Kinetic Molecular Theory (KMT)

Gas molecules travel in straight lines at constant speed until they collide. Because collisions are random in direction, no single direction of motion is preferred. This random motion explains why a gas fills its container uniformly and accounts for the phenomenon of diffusion — the spontaneous mixing of two gases without any stirring.

The actual diameter of a gas molecule (~0.1–0.5 nm) is tiny compared to the average distance between molecules (~3–10 nm at room conditions). Therefore the molecules themselves occupy a negligible fraction of the total container volume. Mathematically, this means the volume available for motion ≈ the container volume V, which is the assumption built into the ideal gas law PV = nRT. Real gases deviate from this at high pressures, where molecules are forced close together.

Between collisions, gas molecules exert no attractive or repulsive forces on one another. This means the internal energy of an ideal gas is purely kinetic — there is no potential energy term from intermolecular interactions. Real gases deviate from this at low temperatures and high pressures, where van der Waals attractions become significant, which is why the van der Waals equation $(P + \frac{n^2 a}{V^2})(V - nb) = nRT$ corrects for both molecular volume (b) and intermolecular attractions (a).

When molecules collide with each other or with the container walls, kinetic energy is transferred between molecules but none is lost to heat or vibration. The total kinetic energy of the gas remains constant as long as temperature is constant. It is these elastic collisions with container walls that give rise to gas pressure: each collision exerts a tiny force, and the cumulative effect of countless collisions per second per unit area equals the macroscopic pressure P.

The average translational kinetic energy $\bar{E}_k$ of a gas molecule is directly proportional to absolute temperature: $\bar{E}_k = \frac{3}{2}k_BT$, where $k_B = 1.381 \times 10^{-23}$ J K⁻¹ is Boltzmann's constant. This is the single most powerful postulate of KMT. It tells us that temperature is literally a measure of molecular motion — at 0 K, all translational motion ceases. Doubling the Kelvin temperature doubles the average kinetic energy of every molecule in the sample, regardless of the identity of the gas.

Because molecules have a distribution of speeds, we need a representative speed. The root mean square speed $u_{rms}$ is defined as the square root of the average of the squared speeds: $u_{rms} = \sqrt{\bar{u^2}}$. Combining $\bar{E}_k = \frac{1}{2}m\bar{u^2} = \frac{3}{2}k_BT$ and using $R = N_A k_B$ (where $N_A$ is Avogadro's number and M is molar mass in kg mol⁻¹), we arrive at the famous KMT equation: $u_{rms} = \sqrt{\frac{3RT}{M}}$. Lighter molecules (small M) move faster; heavier molecules move more slowly at the same temperature. For example, at 298 K, hydrogen molecules ($\ce{H2}$, M = 0.002 kg mol⁻¹) have $u_{rms}$ ≈ 1930 m s⁻¹, while oxygen molecules ($\ce{O2}$, M = 0.032 kg mol⁻¹) have $u_{rms}$ ≈ 484 m s⁻¹.

• u_{rms} — Root Mean Square Speed (m s⁻¹): The square root of the mean of the squares of the individual molecular speeds; it is slightly larger than the simple average speed.
• R — Universal Gas Constant (J mol⁻¹ K⁻¹): R = 8.314 J mol⁻¹ K⁻¹. It connects energy to temperature on a per-mole basis and appears in the ideal gas law PV = nRT.
• T — Absolute Temperature (K): Temperature measured on the Kelvin scale (K = °C + 273.15). Must always be in Kelvin in this equation; using Celsius gives a wrong answer.
• M — Molar Mass (kg mol⁻¹): The mass of one mole of the gas in kilograms. Note: molar mass is usually listed in g mol⁻¹, so you must divide by 1000 before substituting into this equation.

Problem. Calculate the root mean square speed of nitrogen gas ($\ce{N2}$) molecules at 25 °C. (M for $\ce{N2}$ = 28.02 g mol⁻¹, R = 8.314 J mol⁻¹ K⁻¹)

Answer. The rms speed of nitrogen molecules at 25 °C is approximately 515 m s⁻¹ (about 1854 km h⁻¹ — faster than a rifle bullet!).

• Ideal Gas Law (PV = nRT) — KMT provides the molecular-level justification for the ideal gas law; the five postulates are the microscopic assumptions from which PV = nRT can be mathematically derived.
• Real Gases and the van der Waals Equation — Real gases deviate from KMT predictions at high pressure and low temperature because the negligible-volume and negligible-forces assumptions break down; the van der Waals equation corrects for these failures.
• Graham's Law of Effusion and Diffusion — Graham's Law (rate ∝ 1/√M) is a direct consequence of the KMT rms speed equation — lighter gases have higher average speeds and therefore effuse faster through a pinhole.
• Entropy and the Second Law of Thermodynamics — Boltzmann's statistical interpretation of entropy as molecular disorder (S = k_B ln W) grew directly from his work on KMT, connecting gas behaviour to the broader field of thermodynamics.