Problem — Ranking mean kinetic energy

Problem

Three gas samples:

• A: $1$ mole of helium at $300$ K;
• B: $1$ mole of nitrogen at $300$ K;
• C: $1$ mole of helium at $600$ K.

Order the mean kinetic energy per molecule from smallest to largest.

Solution

The mean translational kinetic energy per molecule depends only on temperature, not on mass or chemical species: $\bar E = \tfrac{3}{2}k_B T$ Comparing the three samples:

• A (helium, $300$ K) and B (nitrogen, $300$ K): same temperature, so same $\bar E$. The type of gas is irrelevant: $E_A = E_B$.
• C (helium, $600$ K): double the temperature, so double the $\bar E$ compared with A and B.

Hence $E_A = E_B$ and $E_C = 2E_A$, and ordering from smallest to largest $E_A = E_B < E_C$: $\ev{E_C > E_B = E_A}$

Links

Topics: Kinetic theory of gases Concepts: Kinetic energy · Temperature Skills: Ranking reasoning