CHM-14

Boltzmann factor

n_i/n_j = exp(−ΔE / kT). Population of two levels.

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StatisticalBoltzmann

Governing equation

\dfrac{n_i}{n_j}=\exp(-\Delta E/kT)

$T \in \left[10, 3000\right]$

where

\Delta E: Level gap (eV)
T: Temperature (K)
n_i/n_j: Population ratio (—)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

\dfrac{n_i}{n_j}=\exp(-\Delta E/kT)

$T \in \left[10, 3000\right]$

$T = 10$ $n_i/n_j = 4\times 10^{-51}$

Along the curve

②

Domain of the variable — blue

T_{\min} = 10,\quad T_{\max} = 3000

$T \in \left[10, 3000\right]$

$T = 10$ $T = 10$

Along the curve

③

Response curve — green

\dfrac{n_i}{n_j}=\exp(-\Delta E/kT)

$T \in \left[10, 3000\right]$

$T = 10$ $n_i/n_j = 4\times 10^{-51}$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}n_i/n_j}{\mathrm{d}T}

$T \in \left[10, 3000\right]$

$T = 28.81$ $n_i/n_j^{\prime} = 6.91\times 10^{-13}$

Along the curve

Lecture brief

Historical brief

Ideal-gas law, van ’t Hoff, Nernst, Michaelis–Menten and Clausius–Clapeyron are physical chemistry’s working equations of equilibrium and rate. The lab is pressure, potential and kinetics. This sheet (CHM-14 — Boltzmann factor) is the form associated with Boltzmann. Working symbols: $\Delta E$ , $T$ $\rightarrow$ $n_i/n_j$ . The canonical probability of a state is ∝ e^{−E/kT}. Degeneracy g would multiply the factor.

Purpose

Purpose: compute

n_i/n_j

from

\Delta E

T

in Physical chemistry via

\dfrac{n_i}{n_j}=\exp(-\Delta E/kT)

n_i/n_j = exp(−ΔE / kT). Population of two levels. Use it when a real physical chemistry question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

\Delta E = 0.100\,\mathrm{eV}

T = 300.000\,\mathrm{K}

, the governing relation

\dfrac{n_i}{n_j}=\exp(-\Delta E/kT)

yields

n_i/n_j = 0.0209\,\mathrm{—}

. Two non-degenerate levels, ΔE in eV. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Level gap \Delta E0.1 eVTemperature T300 K

Outputs

Population ratio n_i/n_j0.0209 —

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Narration of this film

Two non-degenerate levels, ΔE in eV.

The canonical probability of a state is ∝ e^{−E/kT}. Degeneracy g would multiply the factor.

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