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The Maxwell-Boltzmann speed distribution $f(v) = 4\pi n(m/(2\pi k_BT))^{3/2} v^2 e^{-mv^2/(2k_BT)}$ gives the number of molecules with speeds between $v$ and $v + dv$. The distribution has three key features: (1) rises as $v^2$ at low speeds (phase space factor), (2) decays exponentially at high speeds (Boltzmann factor), and (3) peaks at $v_{\text{mp}} = \sqrt{2k_BT/m}$.

As temperature increases: the peak shifts right (higher $v_{\text{mp}}$), the distribution broadens, and the peak height decreases (area under curve is conserved = total $N$). As molecular mass increases: the peak shifts left and narrows (heavier molecules are slower and more uniform).

The distribution is not symmetric — it has a long tail at high speeds, which is why $v_{\text{avg}} > v_{\text{mp}}$ and $v_{\text{rms}} > v_{\text{avg}}$. About 37% of molecules exceed $v_{\text{rms}}$, and a small but significant fraction have speeds far exceeding $v_{\text{mp}}$.

For JEE, the qualitative shape and how it changes with $T$ and $M$ are more important than the formula itself. Know the ordering $v_{\text{mp}} < v_{\text{avg}} < v_{\text{rms}}$ and the universal ratio.