Kinetic theory – ratio of rms speed to mean speed For an ideal gas at a given temperature, what is the ratio of root-mean-square molecular speed to average (mean) molecular speed?

Introduction / Context:Kinetic theory provides several characteristic molecular speeds for gases: most probable speed, mean (average) speed, and root-mean-square (rms) speed. Their ratios are constants independent of gas species at a fixed temperature.

Given Data / Assumptions:

• Maxwell–Boltzmann speed distribution applies.
• Definitions: v_mp = most probable, v_avg = average, v_rms = root-mean-square.
• Ideal-gas behavior and continuum of speeds.

Concept / Approach:The standard formulas are v_mp = sqrt(2RT/M), v_avg = sqrt(8RT/(pi M)), and v_rms = sqrt(3RT/M), where R is the universal gas constant and M is molar mass. The ratio v_rms/v_avg = sqrt(3RT/M) / sqrt(8RT/(pi M)) = sqrt(3*pi/8) ≈ 1.086.

Step-by-Step Solution:Write v_avg = sqrt(8RT/(pi M)).Write v_rms = sqrt(3RT/M).Form the ratio: v_rms / v_avg = sqrt(3RT/M) / sqrt(8RT/(pi M)).Simplify: v_rms / v_avg = sqrt(3 * pi / 8) ≈ sqrt(1.1781) ≈ 1.086.

Verification / Alternative check:Other useful ratios: v_avg/v_mp = sqrt(pi/2) ≈ 1.253 and v_rms/v_mp = sqrt(3/2) ≈ 1.225; these constants corroborate the internal consistency of Maxwellian statistics.

Why Other Options Are Wrong:

• 0.086, 3.086, 4.086 are numerically inconsistent and dimensionless outliers.
• 0.913 would be the inverse of 1.095-ish, not the correct ratio here.

Common Pitfalls:Confusing “most probable” with “average” or mixing unit systems; these ratios are dimensionless constants for the Maxwell distribution.

Final Answer:1.086