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A SMALL EXPERIMENT / Statistical thermodynamics

Thermal populations.

Heating a sample changes how its particles occupy energy states. See what the energy gap has to do with it.

02 / TWO ENERGY STATESEquilibrium model

Where the population goes.

State populations and the effect of temperaturePlots load when JavaScript is enabled.

Solid blue: lower state. Dashed brick red: upper state. Bars show the current populations; the curve tracks the upper-state fraction as temperature rises.

Turn up the temperature.

These are illustrative systems with different fixed energy gaps, not measurements of a particular compound.

The relationship

Heating increases the upper-state population. A larger gap suppresses it. Their balance is set by the dimensionless ratio ΔE/RT.

Lower state
—
Upper state
—
Gap / thermal energy
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What changes

Compare the energy gap with RT, the thermal energy scale per mole.

The model, explained.

N1N0=e−ΔE/(RT)\frac{N_1}{N_0} = e^{-\Delta E/(RT)}
p1=e−ΔE/(RT)1+e−ΔE/(RT)p_1 = \frac{e^{-\Delta E/(RT)}}{1+e^{-\Delta E/(RT)}}

ΔE is the molar energy gap, T is temperature in kelvin, and R is the molar gas constant. The two probabilities sum to one. Each level contains one state, so their degeneracies are equal.

This model describes an ensemble at thermal equilibrium with exactly two available states. It does not show transition rates or a reaction barrier. At high temperature the populations approach 50:50; heating alone does not invert them. Real systems can have additional states and unequal degeneracies.

Explore Boltzmann factors with the University of Minnesota