Q.Explain in detail the Maxwell Boltzmann distribution function.

Step 1. Even at fixed macroscopic temperature and pressure, individual molecules constantly exchange speed through collisions, so what stays fixed at equilibrium is the number of molecules in each speed range, not any individual molecule's speed.

Step 2. Maxwell and Boltzmann derived the formula for this steady population count: the number of molecules with speed between $v$ and $v+dv$ is $dN_v=4\pi N\left(\dfrac{m}{2\pi kT}\right)^{3/2}v^2e^{-mv^2/2kT}\,dv$.

Step 3. Plotted against $v$, this curve starts at zero, rises roughly as $v^2$ at low speeds, reaches a single peak exactly at the most probable speed $v_{mp}$, then falls off sharply due to the exponential factor at high speeds.

Step 4. The area under a thin strip of width $dv$ at speed $v$ gives the number of molecules in that narrow range; the area under the WHOLE curve equals the total molecule count $N$.

Step 5. The three characteristic speeds sit at fixed points on this one curve, in the order $v_{mp}<\bar v<v_{rms}$. …

State the distribution function, describe its shape and peak, and explain t …

• Claiming the area under the curve changes with temperature -- it stays fixed at N; only the …