Adiabatic Free Expansion Calculator

Physics Thermodynamics • First Law of Thermodynamics

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

5. Adiabatic Free Expansion

Free expansion into vacuum in an insulated system: $$Q=0$$ , $$W=0$$ ⇒ $\Delta U=0$ . For an ideal gas $$U=U(T)$$ , so $\Delta U=0 \Rightarrow T_2=T_1$ . Endpoints are well-defined, but the path is not quasistatic.

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

Inputs

Amount $n$ (mol): Initial temperature $T_1$ (K): Initial volume $V_1$: Final volume $V_2$: Volume unit: Pressure unit (display): Entropy preview: Gas model (endpoint + preview):

For ideal-gas free expansion into vacuum: $W=0,\ Q=0,\ \Delta U=0,\ T_2=T_1$ , and $P\propto 1/V$ at the endpoints.

Real-gas teaser (optional): van der Waals constants

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

Used only when “van der Waals” model is selected. This preview affects endpoint pressures (not the core result $W=Q=\Delta U=0$ for the insulated vacuum expansion model).

Graph + animation controls

Progress $\tau \in [0,1]$: Play speed: Overlay: reversible adiabatic $PV^\gamma=\text{const}$: Adiabatic index $\gamma$: Custom $\gamma$: Overlay: reversible isothermal $P=nRT/V$: Shade reversible work area:

PV graph supports drag-to-pan, wheel-to-zoom, double-click or “Reset view” to restore. The cylinder animation uses $\tau$ only as a progress parameter (not real seconds).

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Steps

Enter values and click Solve.

Cylinder–piston animation (illustrative)

Insulated cylinder expanding into vacuum (particles + piston)

Progress: $\tau=1$

PV graph (endpoints + comparison overlays)

Free expansion: endpoints only (no quasistatic path) + optional overlays

Hover: (V,P)=…

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Theory — Adiabatic Free Expansion (Into Vacuum)

A free expansion occurs when a gas expands into an evacuated region (vacuum) so that the external pressure opposing the expansion is effectively zero. In the classic setup the container is also insulated, so no heat is exchanged with the surroundings.

1) What makes it “free” and “adiabatic”?

Free (into vacuum): $P_{\text{ext}}\approx 0$, so the boundary work is $W=\int P_{\text{ext}}\,dV \approx 0$.
Adiabatic (insulated): $Q=0$.

The process is generally irreversible and non-quasistatic: during the motion the gas is not in equilibrium, so a single well-defined system $P(V)$ curve is not meaningful. However, the initial and final equilibrium states are well-defined.

2) First law consequence: $\Delta U = 0$

With the common sign convention $\Delta U = Q - W$:

\[ Q=0,\quad W=0 \quad\Rightarrow\quad \Delta U = 0 \]

3) Ideal gas result: temperature stays constant

For an ideal gas, the internal energy depends only on temperature: $U=U(T)$. Therefore,

\[ \Delta U = 0 \Rightarrow \Delta T = 0 \Rightarrow T_2 = T_1 \]

The endpoint pressure follows from the ideal gas law:

\[ P_1=\frac{nRT_1}{V_1},\qquad P_2=\frac{nRT_2}{V_2}=\frac{nRT_1}{V_2} \]

4) Entropy increases (irreversible)

Even though $Q=0$, the entropy can increase because the process is irreversible. For an ideal gas, an endpoint formula gives

\[ \Delta S = nR\ln\!\left(\frac{V_2}{V_1}\right) \;>\; 0 \quad (V_2>V_1) \]

Compare this with a reversible adiabatic (quasistatic) expansion, which has $\Delta S=0$.

5) Comparison: free expansion vs reversible adiabatic

Both have $Q=0$, but they differ in work:

Free expansion: $W=0\Rightarrow \Delta U=0\Rightarrow T$ unchanged (ideal gas).
Reversible adiabatic expansion: $W>0\Rightarrow \Delta U=-W<0\Rightarrow T$ decreases (ideal gas).

\[ \text{Reversible adiabatic: } PV^\gamma=\text{const},\qquad TV^{\gamma-1}=\text{const} \]

6) Real-gas note: Joule (constant-$U$) effect and Joule–Thomson (extension)

Real gases do not obey $U=U(T)$ exactly. In a free expansion with $\Delta U=0$, the temperature can change slightly. A common “teaser” model uses a van der Waals-type internal energy $U(T,V)=nC_vT-\dfrac{a n^2}{V}$, which can predict a small $T$ decrease as $V$ increases.

University extension: the Joule–Thomson effect concerns temperature change in throttling at constant enthalpy ($\Delta H=0$), which is a different process than free expansion ($\Delta U=0$).

Steps

Cylinder–piston animation (illustrative)

PV graph (endpoints + comparison overlays)

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