Molecular kinetic energy comparison:\nAt the same temperature and pressure, which gas sample has the highest kinetic energy per mole?

Introduction / Context:
In kinetic theory, the average translational kinetic energy per molecule depends only on absolute temperature, not on the molecular identity. Expressed per mole, this leads to a universal value proportional to temperature, which is critical for understanding equipartition and temperature scales.

Given Data / Assumptions:

• Ideal-gas assumption.
• Same temperature for all gases being compared.
• Per mole basis.

Concept / Approach:
The mean translational kinetic energy per molecule is (3/2) k_B T. On a per mole basis, multiply by Avogadro’s number to obtain (3/2) R T, which is identical for any ideal gas at the same temperature. Pressure does not alter this equality for ideal gases; it affects density and mean free path, not the per-mole kinetic energy at a given T.

Step-by-Step Solution:
KE_molar = (3/2) R T for an ideal gas.At fixed T, KE_molar is the same for Cl2, N2, C2H6, He, etc.Therefore, all gases have the same kinetic energy per mole.

Verification / Alternative check:
Although heavier molecules move more slowly (lower rms speed), their higher mass exactly compensates to give the same average translational energy at the same temperature.

Why Other Options Are Wrong:
Picking a specific gas ignores the temperature-only dependence of KE in the ideal-gas model.

Common Pitfalls:
Confusing speed with kinetic energy; rms speed varies with molecular mass, while KE per mole at fixed T does not.

Final Answer:
all gases have the same kinetic energy per mole