10.2 Kinetic theory of gases

Kinetic Theory of Gases

Kinetic theory connects the microscopic behavior of gas molecules to the macroscopic properties you can measure, like pressure, volume, and temperature. By modeling gas molecules as tiny particles in constant random motion, the theory derives the ideal gas law and explains why heating a gas increases its pressure. These ideas show up everywhere, from predicting how weather balloons expand at altitude to understanding why tire pressure rises on a hot day.

Postulates of Kinetic Theory

The kinetic theory of gases rests on a set of simplifying assumptions. Each one strips away complexity so the math becomes tractable, and together they define what we mean by an "ideal" gas.

• Large number of molecules in constant, random motion. Gas molecules travel in straight lines at high speed until they collide with another molecule or a container wall. The randomness of their motion is what produces Brownian motion when you observe tiny particles suspended in a gas.
• Perfectly elastic collisions. Total kinetic energy is conserved in every collision. Molecules bounce off each other and off walls without losing energy to heat or deformation. Think of idealized billiard balls that never slow down.
• Negligible molecular volume. The actual volume occupied by the molecules themselves is tiny compared to the volume of the container. Picture a few ping-pong balls inside a large room: the balls take up almost none of the room's volume.
• No intermolecular forces except during collisions. Between collisions, molecules exert zero attractive or repulsive forces on each other. They only interact during the brief instant of a collision itself.
• Average kinetic energy is proportional to absolute temperature. When you raise the temperature, molecules move faster and carry more kinetic energy on average. This is the direct link between the microscopic world (molecular speed) and the macroscopic world (temperature reading on a thermometer).

Relationships in Kinetic Theory

Pressure from molecular collisions

Pressure arises because billions of molecules slam into the container walls every second. Each collision exerts a tiny force; the sum of all those forces, spread over the wall's area, is the pressure you measure.

The kinetic theory expression for pressure is:

$P = \frac{1}{3} n m \overline{v^2}$

where $n$ is the number density (molecules per unit volume), $m$ is the mass of a single molecule, and $\overline{v^2}$ is the mean square velocity. Either increasing the number of collisions (more molecules or a smaller container) or increasing collision intensity (faster molecules) raises the pressure. That's why your tire pressure climbs after a long highway drive: the air inside heats up, molecules move faster, and each wall collision delivers more impulse.

Ideal gas law

$PV = nRT$

Here $P$ is pressure, $V$ is volume, $n$ is the number of moles, $R$ is the universal gas constant ($8.314 \text{ J mol}^{-1}\text{K}^{-1}$), and $T$ is the absolute temperature in kelvins. This single equation ties together all four macroscopic gas variables. A weather balloon, for example, expands as it rises because external pressure drops while the amount of gas and temperature stay roughly constant, so $V$ must increase.

Kinetic energy and temperature

The average translational kinetic energy of a gas molecule is:

$\frac{1}{2} m \overline{v^2} = \frac{3}{2} k_B T$

where $k_B$ is the Boltzmann constant ($1.381 \times 10^{-23} \text{ J K}^{-1}$). Notice that this depends only on temperature, not on the type of gas. At a given temperature, every ideal gas has the same average molecular kinetic energy. Heavier molecules compensate by moving more slowly.

Velocity and Energy of Gas Molecules

Root mean square (rms) velocity

Because molecular speeds vary widely, we need a single representative value. The rms velocity is defined as:

$v_{rms} = \sqrt{\overline{v^2}} = \sqrt{\frac{3RT}{M}}$

where $M$ is the molar mass (in kg/mol). For nitrogen ($M = 0.028 \text{ kg/mol}$) at 300 K, this gives roughly 517 m/s. Lighter gases move faster: hydrogen at the same temperature has an rms speed of about 1920 m/s.

Average kinetic energy per molecule

$\overline{KE} = \frac{1}{2} m \overline{v^2} = \frac{3}{2} k_B T$

Two key takeaways from this equation:

• At the same temperature, all ideal gas molecules share the same average kinetic energy regardless of their mass.
• Heavier molecules must therefore have lower average speeds to keep the kinetic energy the same. Oxygen molecules ($M = 0.032 \text{ kg/mol}$) move noticeably slower than helium atoms ($M = 0.004 \text{ kg/mol}$) at room temperature.

Maxwell-Boltzmann Velocity Distribution

Not every molecule in a gas travels at the same speed. The Maxwell-Boltzmann distribution gives the fraction of molecules with speeds in any given range at thermal equilibrium. When plotted, it produces an asymmetric curve that rises steeply, peaks, and then tails off gradually toward high speeds.

How temperature and mass affect the distribution

• Higher temperature shifts the peak to the right and flattens the curve. The distribution broadens because more molecules reach higher speeds.
• Heavier molecules produce a narrower, taller peak centered at a lower speed. Compare oxygen and hydrogen at room temperature: hydrogen's curve is much broader and peaks at a much higher speed.

Three characteristic speeds

The distribution defines three commonly used speeds, and their ordering is always the same:

1. Most probable velocity $v_p = \sqrt{\frac{2RT}{M}}$ This is the speed at the peak of the distribution curve, where the largest fraction of molecules is found.

2. Mean (average) velocity $\overline{v} = \sqrt{\frac{8RT}{\pi M}}$ The arithmetic average of all molecular speeds. It sits slightly to the right of $v_p$.

3. Root mean square velocity $v_{rms} = \sqrt{\frac{3RT}{M}}$ Always the largest of the three because squaring gives extra weight to the fastest molecules.

The relationship is always $v_p < \overline{v} < v_{rms}$. All three increase with temperature and decrease with molar mass, but their ratios to each other stay constant for any ideal gas.