Distribution of Molecular Speeds

The Maxwell-Boltzmann distribution describes the predictable spread of molecular speeds in a gas, even though individual molecules move randomly. This distribution is fundamental because it explains how molecular speeds vary at different temperatures and enables predictions about gas properties like vapor pressure and chemical reaction rates.

Root-mean-square (RMS) speed is the square root of the mean-square speed derived from average kinetic energy and temperature. The most probable velocity, shown as the peak of the Maxwell-Boltzmann distribution curve, is always less than the RMS speed, reflecting the asymmetric shape of the molecular speed distribution.

At higher temperatures, gas molecules possess greater average kinetic energy, causing them to move faster and with greater variation in speeds. This increased energy spreads the velocity distribution curve wider and shifts the RMS speed to higher values, creating a broader, flatter distribution curve.

Molecules must reach a minimum escape velocity to leave a liquid surface and enter the vapor phase. The Maxwell-Boltzmann distribution shows that more molecules exceed this threshold at higher temperatures, increasing evaporation rates. When evaporation and condensation rates balance, phase equilibrium is established, determining vapor pressure.

Chemical reaction rates depend on the Maxwell-Boltzmann distribution because only molecules with sufficient kinetic energy can overcome activation barriers and react. As temperature increases, the distribution shifts to higher speeds, exponentially increasing the fraction of molecules with enough energy to react.

With every collision between gas molecules, the velocity of individual particles changes in both magnitude and direction. However, despite these constant changes at the molecular level, the collection of many molecules maintains a stable, predictable speed distribution characteristic of the Maxwell-Boltzmann curve.

Kinetic molecular theory establishes that average translational kinetic energy per molecule depends only on temperature, not on molecular mass or pressure. This relationship allows us to derive root-mean-square speed from temperature and explains why all gases at the same temperature have the same average kinetic energy.