Maxwell-Boltzmann Speed Distribution Function

Why a distribution function is needed. Even though macroscopic quantities like temperature and pressure are fixed for a gas sample, individual molecules are constantly colliding and exchanging speed with one another, so no single molecule keeps the same speed for long. What genuinely stays fixed, once the gas reaches thermal equilibrium, is the number of molecules whose speed falls within any given range -- if one molecule slows down in a collision, some other molecule speeds up to compensate, so the overall population count in each speed range stays steady. The Maxwell-Boltzmann speed distribution function is the formula that gives this steady population count.

The distribution function. The number of molecules $dN_v$ whose speed lies between $v$ and $v+dv$, out of a total of $N$ molecules, is given by

Graphically (Figure 9.3), this function starts at zero, rises, reaches a single peak at the most probable speed $v_{mp}$, and then falls back toward zero at very high speeds; the area of a thin vertical strip of width $dv$ under the curve at speed $v$ equals exactly $N_v\,dv$, the number of molecules in that narrow speed range. The area under the entire curve equals the total number of molecules $N$ in the sample.

Shape of the curve. For low speeds, the curve rises roughly as $v^2$ (a parabolic-looking rise), because there are progressively more ways for a molecule's velocity components to combine into a slightly larger speed; but for high speeds, the exponential factor $e^{-mv^2/2kT}$ dominates and forces the curve to fall off sharply. The three characteristic speeds from earlier sections all sit on this one curve, in the fixed order $v_{mp}<\bar v<v_{rms}$ (Figure 9.3).

Effect of temperature (Figure 9.4). Plotting the distribution at two different temperatures shows that raising the temperature shifts the peak of the curve to the right (toward higher speeds) and flattens/broadens the curve -- meaning the average speed of the molecules increases and the spread of speeds widens. Importantly, the area under both curves stays exactly the same, since it always represents the same fixed total number of molecules; a higher temperature simply redistributes that fixed population toward higher speeds, it never creates or destroys molecules. …