Kinetic theory of gases — mean translational kinetic energy per kilomole For an ideal gas at absolute temperature T, what is the mean translational kinetic energy per kilogram-mole (kmol) of gas? (Ru denotes the universal gas constant.)

Introduction / Context:
The kinetic theory links temperature to molecular motion. For monatomic ideal gases (and for the translational component of any ideal gas), the average translational kinetic energy depends linearly on absolute temperature. Converting per-molecule results to per-kilomole uses the universal gas constant Ru.

Given Data / Assumptions:

• Ideal gas behavior.
• Translational degrees of freedom are three in three-dimensional space.
• Ru is the universal gas constant per kilomole.

Concept / Approach:
Per molecule, mean translational kinetic energy is (3/2) * k_B * T. Multiplying by Avogadro’s number N_A converts to per mole: (3/2) * N_A * k_B * T = (3/2) * R, where R is the gas constant per mole. Per kilomole, the corresponding constant is Ru, hence energy becomes (3/2) * Ru * T.

Step-by-Step Solution:
Start with E_trans,per molecule = (3/2) * k_B * T.Multiply by Avogadro’s number: E_trans,per mole = (3/2) * R * T.Per kmol, replace R with Ru: E_trans,per kmol = (3/2) * Ru * T.Therefore, the correct choice is 1.5 * Ru * T.

Verification / Alternative check:
Dimensional check: Ru has units kJ/(kmol·K) in engineering units, so energy per kmol is kJ/kmol as expected for a temperature-proportional energy.

Why Other Options Are Wrong:
RuT underestimates by a factor of 1.5; 2RuT and 3RuT overestimate; 0.5Ru*T is not supported by kinetic theory.

Common Pitfalls:
Mixing molar (R) with universal (Ru) constants; confusing total kinetic energy with internal energy for polyatomic gases—here we refer specifically to translational component that scales as (3/2) * Ru * T per kmol.

Final Answer:
1.5 * Ru * T