Free expansion of a gas

Problem

$3\ \text{kg}$ of nitrogen ($n = 107\ \text{mol}$) are in a $2\ \text{m}^3$ container at $T_A = 300\ \text{K}$, connected by a valve to another identical, empty box, the whole assembly thermally insulated. The valve is opened. What is the final state?

Solution

Energy balance. Let us analyse the three terms:

• Insulated container $\Rightarrow Q = 0$.
• The gas expands into a vacuum: it pushes against nothing $\Rightarrow L_\text{gas} = 0$.
• No surrounding atmosphere in contact $\Rightarrow L_\text{atm} = 0$.

First law. $\Delta E_\text{int} = Q - L_\text{gas} + L_\text{atm} = 0 \ \Rightarrow\ T_B = T_A = 300\ \text{K}$

Final volume. The gas fills both boxes: $V_B = 2\cdot 2 = 4\ \text{m}^3$.

Final pressure. At constant temperature: $P_B = P_A\cdot\frac{V_A}{V_B} = \frac{P_A}{2}$

$\ev{T_B = 300\ \text{K},\ V_B = 4\ \text{m}^3,\ P_B = P_A/2}$

The gas expands but does NOT cool: this is an irreversible process, which cannot be traced on the $P$-$V$ plane.

Links

Topics: Thermodynamics Concepts: First law of thermodynamics · Thermodynamic processes Objects: Ideal gas