Root-mean-square (rms) molecular speed: Which expression correctly gives the rms speed of a gas molecule in terms of Boltzmann’s constant k, absolute temperature T, and molecular mass m (mass of one molecule)?

Introduction / Context:
Kinetic theory links macroscopic thermodynamic properties to microscopic molecular motion. The root-mean-square (rms) speed is particularly important because it appears in relations for pressure and temperature and connects directly to energy per molecule.

Given Data / Assumptions:

• Ideal gas behavior with non-interacting point molecules.
• Equipartition of energy for translational degrees of freedom.
• m is the mass of one molecule; k is Boltzmann’s constant.

Concept / Approach:
For an ideal gas, the average translational kinetic energy per molecule is (1/2) * m * c_rms^2 = (3/2) * k * T. Solving for c_rms gives c_rms = sqrt(3 * k * T / m). Using molar quantities leads to an equivalent form c_rms = sqrt(3 * R_u * T / M), where M is molar mass and R_u is the universal gas constant.

Step-by-Step Solution:

Verification / Alternative check:
For nitrogen at room temperature, plugging typical values yields speeds of several hundred m/s, consistent with experimental data and kinetic-theory predictions.

Why Other Options Are Wrong:

• sqrt(kT/(3m)) and sqrt(2kT/m): Incorrect numerical factors from misapplying degrees of freedom.
• 3k*T/m (no square root): Wrong dimensionality; gives m/s^2 units under the rootless form.

Common Pitfalls:
Confusing most probable, mean, and rms speeds; mixing molar and molecular mass forms without consistent constants.

Final Answer:
c_rms = sqrt(3 * k * T / m)