Physics · Molecular physics and thermodynamics · Grade 9

Brownian motion and diffusion

Why does a particle under the microscope jitter endlessly, and how does this tell us the size of atoms?

The situation

A drop of ink in water spreads slowly but without stopping; the scent of perfume opened in one corner of a room is soon noticed in the other. In 1827 the botanist Robert Brown saw under a microscope that tiny particles released from pollen grains in water perform a ceaseless, random “dance”. These particles are pushed from every side by invisible water molecules. Do molecules really exist, and how small are they? How can we find the Avogadro constant — the number of molecules in 1 mole — by measuring the random path of a particle?

Topic

Brownian motion and diffusion. Particles under a microscope, ink dropped into water and an oil film

Key ideas

Brownian motion
Diffusion
Avogadro constant

Formulas

⟨ r^{2} ⟩ = 4 D t

The mean square displacement of a Brownian particle is proportional to time (in a plane; along each axis $⟨ x^{2} ⟩ = 2 D t$ ).

D = \frac{k T}{6 π η a} = \frac{R T}{6 π η a N_{A}}

The Einstein–Stokes formula: $a$ is the radius of the particle and $η$ is the viscosity of water.

J = - D \frac{d c}{d x}

Fick's law: a substance flows in the direction in which its concentration decreases.

d = \frac{V}{S}

The oil film is one molecule thick: the volume of oil in the drop / the area of the film.

U (r) = 4 ε [{(\frac{σ}{r})}^{12} - {(\frac{σ}{r})}^{6}], ν = \frac{N}{N_{A}} = \frac{m}{M}

The interaction energy of molecules (Lennard-Jones) and the amount of substance.

Theory

Molecular-kinetic theory

The three principles of the molecular-kinetic theory: 1) matter consists of particles — atoms and molecules; 2) they move continuously and at random; 3) they attract and repel one another. Experimental evidence: Brownian motion, diffusion, gas pressure, the change in volume when substances are mixed. The idea of atoms was voiced as early as ancient Greece, but their existence was proved at the start of the 20th century by Einstein's theory and Perrin's measurements. In the classical model a molecule is a small hard sphere; in quantum physics, however, the energy of a molecule is quantised (electronic, vibrational and rotational levels show up in spectra), and chemical bonding and valency are quantum in nature too.

Brownian motion and Perrin's experiment

The molecules around a particle hit it about $10^{20}$ times a second; the blows are never perfectly balanced, so the particle makes a random walk. You will see this in the magnified cube (not to scale: in reality a molecule is a thousand times smaller than the particle). The displacement is proportional not to time but to $\sqrt{t}$ : $⟨ r^{2} ⟩ = 4 D t$ gives a straight line on the graph, while the $Δ x$ values follow a Gaussian distribution. Einstein (1905) related $D$ to $N_{A}$ ; Perrin (1908) marked the positions of particles at equal time intervals and found $N_{A}$ — you will do exactly the same. The particle itself also has an average kinetic energy of $\frac{3}{2} k T$ (the equipartition of energy among the degrees of freedom). The probability density obeys $\partial P / \partial t = D \nabla^{2} P$ ; with an external force we have the Smoluchowski (Fokker–Planck) equation, whose equilibrium solution is the Boltzmann distribution (detailed balance), and the mobility is $μ = D / k T$ .

Diffusion and Fick's law

Diffusion is the spontaneous mixing of substances caused by thermal motion. Fick's first law $J = - D d c / d x$ : the flux is proportional to the concentration gradient. If the source and the sink are constant, we have steady-state diffusion (the profile is linear). When a single drop spreads, it is non-steady: $\partial c / \partial t = D \partial^{2} c / \partial x^{2}$ , the spot widens in a Gaussian shape, $σ^{2} = σ_{0}^{2} + 2 D t$ . In a gel there is no convection — only diffusion remains, and it is very slow: several hours for 1 cm (time is sped up 1800 times). It is faster when hot: $D \propto T / η$ . Two substances penetrating each other is mutual diffusion; a temperature difference producing a concentration difference is thermal diffusion. In a plasma electrons and ions are linked by the electric field and diffuse together (ambipolar diffusion). In solids atoms jump through vacancies: $D$ is very small and rises sharply with temperature — in the “Crystalline and amorphous solids” experiment.

Molecule size: the oil film

An oleic acid molecule sticks to the water by its “head”, with its “tail” pointing up: a film one molecule thick forms on the water surface. A very dilute solution in alcohol is used: the alcohol dissolves in the water, the oil spreads out and pushes the powder to the edges. The volume of oil is $V = c \cdot n \cdot V_{d r o p}$ , the area of the film is $S = π D^{2} / 4$ ( $D$ is the mean of two measured diameters), and the thickness is $d = V / S \approx 10^{- 9}$ m. From this the volume of a molecule is ≈ $d^{3}$ and its mass is $m_{0} = M / N_{A}$ . For a water molecule $m_{0} = 0.018 / 6.02 \cdot 10^{23} \approx 3 \cdot 10^{- 26}$ kg and its size is ≈ 0.3 nm.

Amount of substance and Avogadro's law

Amount of substance $ν = N / N_{A} = m / M$ , in mol. The Avogadro constant is $N_{A} = 6.022 \cdot 10^{23}$ mol⁻¹. The relative molecular mass $M_{r}$ is the ratio of the mass of a molecule to 1/12 of the mass of a carbon-12 atom; the molar mass is $M = M_{r} \cdot 10^{- 3}$ kg/mol (water: $M_{r} = 18$ ). Avogadro's law: equal volumes of gases at the same pressure and temperature contain equal numbers of molecules; at standard conditions 1 mol of gas occupies ≈ 22.4 L. The basic equation of the kinetic theory of an ideal gas is $p = \frac{2}{3} n ⟨ E_{k} ⟩$ , and each degree of freedom has energy $\frac{1}{2} k T$ — in the “Gas under a piston” experiment.

Interaction and the three states of matter

At short distances the electron shells of molecules repel each other, and farther apart the van der Waals attraction dominates. In the Lennard-Jones potential the lowest energy $- ε$ is at the distance $r_{0} = 2^{1 / 6} σ$ . The state depends on the ratio of the thermal energy $k T$ to the well depth $ε$ : $k T ≪ ε$ — atoms vibrate in place (solid, crystal lattice); $k T \sim ε$ — short-range order remains but atoms swap places (liquid: keeps its volume but not its shape); $k T ≫ ε$ — atoms fly freely (gas). The model: 156 argon atoms in a plane, weak gravity and a thermostat. The transition temperatures of the 2D model differ from those of real argon (melting 84 K, boiling 87 K).

Did you know?

Jean Perrin confirmed Einstein's theory and determined the Avogadro constant by measuring Brownian motion; for this work he received the Nobel Prize in Physics in 1926.
In the 1770s Benjamin Franklin observed that one teaspoon of oil spread over about half an acre (≈ 2000 m²) of a pond. From this it follows that the film is only 1–2 nm thick.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

More experiments in this section

Tandoor and sandal: heat — Grade 7
Gas under a piston — Grade 9
Humidity and dew — Grade 9
Heat engine: the Otto cycle — Grade 9
Entropy and the arrow of time — Grade 9
Surface tension and capillarity — Grade 9
Crystalline and amorphous solids — Grade 9
Phase transitions — Grade 10
Thermal expansion — Grade 10
Refrigerator and heat pump — Grade 10