Maxwell Speed Distribution

Defining three characteristic speeds, the Maxwell–Boltzmann distribution gives the most probable speed $v_{mp}=\sqrt{2RT/M}$, the average speed $v_{avg}=\sqrt{8RT/\pi M}$, and the root-mean-square speed $v_{rms}=\sqrt{3RT/M}$. These differ because the distribution of molecular speeds is asymmetric, with a long tail toward high speeds that pulls the average and rms values above the peak of the curve. Their ratio depends only on the numerical coefficients, since $R$, $T$, and $M$ cancel. Thus $\frac{v_{rms}}{v_{mp}}=\sqrt{\frac{3}{2}}=1.225$. The value 1.128 is the ratio $v_{avg}/v_{mp}=\sqrt{4/\pi }$, not the rms-to-mp ratio. The value 1.085 equals $v_{rms}/v_{avg}=\sqrt{3\pi /8}$, a different comparison. The value 1.414 mistakenly takes $\sqrt{2}$, ignoring the factor of 3 in the rms expression. This ordering $v_{mp}<v_{avg}<v_{rms}$ is a key NCERT feature of the speed distribution, reflecting that squaring weights faster molecules more heavily before averaging. As temperature rises the whole distribution broadens and shifts to higher speeds, but the fixed ratios among these three speeds remain unchanged because temperature and molar mass cancel. As a consistency check, all three speeds follow the fixed hierarchy relative to the most probable speed, confirming that the rms speed is the largest of the three and that the computed value 1.225 is correct.