Diffusion and Brownian motion

Particles of a gas, liquid, or solution move continuously and chaotically; their mean kinetic energy increases with temperature, for example $\langle E_k \rangle = \frac{3}{2}k_B T$ for an ideal gas.

Diffusion is the spontaneous transport of a substance from a region of higher concentration to a region of lower concentration: random walks carry particles out of a dense region more often than back into it. Brownian motion is the jitter and displacement of a large suspended particle (for example, a dust grain in water) seen under a microscope, caused by unbalanced impacts of many molecules of the medium.

Both phenomena are direct consequences of the molecular-kinetic model: visible order (equalized concentration, steady chaotic jitter) is born from disordered collisions. The diffusion coefficient $D$ increases with temperature and decreases as particle size and the viscosity of the medium increase. For a random walk the mean square displacement grows linearly with time, $\langle x^2 \rangle = 2Dt$, so diffusion over large distances is very slow, $t \sim L^2/D$.

A drop of ink is placed in still water at $20\ ^\circ\text{C}$. After some time the water is uniformly tinted. Explain the spreading of the ink in terms of particle motion, and state what kind of motion a single ink particle undergoes.

The diffusion coefficient of oxygen in water at $20\ ^\circ\text{C}$ is about $D = 2.1\times10^{-9}\ \text{m}^2/\text{s}$. Estimate the time for oxygen molecules to diffuse a mean distance $L = 1\ \text{mm}$, using $t \sim L^2/(2D)$.

A pollen grain of radius $r$ is suspended in water. Two samples are prepared: one at $20\ ^\circ\text{C}$ and one at $40\ ^\circ\text{C}$. (a) Predict how the observed Brownian jitter differs between the samples. (b) A second grain of twice the mass is placed in the warm sample; explain whether its jitter is larger or smaller than that of the first grain at the same temperature.

Observation (household, safe). Pour water into two clean glasses: one at room temperature, one warm from the tap (not boiling). Add one drop of food colouring to each and, without stirring, record every 30 s the time until the colour is uniform; sketch the colour front. Compare the times and explain the difference by temperature.

Simulation (for the molecular scale). Open an interactive simulation (for example, PhET “Diffusion” or “Gas Properties”) and track a single particle’s path: check that it is jagged and random, and that the concentration equalizes by itself. Do not drink the coloured water; do not heat the water above warm tap water.

Show how disordered microscopic walks produce a directed macroscopic flow in diffusion, and how unbalanced molecular impacts make a large particle jitter.

Two-panel sketch.

(a) On the left, a region of high concentration of small circles (molecules), on the right a region of low concentration; arrows of random walks point in all directions, but the net flow arrow points to the right, and below the sketch a graph of concentration $c(x)$ decreases from left to right and flattens with time.

(b) A large circle (the Brownian particle) is surrounded by small molecules; impact arrows from different sides have unequal lengths, and their resultant $\vec F$ is shown separately; below, a jagged particle trajectory (random walk) is drawn in a plane.

A 4 g sugar cube is dropped into 200 mL of tea at 80 °C. The sweetness is noticeable after about a minute even without stirring, while in the same volume of iced tea at 5 °C it takes minutes. Under a magnifier, the edge of a sugar grain blurs into the water long before the cube disappears.

Sugar molecules leave the crystal and cross the thin unstirred water layer about $0.1\ \text{mm}$ thick next to its surface by diffusion alone. With $D \approx 5\cdot10^{-10}\ \text{m}^2/\text{s}$ for sucrose in water, the random-walk relation $\langle x^2\rangle = 2Dt$ gives $t \approx x^2/(2D) = (10^{-4})^2/(2\cdot5\cdot10^{-10}) \approx 10\ \text{s}$ — exactly the timescale on which the taste appears. Heating the tea raises the mean kinetic energy of the water molecules and lowers viscosity, so $D$ grows and the same layer is crossed faster. Mixing the remaining 200 mL in the cup is done by convection when you stir: diffusion alone over 5 cm would need hours.

A lithium-ion phone battery stores lithium in graphite particles about $2\ \mu\text{m}$ in radius. Charging pushes lithium ions into those particles; a full charge takes about 1 hour at 20 °C and about 40 minutes at 35 °C, while below 0 °C the phone may refuse to fast-charge at all.

Inside a graphite particle there is no stirring: each lithium ion moves by random hops between sites, so the filling time of a particle is set by the random-walk relation $t \approx r^2/(2D)$. With $r = 2\ \mu\text{m}$ and $D \approx 10^{-14}\ \text{m}^2/\text{s}$ at room temperature, $t \approx (2\cdot10^{-6})^2/(2\cdot10^{-14}) \approx 200\ \text{s}$ — one of the steps that limits the charging current. Because $D$ grows with temperature, warming the cell speeds up the hops; the same relation explains why cold slows charging and why manufacturers limit the current.

Steel reinforcement in a road bridge lies about 50 mm below the concrete surface. In winter, salt from the road dissolves and forms chloride ions at the surface. With a chloride diffusion coefficient of about $10^{-12}\ \text{m}^2/\text{s}$ in sound concrete, the ions need roughly 40 years to reach the steel in noticeable concentration — which is why bridges are designed for a 50–100 year service life and are inspected for corrosion.

This is diffusion in a solid: chloride ions do not flow as a stream but perform a random walk through the pore water between cement grains. The penetration depth follows from $\langle x^2\rangle = 2Dt$, so $t = x^2/(2D) = (0.05)^2/(2\cdot10^{-12}) \approx 1.25\cdot10^9\ \text{s} \approx 40\ \text{years}$. Because depth grows only as $\sqrt{t}$, doubling the service life needs four times the time — thick concrete cover or waterproofing membranes, rather than a slightly thicker surface layer, are what actually delay corrosion.