Speed Distribution Types

Question

For a sample of ideal gas at a given temperature, arrange the most probable speed, the average speed and the rms speed in order of increasing magnitude.

RankGuru · Accepted Answer

The ordering of the three characteristic speeds follows from the Maxwell distribution discussed in NCERT Class 11, Chapter 13 (Kinetic Theory), where the most probable speed $v_p = \sqrt{\frac{2RT}{M}}$, the average speed $\overline{v} = \sqrt{\frac{8RT}{\pi M}}$, and the rms speed $v_{rms} = \sqrt{\frac{3RT}{M}}$. Comparing the numerical coefficients under the square roots, we have $2$ for the most probable, $\frac{8}{\pi} \approx 2.55$ for the average, and $3$ for the rms speed. Since $2 < 2.55 < 3$, the speeds increase in the order most probable, then average, then rms. The option reversing this order mistakes the largest coefficient for the smallest. The option placing average below most probable contradicts the value $\frac{8}{\pi} > 2$. The option that all speeds are equal ignores the asymmetric shape of the distribution. These three measures describe the same distribution differently: the most probable speed marks the curve's peak, the average speed is the arithmetic mean, and the rms speed weights faster molecules more heavily by squaring their speeds. A plausibility check confirms the ordering: the distribution has a long high-speed tail that pulls the average and especially the rms value above the peak (most probable) speed, so $v_p < \overline{v} < v_{rms}$ is physically expected.

Speed Distribution Types

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