Kinetic Theory of Gases – Translational Kinetic Energy vs. Absolute Temperature According to the kinetic theory, the average translational kinetic energy of a gas molecule is directly proportional to the absolute temperature (on the Kelvin scale).

Introduction / Context:
The kinetic theory of gases links microscopic molecular motion with macroscopic thermodynamic properties. A cornerstone result is that the average translational kinetic energy of molecules scales with absolute temperature, providing a physical meaning to temperature as a measure of molecular agitation.

Given Data / Assumptions:

• Ideal gas with negligible intermolecular potential energy during collisions.
• Random, isotropic molecular motion (no preferred direction).
• Equilibrium state so that averages are meaningful.

Concept / Approach:

For an ideal monatomic gas, the mean translational kinetic energy per molecule is (3/2) * k_B * T, where k_B is Boltzmann’s constant and T is absolute temperature. This shows a linear dependence on T. While the numeric prefactor can change with degrees of freedom in more complex molecules, the proportionality to T for translational modes remains fundamental.

Step-by-Step Solution:

Verification / Alternative check:

Root-mean-square speed u_rms satisfies u_rms ∝ sqrt(T). Since KE ∝ u_rms^2, it follows directly that KE ∝ T. Calorimetric measurements of heat capacity also align with this relationship for gases near ideal behavior.

Why Other Options Are Wrong:

1/T and constant independence contradict the kinetic-theory equations. T^2 is too strong a dependence; the correct law is first power of T. Pressure-only dependence ignores the equation of state and microscopic basis.

Common Pitfalls:

Confusing energy per mole with energy per molecule; mixing total internal energy (which includes all modes) with purely translational kinetic energy; using Celsius instead of Kelvin.

Final Answer:

proportional to T (absolute temperature)