Maxwell-Boltzmann Distribution
The Speed Distribution Function
The Maxwell-Boltzmann speed distribution gives the probability of finding a molecule with a speed between v and v + dv: f(v) = 4 pi n (m/(2 pi kT))3/2 v2 exp(-mv2/(2kT)), where m is the molecular mass, k is Boltzmann constant, T is the absolute temperature, and n is the number density. The v2 factor (from the surface area of a spherical shell in velocity space) ensures that f(v) starts at zero for v = 0, peaks at an intermediate speed, and then decreases exponentially at high speeds.
The distribution has a characteristic asymmetric shape: a gentle rise from zero, a peak at the most probable speed, and a long tail extending to high speeds. The tail is important because even at moderate temperatures, a small but significant fraction of molecules have speeds much higher than the average. These high-speed molecules are disproportionately important in chemical reactions, where only molecules exceeding the activation energy barrier can react.
The distribution depends on only two parameters: the temperature T and the molecular mass m. Higher temperatures shift the distribution to higher speeds and broaden it (molecules have a wider range of speeds). Heavier molecules have lower average speeds at the same temperature. These dependencies follow directly from the equipartition theorem: the average translational kinetic energy per molecule is (3/2)kT, regardless of mass.
Characteristic Speeds
Three characteristic speeds are commonly defined from the Maxwell-Boltzmann distribution. The most probable speed v{sub}mp{/sub} = sqrt(2kT/m) is the speed at the peak of the distribution, where the largest number of molecules are found. The mean speed v{sub}avg{/sub} = sqrt(8kT/(pi m)) is the average of all molecular speeds. The root-mean-square speed v{sub}rms{/sub} = sqrt(3kT/m) is calculated from the average of v2 and relates directly to the average kinetic energy.
These three speeds are related by v{sub}mp{/sub} < v{sub}avg{/sub} < v{sub}rms{/sub}, with the ratios v{sub}mp{/sub} : v{sub}avg{/sub} : v{sub}rms{/sub} = 1 : 1.128 : 1.225. The difference arises from the asymmetric shape of the distribution: the long high-speed tail pulls the mean and RMS speeds above the peak. For nitrogen (N{sub}2{/sub}) at room temperature (300 K), v{sub}mp{/sub} is about 422 m/s, v{sub}avg{/sub} is about 476 m/s, and v{sub}rms{/sub} is about 517 m/s.
The root-mean-square speed is the most useful for connecting to macroscopic properties because the average kinetic energy per molecule is (1/2)m v{sub}rms{/sub}2 = (3/2)kT. This equation directly links molecular speed to temperature, confirming that temperature is a measure of average molecular kinetic energy. It also shows why lighter gases diffuse faster: at the same temperature, hydrogen molecules (m = 2 u) move about 4 times faster than oxygen molecules (m = 32 u).
Effect of Temperature and Molecular Mass
Increasing the temperature broadens the Maxwell-Boltzmann distribution and shifts it to higher speeds. The peak height decreases because the distribution must remain normalized (the total probability is always 1). At higher temperatures, more molecules populate the high-speed tail, which has important consequences for reaction rates: a modest increase in temperature can dramatically increase the fraction of molecules above the activation energy.
Heavier molecules have narrower distributions centered at lower speeds. At 300 K, helium atoms (mass 4 u) have a v{sub}rms{/sub} of about 1370 m/s, while xenon atoms (mass 131 u) have a v{sub}rms{/sub} of about 240 m/s. This mass dependence explains Graham law of effusion: the rate at which gas molecules escape through a small hole is inversely proportional to the square root of their molecular mass.
The ratio of kinetic energy to thermal energy determines how the distribution behaves. The exponential factor exp(-mv2/(2kT)) = exp(-E{sub}k{/sub}/(kT)) shows that the probability of finding a molecule with kinetic energy E{sub}k{/sub} decreases exponentially as the energy exceeds kT. This is a special case of the Boltzmann distribution, which applies to all forms of energy, not just translational kinetic energy.
The Maxwell-Boltzmann distribution shows that molecular speeds in a gas follow a specific statistical pattern determined only by temperature and molecular mass.
Applications of the Maxwell-Boltzmann Distribution
In chemical kinetics, the Arrhenius equation k = A exp(-E{sub}a{/sub}/(RT)) for reaction rate constants derives directly from the Maxwell-Boltzmann distribution. The exponential factor represents the fraction of molecules with enough kinetic energy (at least E{sub}a{/sub}) to overcome the activation energy barrier. This explains why reaction rates increase exponentially with temperature: the fraction of molecules in the high-speed tail grows rapidly with T.
In atmospheric science, the Maxwell-Boltzmann distribution determines which gas molecules can escape Earth gravitational field. The escape velocity from Earth is about 11,200 m/s. While the average speed of nitrogen molecules at atmospheric temperatures is far below this, the tail of the distribution extends to very high speeds. For light molecules like hydrogen and helium, a significant fraction exceeds escape velocity, which is why Earth atmosphere has lost most of its primordial hydrogen and helium over geological time.
In spectroscopy, thermal motion of atoms and molecules causes Doppler broadening of spectral lines. Each molecule emits or absorbs light at a frequency shifted by the Doppler effect, and the distribution of shifts follows the Maxwell-Boltzmann speed distribution. The resulting spectral line has a Gaussian shape whose width is proportional to sqrt(T/m). Measuring this width allows astronomers to determine the temperature of distant gas clouds and stellar atmospheres.