Kinetic Theory — Temperature Dependence of Molecular Speeds According to the kinetic theory of gases, how does the average molecular speed change with increasing temperature (for a given gas)?

Introduction / Context:Kinetic theory links macroscopic temperature to microscopic molecular motion. Understanding this relationship is essential for diffusion, viscosity, and speed of sound calculations in gases.

Given Data / Assumptions:

• Ideal gas approximation.
• Molecular mass constant for the gas.
• Temperature measured on an absolute scale (kelvin).

Concept / Approach:The root-mean-square (rms) speed is given by c_rms = sqrt(3 * R_specific * T) = sqrt(3 * k_B * T / m). As temperature T increases, c_rms increases with the square root of T for a fixed molecular mass m, meaning molecules move faster on average at higher temperatures.

Step-by-Step Solution:Write c_rms = sqrt(3 * R_specific * T).At constant gas species (R_specific fixed), c_rms ∝ sqrt(T).Therefore, as T increases, average molecular speed increases.

Verification / Alternative check:Maxwell–Boltzmann distributions broaden and shift to higher speeds with temperature, consistent with the square-root dependence of characteristic speeds (most probable, mean, rms) on T.

Why Other Options Are Wrong:“Remains constant” would require no temperature effect; “decreases” contradicts kinetic theory relations.

Common Pitfalls:Confusing effects of changing gas species (m changes) with changing temperature for a fixed gas; mixing Celsius changes with absolute Kelvin changes in proportional formulas.

Final Answer:increases