13.4 Kinetic Theory: Atomic and Molecular Explanation of Pressure and Temperature

Kinetic Theory of Gases

Kinetic theory connects the invisible motion of individual gas molecules to the large-scale properties you can measure, like pressure, volume, and temperature. It's the bridge between the microscopic world and the macroscopic readings on your instruments, and it forms the foundation for much of thermodynamics.

Ideal Gas Law and Molecular Properties

The familiar ideal gas law ($PV = Nk_BT$) can be rewritten in a form that directly connects pressure to molecular motion:

$PV = \frac{1}{3}Nm\overline{v^2}$

• $P$ = pressure of the gas
• $V$ = volume of the gas
• $N$ = total number of molecules
• $m$ = mass of a single molecule
• $\overline{v^2}$ = mean of the squared molecular speeds

This version tells you something powerful: pressure comes from molecules slamming into the walls of their container. More molecules, heavier molecules, or faster molecules all mean higher pressure.

The kinetic theory model rests on a few simplifying assumptions:

• Gas molecules are point particles (negligible volume compared to the container).
• There are no attractive or repulsive forces between molecules.
• All collisions (molecule-molecule and molecule-wall) are perfectly elastic, meaning kinetic energy is conserved.

One related concept is mean free path, the average distance a molecule travels between collisions with other molecules. In denser gases, the mean free path is shorter because molecules are packed more closely together.

Thermal Energy and Molecular Motion

Thermal energy is the total kinetic energy of all the molecules in a substance. For a gas, this kinetic energy is primarily translational (molecules moving through space), though molecules with more than one atom can also rotate and vibrate.

The thermal energy of a gas depends on two things: how many molecules you have, and how much kinetic energy each one carries on average. Raising the temperature increases the average kinetic energy per molecule, so thermal energy goes up. Adding more gas molecules at the same temperature also increases total thermal energy.

Brownian motion provides visible evidence of kinetic theory. If you watch tiny particles (like pollen grains) suspended in a fluid under a microscope, they jiggle around randomly. That jiggling is caused by countless collisions with the surrounding molecules, and it was one of the first pieces of experimental evidence that molecules are real and in constant motion.

Gas Molecule Kinetic Energy Calculation

The average translational kinetic energy of a single gas molecule is directly proportional to absolute temperature:

$\bar{K} = \frac{3}{2}k_BT$

• $\bar{K}$ = average kinetic energy per molecule (in joules)
• $k_B$ = Boltzmann constant = $1.38 \times 10^{-23}$ J/K
• $T$ = absolute temperature (in Kelvin)

Notice that molecular mass does not appear in this equation. At the same temperature, a lightweight helium atom and a heavy xenon atom have the same average kinetic energy. The lighter molecule just moves faster to compensate.

Absolute temperature is measured from absolute zero (0 K, or $-273.15$ °C), the point where molecular motion reaches its minimum. You must use Kelvin in this equation; using Celsius will give you a wrong answer.

Temperature Effects on Gas Kinetics

Temperature is really a measure of the average kinetic energy of the molecules in a gas. Here's how changing the temperature plays out:

When temperature increases:

1. Molecules speed up, raising their average kinetic energy.
2. They hit the container walls more often and with greater force.
3. If the volume is fixed, pressure rises. If the pressure is fixed, the gas expands.

When temperature decreases:

1. Molecules slow down, lowering their average kinetic energy.
2. They hit the walls less often and with less force.
3. If the volume is fixed, pressure drops. If the pressure is fixed, the gas contracts.

This is the molecular-level explanation behind Gay-Lussac's law (pressure proportional to temperature at constant volume) and Charles's law (volume proportional to temperature at constant pressure).

Distribution of Molecular Speeds

Not every molecule in a gas moves at the same speed. The Maxwell-Boltzmann distribution describes the spread of molecular speeds at a given temperature. The curve is asymmetric: it rises steeply on the left, peaks, and then has a long tail stretching toward high speeds. A few molecules are nearly stationary, and a few are moving very fast, but most cluster around the peak.

Three characteristic speeds are defined from this distribution:

The ordering is always $v_p < \bar{v} < v_{rms}$.

Two factors shift the distribution:

• Higher temperature broadens the curve and shifts the peak to higher speeds. The molecules have more energy, so the range of speeds widens.
• Higher molecular mass narrows the curve and shifts the peak to lower speeds. Heavier molecules move more slowly at the same temperature.

Energy Distribution and Molecular Properties

The equipartition theorem states that, at thermal equilibrium, energy is shared equally among all available degrees of freedom, with each degree of freedom receiving $\frac{1}{2}k_BT$ of energy per molecule on average.

Degrees of freedom are the independent ways a molecule can store energy:

• A monatomic gas (like helium) has 3 translational degrees of freedom (motion in the x, y, and z directions), giving it an average energy of $\frac{3}{2}k_BT$ per molecule.
• A diatomic gas (like $N_2$) at moderate temperatures has 5 degrees of freedom (3 translational + 2 rotational), giving it $\frac{5}{2}k_BT$ per molecule.
• At very high temperatures, vibrational modes can activate, adding more degrees of freedom.

This is why the $\frac{3}{2}$ factor appears in the kinetic energy equation: it comes from the 3 translational degrees of freedom.

Avogadro's number ($N_A = 6.022 \times 10^{23}$) connects the molecular scale to the macroscopic scale. One mole of any substance contains $N_A$ particles, which lets you convert between per-molecule quantities (using $k_B$) and per-mole quantities (using the gas constant $R = N_A k_B = 8.314$ J/(mol·K)).