Physics · Molecular physics and thermodynamics · Grade 10

Maxwell–Boltzmann distribution

How do random collisions give rise to an exact law — the Gaussian curve?

The situation

Every molecule of air collides billions of times a second, and its speed cannot be predicted in advance. But trillions of molecules together obey a strict law: their speeds are distributed along the Maxwell curve, and with height the air thins out exponentially — on the summit of Big Chimgan (3309 m) the air pressure is about a third lower than at sea level. On a Galton board, too, each ball chooses its path at random, yet thousands of balls form a beautiful bell-shaped curve. How does an exact law arise out of chance?

Topic

Maxwell and Boltzmann distributions. Galton board and thin air in the mountains: order from chance

Key ideas

Gaussian and Maxwell distributions
Barometric formula
Degrees of freedom

Formulas

P_{n} (k) = C_{n}^{k} p^{k} q^{n - k}, ⟨ k ⟩ = n p, σ^{2} = n p q

Binomial distribution: $k$ times “to the right” in $n$ independent trials. For large $n$ it turns into the Gaussian curve, for small $p$ into the Poisson distribution ( $λ = n p$ ).

f (v) = 4 π {(\frac{m_{0}}{2 π k T})}^{3 / 2} v^{2} e^{- m_{0} v^{2} / 2 k T}

Maxwell distribution: $f (v) d v$ is the fraction of molecules with speeds between $v$ and $v + d v$ .

v_{e} = \sqrt{\frac{2 k T}{m_{0}}}, ⟨ v ⟩ = \sqrt{\frac{8 k T}{π m_{0}}}, v_{k v} = \sqrt{\frac{3 k T}{m_{0}}}

The most probable, mean and root-mean-square speeds: $1 : 1.128 : 1.225$ .

n (h) = n_{0} e^{- m_{0} g h / k T}, p (h) = p_{0} e^{- M g h / R T}

The Boltzmann distribution and the barometric formula (at constant temperature).

⟨ ε ⟩ = \frac{i}{2} k T, C_{V} = \frac{i}{2} R

Equipartition of energy among the degrees of freedom: $i$ is the number of “active” degrees of freedom.

Theory

Chance and law: the statistical method

A macroscopic system can be studied in three ways: dynamically (Newton’s equations for every particle — practically impossible for 10²³ particles), statistically (probabilities and mean values) and thermodynamically (ignoring molecules, using only macroscopic laws). On a Galton board, at each peg the ball veers to the right or to the left with probability $p$ and $q = 1 - p$ ; the decisions are statistically independent. The result is the binomial distribution, and for large $n$ the Gaussian curve (the sum of many independent random events always looks like this — the central limit theorem). The width of the distribution is $σ = \sqrt{n p q}$ , the relative fluctuation is $σ / ⟨ k ⟩ \propto 1 / \sqrt{n}$ ; the error of the mean value measured over $N$ balls is also $\propto 1 / \sqrt{N}$ — see it on the graph. If $p$ is very small (a rare event — for example, the decay of nuclei in 1 s), the distribution turns into the Poisson law.

Maxwell distribution

Each velocity component ( $v_{x}$ , $v_{y}$ , $v_{z}$ ) is independently Gaussian-distributed. For the magnitude of the velocity, the volume of a spherical shell $4 π v^{2} d v$ is added, and the Maxwell curve results: there are few slow molecules (there is little “room” for them), and very fast ones are also few (the exponential). When the temperature rises or the molecule becomes lighter, the peak shifts to the right and gets lower (the area under the curve is always 1). In the relative speed $u = v / v_{e}$ the distribution is the same for all gases and temperatures: $f (u) = \frac{4}{\sqrt{π}} u^{2} e^{- u^{2}}$ . The basic equation of the molecular-kinetic theory, $p = \frac{1}{3} n m_{0} ⟨ v^{2} ⟩ = n k T$ , links the mean square speed to the pressure. Model: 1000 “representative” molecules with random pairwise collisions (DSMC); the motion on the screen is slowed down many thousands of times.

Stern's experiment

In 1920 Otto Stern heated a silver-coated platinum wire in a vacuum. The evaporated silver atoms leave through the slit in the inner cylinder, strike the outer cylinder and leave a mark. If the apparatus is at rest, the mark is opposite the slit. If the apparatus rotates with angular velocity $ω$ , then while the atom is in flight ( $t = (R - r) / v$ ) the cylinder has time to turn, and the mark is displaced by $s = ω R (R - r) / v$ — from this $v$ is found. The smearing of the mark shows that the speeds of the atoms differ. Later, precise experiments with rotating toothed discs (a velocity selector; for example Miller and Kusch, 1955) confirmed the Maxwell distribution with high accuracy. The flux of atoms in the beam is proportional to $v^{3} e^{- m_{0} v^{2} / 2 k T}$ — fast atoms pass through the slit more often.

The Boltzmann distribution and the barometric formula

In a force field the concentration of particles depends on the potential energy: $n \propto e^{- E_{p} / k T}$ (the Boltzmann distribution). In a gravitational field $E_{p} = m_{0} g h$ : the air density falls by a factor of $e$ every $H = k T / (m_{0} g) \approx 8$ km. A heavy gas (xenon) collects at the bottom, light ones (hydrogen, helium) rise high; when it is heated, the column “stretches”. In the real atmosphere the temperature falls with height and the wind mixes the gases, but the formula is a good approximation. Speed and position together give the Maxwell–Boltzmann distribution $\propto e^{- (E_{k} + E_{p}) / k T}$ . In a rotating system the centrifugal potential is $- m_{0} ω^{2} r^{2} / 2$ : heavy molecules gather at the wall — this is how gas centrifuges separate uranium isotopes. Each layer in the column exchanges particles (the grand canonical ensemble): in equilibrium the chemical potential $μ = k T \ln n + m_{0} g h$ is the same everywhere — this also gives the barometric formula.

The Gibbs distribution and ensembles

In an isolated system (energy exactly fixed — the microcanonical ensemble) all possible microstates are equally probable. A small part of it — for example one molecule — exchanges heat with the rest, and its energy obeys the Gibbs canonical distribution: $w_{i} \propto e^{- E_{i} / k T}$ . The normalising partition function $Z = \sum_{i} e^{- E_{i} / k T}$ gives all of thermodynamics: the free energy is $F = - k T \ln Z$ . In the classical (quasi-classical) limit the sum becomes an integral over phase space — the Maxwell and Boltzmann distributions follow from this; for quantum systems the ensemble is described by a density matrix. The fundamental (ergodic) hypothesis of statistical physics: the time average for one system equals the average over many copies — the ensemble average. On a Galton board, both the fraction of “rights” among the 12 decisions of a single ball and the fraction over many balls tend to $p$ .

Degrees of freedom and heat capacity

A monatomic molecule has 3 translational degrees of freedom: $⟨ ε ⟩ = \frac{3}{2} k T$ , $C_{V} = \frac{3}{2} R$ . A diatomic molecule also rotates about 2 axes: $C_{V} = \frac{5}{2} R$ . Vibration gives two quadratic terms — kinetic and potential energy (for a harmonic oscillator, the virial theorem: their averages are equal), hence $k T$ and $C_{V} = \frac{7}{2} R$ . But according to quantum mechanics energy comes in steps: if $k T$ is smaller than the step, the degree of freedom “freezes out”. The rotation of hydrogen freezes below about 100 K, the vibration of nitrogen is frozen even at room temperature ( $θ_{vib} = 3374$ K), while the vibration of chlorine is partly active. Laboratory: heat the gas at constant volume with an amount of heat $Q$ , and when the thermometer has settled measure $Δ T$ : $C_{V} = Q / (ν Δ T)$ . The thermometer reading “trembles” slightly: among the 2000 representative molecules, energy is randomly redistributed between translational and rotational motion. This fluctuation obeys the Gaussian law and its relative size is $\propto 1 / \sqrt{N}$ — in a real gas of 10²³ molecules it is imperceptible. Model: ν = 1 mol, V = 27 L; for rotation, a formula fitted to the measured curve of hydrogen; for vibration, an Einstein oscillator; energy exchange — pairwise collisions in the Larsen–Borgnakke manner.

Did you know?

Otto Stern was awarded the Nobel Prize in Physics in 1943 — for developing the molecular beam method and discovering the magnetic moment of the proton.
The composition of air (about 78 % nitrogen and 21 % oxygen) hardly changes up to a height of 100 km: wind and convection keep mixing the gases, and the heavier gases do not collect separately at the bottom.

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.

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Crystalline and amorphous solids — Grade 9
Phase transitions — Grade 10
Thermal expansion — Grade 10
Refrigerator and heat pump — Grade 10
Real gases — Grade 10
Transport phenomena — Year 1
Quantum statistics — Year 4
Phase transition and fluctuations — Year 4