13.1 Kinetic theory of gases and Maxwell-Boltzmann distribution

Kinetic Theory Assumptions

Kinetic theory explains macroscopic gas properties (temperature, pressure, volume) by modeling what happens at the molecular level. The theory rests on a set of idealizing assumptions that define what we call an ideal gas.

Particle Motion and Collisions

Gas particles are in constant, random motion, traveling in straight lines between collisions. All collisions are perfectly elastic, meaning total kinetic energy is conserved. The particles do exert forces on each other during a collision, but those forces are assumed to be negligible at all other times.

Particle Volume and Interactions

• The volume of individual gas particles is negligible compared to the volume of the container. This lets us treat particles as point-like objects that move freely without crowding each other.
• Between collisions, particles experience no intermolecular forces. Each particle moves independently in a straight line until it hits another particle or a wall.

These two assumptions break down for real gases at high pressures (particles crowd together) or low temperatures (intermolecular attractions become significant). That's exactly what the van der Waals equation corrects for, but for this topic we stick with the ideal case.

Kinetic Energy and Temperature

The average translational kinetic energy of gas particles is directly proportional to the absolute temperature of the gas. As temperature rises, particles move faster on average and collide more often and more forcefully.

The theory also assumes the gas is in thermal equilibrium: temperature is uniform throughout the sample. Under this condition, particle speeds follow the Maxwell-Boltzmann distribution.

Maxwell-Boltzmann Distribution

Probability Distribution of Molecular Speeds

The Maxwell-Boltzmann distribution gives the probability of finding a gas particle with a particular speed at a given temperature. It comes from applying the Boltzmann distribution to the translational kinetic energy $\frac{1}{2}mv^2$ of the particles.

The speed distribution function is:

$f(v) = 4\pi \left(\frac{m}{2\pi k T}\right)^{3/2} v^2 \exp\!\left(-\frac{mv^2}{2kT}\right)$

where $m$ is the particle mass, $k$ is the Boltzmann constant, $T$ is the absolute temperature, and $v$ is the speed.

Two features of this function are worth noting:

• The $v^2$ factor means the distribution starts at zero, rises to a peak, and then falls off. Very low speeds are unlikely (small $v^2$), and very high speeds are unlikely (the exponential decay dominates).
• The exponential term $\exp(-mv^2/2kT)$ acts as a "Boltzmann penalty" for high kinetic energies. Higher temperature weakens this penalty, spreading the distribution toward faster speeds.

Factors Influencing the Distribution

Temperature: Increasing $T$ shifts the peak to higher speeds and broadens the curve. The distribution flattens out because more particles can access higher kinetic energies. For example, the speed distribution of $N_2$ at 300 K is noticeably broader and has a higher peak speed than the same gas at 100 K.

Particle mass: At the same temperature, lighter particles have a broader, faster distribution than heavier ones. Helium atoms ($m \approx 4$ u) spread across a much wider range of speeds than xenon atoms ($m \approx 131$ u) at the same $T$. This is because lighter particles need less energy to reach a given speed.

Molecular Speed Calculations

Three Characteristic Speeds

The Maxwell-Boltzmann distribution defines three commonly used average speeds. All three depend on $kT/m$, but they weight the distribution differently.

• Most probable speed $v_{mp}$: the speed at the peak of the distribution.

$v_{mp} = \sqrt{\frac{2kT}{m}}$

• Mean (average) speed $v_{avg}$: the arithmetic mean over all particles.

$v_{avg} = \sqrt{\frac{8kT}{\pi m}}$

• Root-mean-square speed $v_{rms}$: the square root of the mean of $v^2$.

$v_{rms} = \sqrt{\frac{3kT}{m}}$

For $N_2$ ($m \approx 4.65 \times 10^{-26}$ kg) at 300 K, these work out to roughly 422 m/s, 475 m/s, and 517 m/s respectively.

Why the Three Speeds Differ

The ordering $v_{mp} < v_{avg} < v_{rms}$ always holds, at any temperature, for any gas. The reason is the asymmetric shape of the Maxwell-Boltzmann curve: it has a long tail extending toward high speeds.

• $v_{mp}$ just locates the peak, so it's the lowest of the three.
• $v_{avg}$ averages over all speeds, and the high-speed tail pulls the mean above the peak.
• $v_{rms}$ squares each speed before averaging, which gives extra weight to fast particles, making it the largest.

The ratios between them are fixed by the numerical prefactors: $v_{mp} : v_{avg} : v_{rms} = \sqrt{2} : \sqrt{8/\pi} : \sqrt{3} \approx 1 : 1.128 : 1.225$.

Kinetic Energy and Temperature

The Core Relationship

The average translational kinetic energy of a gas particle is:

$\langle KE \rangle = \frac{3}{2}kT$

This comes from the equipartition theorem: each translational degree of freedom contributes $\frac{1}{2}kT$ to the average energy, and there are three translational degrees of freedom (motion along $x$, $y$, and $z$).

• For monatomic gases (He, Ne, Ar), translation is the only form of kinetic energy, so total average kinetic energy per particle is $\frac{3}{2}kT$.
• For diatomic ($N_2$, $O_2$) and polyatomic ($CO_2$, $CH_4$) gases, rotational and vibrational modes add to the total energy. But the translational kinetic energy is still $\frac{3}{2}kT$ regardless of molecular complexity.

A key takeaway: at the same temperature, every ideal gas has the same average translational kinetic energy per particle. Heavier molecules just move more slowly to carry that energy.

Connecting to Macroscopic Properties

Pressure arises from particles colliding with container walls. The kinetic theory result is:

$P = \frac{2}{3}\frac{N}{V}\langle KE \rangle$

Substituting $\langle KE \rangle = \frac{3}{2}kT$ recovers the ideal gas law $PV = NkT$. Higher temperature means faster particles, harder and more frequent wall collisions, and therefore higher pressure.

Volume at constant pressure grows with temperature because faster-moving particles push the container walls outward (or, in a flexible container, the gas expands). This is Charles's law: $V \propto T$ at constant pressure.