Velocity Distribution

Maxwell-Boltzmann distribution of molecular speeds in ideal gases. Theory, formulas, worked examples and interactive graphs.

Conceptual context

An ideal gas does not contain molecules moving at one common speed. Even at a single $T$, each particle has its own $v$ because collisions constantly redistribute microscopic motion. The Maxwell-Boltzmann speed distribution describes that population: few molecules are very slow, many gather around a typical range, and few are very fast. This is why a gas can have reproducible pressure while its molecules remain individually unpredictable. The curve is therefore not decorative; it is the object that links microscopic mechanics to thermodynamic observation. The central question is how $T$ shapes the whole distribution, not which molecule happens to be fastest. Therefore the graph of $f_v$ versus $v$ is a statistical map of molecular agitation. It explains why repeated measurements on a large gas sample can be stable even though no molecule keeps a fixed speed.

🟢 Essential level

The $T$ sets the energy scale of molecular motion. When it increases, the distribution of $v$ shifts toward larger values and becomes broader: more molecules occupy the fast tail, although slow molecules still exist. When it decreases, the distribution narrows and moves toward lower speeds. The $m$ also matters. At the same $T$, lighter particles reach larger characteristic speeds because the same thermal energy corresponds to more motion for a smaller mass.

The $f_v$ is not a direct number of molecules. A probability comes from an interval of $v$, so the area between two limits gives $P$, while the curve height only shows which speeds are locally more frequent.

Maxwell derived this distribution in 1860 from symmetry and statistical-independence arguments, decades before any direct experimental evidence of individual molecular motion existed. Boltzmann later generalized it to systems with potential energy, giving rise to the Boltzmann factor that appears throughout modern statistical physics, from chemical kinetics to semiconductor devices.

🔵 Formal level

Basic formulas

The model uses the Maxwell-Boltzmann speed distribution. The horizontal variable is $v$, the vertical variable is $f_v$, and the shape depends on $T$, $m$, and $k_B$.

Maxwell-Boltzmann speed distribution

Three characteristic speeds summarize different readings of the same curve. The $v_p$ marks the maximum of $f_v$. The $v_{media}$ is the average speed. The $v_{rms}$ weights fast molecules more strongly because it uses squared speed.

Most probable speed

Mean speed

Root-mean-square speed

Real student questions

Why does the curve start at zero if very slow molecules can exist? Because the probability of exactly zero speed is negligible in three dimensions; a small interval of $v$ is what matters.

Is $v_p$ the same as $v_{media}$? No. The fast tail pushes $v_{media}$ above the local maximum.

Does higher $T$ accelerate every molecule equally? No. It changes the whole distribution: some molecules remain slow and others become much faster.

What does $f_v$ measure? It measures probability per unit speed, so a real probability needs an interval.

Final synthesis

The $T$ does not assign one speed to the gas, and that is the central lesson. It fixes the shape of a population, so physical interpretation comes from the curve, its characteristic speeds, and the areas associated with intervals. That viewpoint is statistical physics in miniature. It shapes a distribution. Maxwell-Boltzmann lets us read typical speeds, mean energy, and interval probabilities while keeping the statistical nature of molecular motion visible.