1-nb}\right). \]

For the ideal limit (with $T_2=T_1$), this reduces to $\Delta S = nR\ln(V_2/V_1)$.

5) Joule vs. Joule–Thomson (throttling)

Joule expansion (this calculator) is free expansion into vacuum in an insulated rigid container with $\Delta U=0$. Joule–Thomson throttling (flow through a valve/porous plug) is a different process where $\Delta H\approx 0$ (enthalpy is approximately conserved). Both can produce temperature changes for real gases, but the governing conserved quantity differs.

6) Parameter table tip

Van der Waals constants $a$ and $b$ depend on the gas and the chosen unit system. This calculator expects SI-based constants: $a$ in Pa·m$^6$/mol$^2$ and $b$ in m$^3$/mol. The built-in CO$_2$ preset uses common sample values for demonstration.

Preset	$a$ (Pa·m$^6$/mol$^2$)	$b$ (m$^3$/mol)	Notes
CO$_2$ (example)	0.364	4.27×10$^{-5}$	Used in sample input; adjust as needed for your reference.