Lecture 21: Distributions | Chemical Thinking

The Hook

In a gas at room temperature, molecules move at different speeds:

Some are slow (~100 m/s)
Most are moderate (~400 m/s for N₂)
Some are very fast (~1000 m/s)
Almost none exceed 2000 m/s

Question: What determines this spread? Why this particular shape?

Answer: The Boltzmann distribution — nature's way of spreading energy across states.

The Boltzmann Distribution

At thermal equilibrium, the probability of a system being in state i with energy E_i is:

$$P_i = \frac{g_i e^{-E_i/k_BT}}{Z}$$

where Z = Σ gᵢ e^-Eᵢ/k_BT is the partition function

The Boltzmann Factor

The heart of the distribution: e^-E/k_BT

Low energy states: e^-E/kT ≈ 1 (high probability)
High energy states: e^-E/kT → 0 (low probability)
Higher T: flatter distribution (more states accessible)

The Boltzmann distribution is the most probable distribution — it maximizes entropy subject to constraints (fixed average energy).

Interactive: Two-Level System

Population vs Temperature: Ground (E=0) and Excited (E=ε) States

P₀ (ground)

P₁ (excited)

P₁/P₀

T where k_BT = ε

Energy gap ε (cm⁻¹): 500

Temperature (K): 300

P₀ (ground)

P₁ (excited)

k_BT = ε

Population Ratio

$$\frac{P_j}{P_i} = e^{-(E_j - E_i)/k_BT}$$

The ratio depends only on the energy difference.

The Maxwell-Boltzmann Speed Distribution

$$f(v) = 4\pi \left(\frac{m}{2\pi k_BT}\right)^{3/2} v^2 e^{-mv^2/(2k_BT)}$$

Molecular Speed Distribution

v_mp (m/s)

⟨v⟩ (m/s)

v_rms (m/s)

Temperature (K): 300

Molecule:

v_mp = √(2RT/M)

⟨v⟩ = √(8RT/πM)

v_rms = √(3RT/M)

Why the v² factor? In 3D, there are more ways to have high speed than low — the number of velocity states with speed v is proportional to the surface area of a sphere: 4πv². This competes with the Boltzmann factor e^-mv²/2kT.

Speed	Formula	Meaning
v_mp	√(2k_BT/m)	Peak of distribution
⟨v⟩	√(8k_BT/πm)	Average speed
v_rms	√(3k_BT/m)	√⟨v²⟩, relates to kinetic energy

Order: v_mp < ⟨v⟩ < v_rms (always!)

The Gaussian (Normal) Distribution

$$f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-(x-\mu)^2/(2\sigma^2)}$$

Gaussian Distribution with 68-95-99.7 Rule

Mean μ: 0

Std dev σ: 1

The 68-95-99.7 Rule

Range	Probability
μ ± 1σ	68.3%
μ ± 2σ	95.4%
μ ± 3σ	99.7%

Central Limit Theorem: The sum of many independent random variables approaches a Gaussian, regardless of the original distributions. This is why Gaussians appear everywhere — measurement errors, thermal fluctuations, diffusion.

Rotational Population Distribution

$$P_J \propto (2J+1) e^{-BJ(J+1)/k_BT}$$

The degeneracy factor (2J+1) causes the peak at J > 0!

Rotational State Populations

J_mp

k_BT (cm⁻¹)

Rotational constant B (cm⁻¹): 2

Temperature (K): 300

Why J_mp > 0?

The degeneracy (2J+1) increases with J, but the Boltzmann factor decreases. These compete:

Low J: low degeneracy but high Boltzmann factor
High J: high degeneracy but low Boltzmann factor
Peak: where they balance

$$J_{mp} \approx \sqrt{\frac{k_BT}{2B}} - \frac{1}{2}$$

Chemistry Applications

Arrhenius Equation

$$k = A e^{-E_a/RT}$$

This is a Boltzmann factor! Only molecules with E ≥ E_a can react.

Spectral Line Intensities

Absorption intensity from level i to j depends on:

$$I_{i \to j} \propto N_i \cdot |\mu_{ij}|^2$$

Population N_i follows the Boltzmann distribution!

Doppler Broadening

Spectral lines are broadened due to molecular motion:

$$I(\nu) \propto e^{-mc^2(\nu - \nu_0)^2/(2k_BT\nu_0^2)}$$

A Gaussian shape with width ∝ √T

Summary: Key Distributions

Distribution	Domain	Key Parameter	Use
Boltzmann	Discrete states	k_BT	Energy state populations
Gaussian	Continuous, unbounded	σ (width)	Errors, fluctuations
Maxwell-Boltzmann	v ≥ 0	T and m	Molecular speeds
Poisson	Non-negative integers	λ (rate)	Rare events
Exponential	t ≥ 0	λ (rate)	Waiting times