Free expansion into a vacuum

A particularly instructive case. A gas initially occupies half of an isolated container; the other half is empty. A valve is opened and the gas fills the whole container. Let us analyse the three exchanges:

• The container is isolated: $Q = 0$.
• The gas expands into a vacuum: it pushes against nothing, so $L_\text{gas} = 0$.
• There is no atmosphere in contact: $L_\text{atm} = 0$.

From the first law it follows immediately that

$\Delta E_\text{int} = 0 \quad\Rightarrow\quad T_B = T_A$

Notable result

In free expansion the gas expands but does not cool. Since the internal energy of an ideal gas depends only on $T$, and no energy enters or leaves, the temperature stays unchanged.

If the volume doubles ($V_B = 2V_A$), from the ideal gas law at constant $T$ the pressure halves:

$P_B = P_A\cdot\frac{V_A}{V_B} = \frac{P_A}{2}$

This is a typically irreversible transformation: during the expansion the gas never has a well-defined equilibrium state (the particles at the front rush into the vacuum before those behind), so it would make no sense to trace a curve from A to B on the $P$-$V$ diagram.

Why it is NOT an isotherm

A student might say: “If $T_A = T_B$, then it’s an isotherm!” Wrong. In a reversible isotherm the gas releases heat $Q = nRT\ln(V_B/V_A)$ to a thermostat and performs an equal amount of work on the surroundings. Here instead $Q = 0$ and $L = 0$. The two transformations have the same initial and final states of the gas, but they change the universe in different ways: one is reversible (it can be inverted), the other is not. The difference will be clarified by entropy, in the next chapter.

The complete numerical calculation (nitrogen, volume doubling) is in Free expansion of a gas.

Links

Topics: Thermodynamics Concepts: First law of thermodynamics · Internal energy · Thermodynamic transformations Objects: Ideal gas

Related exercises: Worked exercise — reversible vs irreversible isothermal expansion · Nitrogen heated in a rigid cylinder · Same temperature, different heat