Describe the shape of a graph of the Boltzmann factor against energy at a fixed temperature.

It starts at one where the energy is zero and falls exponentially, dropping by the same fraction for each equal increase in energy.

How does a graph of Boltzmann factor against energy change when the temperature is raised?

The curve falls away more gradually, so at any given energy the factor is larger.

Which plot straightens Boltzmann factor data taken at one temperature, and what is its gradient?

The natural logarithm of the factor against energy gives a straight line of gradient −1/kT.

Describe the shape of a graph of the Boltzmann factor against temperature at a fixed energy.

It is almost zero at low temperatures, rises steeply once kT becomes comparable with the energy, then flattens as it approaches one.

Why does such a graph never rise above one?

The exponent is never positive, so the higher state can at most be as populated as the lower one.

OCR A-Level Physics B H557

Planning

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Implementing

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Analysis

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Matter: Very Simple

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Matter: Hot or Cold

Topic overview Energy, temperature and the ratio E/kT The Boltzmann factor and populations of energy states Graphs of the Boltzmann factor Activation processes and the effect of temperature

Electromagnetism

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Probing Deep Into Matter

Topic overview Particle accelerators and high-energy beams Scattering and the nuclear atom Paths of scattered particles and closest approach Discrete energy levels in atoms Electron standing waves and the confined electron Quarks, hadrons and leptons Conservation laws and nuclear equations Charged particles in detectors and magnetic fields Relativistic energy and the relativistic factor

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Topic overview The nature of ionising radiations Effects on living tissue and radiation safety Absorbed dose, effective dose and risk Binding energy and nuclear stability The Nuclear Valley and modes of decay Fission, fusion and nuclear power Activity, half-life and the decay constant Investigating absorption and half-life

Matter: Hot or Cold

Graphs of the Boltzmann factor

3 things to know
Components 01, 02 and 03

Overview

Recall questions on its shape against energy and against temperature, why it never exceeds one, which logarithmic plots straighten the data, their gradients and intercepts, and obtaining an activation energy.

What you need to know

Answer each one before you reveal it.

Your recallNot started

Key term

Why plot Boltzmann factors logarithmically, how do two activation energies differ on such a plot, and what does the e⁻¹ point show?

Key term

What energy increase divides the Boltzmann factor by e, how does higher temperature change its curve, which plot straightens it, and why never above one?

Key term

What increase in energy makes the Boltzmann factor fall to 1/e of its value?

Questions from this subtopic

Each one with the model answer underneath.

1
Describe the shape of a graph of the Boltzmann factor against energy at a fixed temperature.
Model answer
It starts at one where the energy is zero and falls exponentially, dropping by the same fraction for each equal increase in energy.
2
What increase in energy makes the Boltzmann factor fall to 1/e of its value?
Model answer
An increase of kT does.
3
How does a graph of Boltzmann factor against energy change when the temperature is raised?
Model answer
The curve falls away more gradually, so at any given energy the factor is larger.
4
Which plot straightens Boltzmann factor data taken at one temperature, and what is its gradient?
Model answer
The natural logarithm of the factor against energy gives a straight line of gradient −1/kT.
5
Describe the shape of a graph of the Boltzmann factor against temperature at a fixed energy.
Model answer
It is almost zero at low temperatures, rises steeply once kT becomes comparable with the energy, then flattens as it approaches one.
6
Why does such a graph never rise above one?
Model answer
The exponent is never positive, so the higher state can at most be as populated as the lower one.

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