Molecular kinetic theory modelAQA A-Level Physics: Revision notes

Brownian motion is the random, zigzag motion of small visible particles (such as smoke or pollen) suspended in a fluid. Smoke in air viewed under a microscope shows bright specks changing direction at random.

The explanation is that the fluid is made of tiny, invisible molecules moving rapidly and randomly, which collide with the visible particle from all sides. At any instant the collisions are unbalanced, giving a resultant force in a random direction. It is therefore evidence for the existence of atoms and molecules and for their random motion.

Heating the fluid makes the molecules faster, so the motion becomes more vigorous; larger particles move less because the collisions average out more.

The kinetic theory model of an ideal gas assumes:

• a very large number of identical molecules moving randomly
• molecules have negligible volume compared with the container
• collisions are perfectly elastic, and last a negligible time
• no intermolecular forces except in collisions
• Newton's laws apply

Explaining the gas laws: pressure is the average force per unit area from molecular collisions. Halving the volume at constant temperature doubles the collision rate with the walls, so $p$ doubles (Boyle). Raising the temperature increases the mean speed, so collisions are harder and more frequent, and $p$ rises at constant volume (pressure law).

The gas laws are empirical (found from experiments), whereas the kinetic theory is a theoretical model from which the gas laws are derived.

Take a cube of side $L$ with $N$ molecules each of mass $m$. For one molecule with velocity component $c_x$:

1. Momentum change at one wall: $2mc_x$.
2. Time between collisions with the same wall: $2L/c_x$.
3. Force on the wall $= 2mc_x/(2L/c_x) = mc_x^2/L$.

For $N$ molecules the total force is $\frac{Nm\overline{c_x^2}}{L}$, and the pressure is force divided by $L^2$: $p = \frac{Nm\overline{c_x^2}}{L^3} = \frac{Nm\overline{c_x^2}}{V}$.

Motion is random, so $\overline{c^2} = \overline{c_x^2} + \overline{c_y^2} + \overline{c_z^2} = 3\overline{c_x^2}$, so $\overline{c_x^2} = \frac{1}{3}\overline{c^2}$ and

$pV = \frac{1}{3}Nm\overline{c^2} = \frac{1}{3}Nmc_{rms}^2$

Worked example: $N = 2.0 \times 10^{22}$, $m = 4.65 \times 10^{-26}$ kg, $V = 1.0 \times 10^{-3}$ m³, $p = 1.00 \times 10^5$ Pa gives $c_{rms} = 5.7 \times 10^2$ m s⁻¹.

Combining $pV = \frac{1}{3}Nmc_{rms}^2$ with $pV = NkT$ gives

$\frac{1}{2}mc_{rms}^2 = \frac{3}{2}kT = \frac{3RT}{2N_A}$

So the mean kinetic energy of a molecule is proportional to the kelvin temperature and does not depend on the gas.

In an ideal gas there are no intermolecular forces, so there is no potential energy, and the internal energy is the total kinetic energy of the molecules: $U = \frac{3}{2}NkT = \frac{3}{2}nRT$.

Worked example: helium at 300 K: mean kinetic energy $= 6.21 \times 10^{-21}$ J, $c_{rms} = 1.37 \times 10^3$ m s⁻¹, and $U$ for 1.00 mol $= 3.74 \times 10^3$ J.

In the 17th to 19th centuries scientists such as Boyle and Charles found the gas laws by experiment. Kinetic theory, developed in the 19th century, offered a theoretical model that explained them from the motion of molecules.

Brown observed pollen grains moving randomly in 1827, which had no explanation. In 1905 Einstein showed that it was caused by molecular bombardment, and Perrin confirmed this experimentally, giving convincing evidence for atoms and molecules, which some scientists had still doubted.

The ideal gas model has limits: real gases deviate at high pressure and low temperature, when molecular volume and forces matter. A theory is accepted when its predictions are confirmed by experiment and checked by other scientists, and refined when it fails.