molecular-kinetic assumptions and Brownian evidence

Molecular-kinetic theory (MKT) rests on three assumptions: matter consists of particles, there are gaps between the particles, and the particles move continuously and chaotically, interacting through attractive and repulsive forces that are noticeable only at short distances.

The first and second assumptions are supported by diffusion: if $1\ \text{cm}^3$ of water is mixed with $1\ \text{cm}^3$ of alcohol, the volume of the mixture is less than $2\ \text{cm}^3$, and a drop of ink colours the whole volume of water without any stirring.

Continuous chaotic motion is revealed by Brownian motion — the erratic jitter of small particles suspended in a liquid or a gas (R. Brown, 1827).

A Brownian particle of radius $r \approx 10^{-6}\ \text{m}$ is bombarded by molecules from all sides; the number of impacts on the left and on the right is random, so their resultant is not zero and changes direction every instant.

The intensity of the motion grows with temperature and falls as the particle size increases, in agreement with the link between the mean kinetic energy of a molecule and the absolute temperature: $\langle E_k \rangle \propto T$.

Knowing mass, density and volume lets us connect observation with calculation of the number of molecules: $N = \dfrac{m}{M}N_A$, where $M$ is the molar mass.

A drop of ink falls into a glass of still water, and after a few hours the water becomes uniformly coloured. Which assumption of molecular-kinetic theory does this observation support? Justify your answer.

A drop of oil of volume $V = 1{,}0\cdot 10^{-11}\ \text{m}^3$ spreads over a water surface into a circular film of diameter $D = 0{,}20\ \text{m}$. Assuming the film is one molecule thick, estimate the diameter of an oil molecule.

Two identical Brownian particles are in water: one at $T_1 = 293\ \text{K}$, the other at $T_2 = 333\ \text{K}$. Assuming the mean square displacement per unit time is proportional to the absolute temperature, find the ratio $\dfrac{\langle x^2\rangle_2}{\langle x^2\rangle_1}$. Using the model of molecular impacts, also explain why the observed Brownian motion cannot be explained by convection and why it is not seen for a particle about $1\ \text{mm}$ in size.

Simulation (the main way to observe). Open a browser simulation of Brownian motion or diffusion.

1. Set the temperature to $T_1$, switch on the trajectory display and record the particle coordinate every $2\ \text{s}$; use $10$–$15$ points to compute the mean square displacement $\langle x^2\rangle$.
2. Repeat at $T_2 > T_1$ and compare $\langle x^2\rangle$; check whether the dependence $\langle x^2\rangle \propto T$ is confirmed.
3. Halve the particle size at the same temperature and note how the jitter amplitude changes.

Safe home diffusion experiment. On a tray, pour cold and warm (not boiling) water into two glasses and carefully add one drop of food colouring to each; compare the times needed for uniform colouring. Do not heat the water to boiling, do not taste the solutions, and wipe up spills at once.

Show that the trajectory of a Brownian particle is random, that it arises from uncompensated molecular impacts, and that its amplitude grows with temperature.

A two-panel sketch: on the left, a Brownian particle in a liquid with a zigzag trajectory and arrows of molecular impacts from different sides; on the right, the same particle at a higher temperature with a larger zigzag; beside them the formulas $\langle x^2\rangle \propto T$ and $r \approx 10^{-6}\ \text{m}$ are written, with a displacement scale under the panels.

Before a match the ball is pumped so that the absolute air pressure inside is $1.6\times10^5$ Pa, the ball volume is $V=5.0$ L $=5.0\times10^{-3}$ m$^3$, and the temperature is $T=293$ K. The number of air molecules inside is about $N=\frac{pV}{k_BT}\approx2.0\times10^{23}$, and the mean molecular speed at $293$ K is of order $500$ m/s. The ball feels springy because its shell continuously receives many molecular impacts.

The scenario uses the molecular-kinetic assumptions: gas molecules are small compared with the distances between them, move chaotically, and collide elastically with the wall; pressure arises as the average force of impacts divided by area. Brownian evidence is not directly observed here, but the same chaotic motion explains why the pressure does not drop by itself.

Into a glass with $200$ mL of water at $20$ °C falls a drop of food dye with volume about $0.05$ mL. Without stirring, the coloured region spreads over several centimetres in $10$ min; individual dye microparticles are seen under a microscope as chaotically jittering dots. The speed of water molecules at $20$ °C is of order $600$ m/s.

This is direct Brownian evidence: water molecules continuously and randomly strike the dye particle from all sides; at each instant the numbers of impacts from left and right are not equal, so the particle performs a random walk, and on average matter is transported from a region of high concentration to a region of low concentration. The scenario tests the molecular-kinetic assumption of continuous chaotic molecular motion.

On a calm warm day at $30$ °C, a pollen grain of diameter $10$ μm in a light beam shifts by $1$–$2$ μm in $1$ s, although the air appears still. Air molecules at $30$ °C have a mean speed of about $510$ m/s, and the number of their impacts on such a grain is about $10^{18}$ per second.

This is the gas version of Brownian evidence: the impacts of air molecules on the pollen are not balanced at each instant, so the grain jitters and shifts randomly. The scenario rests on the molecular-kinetic assumption that molecules are in continuous thermal motion and that their impacts are statistically distributed.