# Learner lab record: Two-level ensemble population and heat-capacity map

Course: Statistical mechanics and ensembles

Name: ____________________  Date: ____________________  Group: ____________________

## Investigation question

How do temperature, energy gap, and degeneracy turn microscopic possibilities into macroscopic averages?

## Setup

Use the canonical ensemble laboratory. Hold degeneracies fixed while sweeping temperature, then change the excited-state multiplicity at one temperature.

## Variables

| Variable | Role | Unit |
| --- | --- | --- |
| Temperature | independent | K or declared scale |
| Energy gap | independent | eV or model energy |
| State degeneracies | controlled, then independent | count |
| Population, mean energy, entropy, heat capacity | dependent | %, energy, entropy, energy/K |

## Predict before changing controls

1. Predict the excited population as temperature approaches zero.

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2. Predict how added excited-state degeneracy competes with a fixed energy penalty.

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## Observation table

| temperature | gap | g0 | g1 | excited % | mean E | entropy | heat capacity |
| --- | --- | --- | --- | --- | --- | --- | --- |
|   |   |   |   |   |   |   |   |
|   |   |   |   |   |   |   |   |
|   |   |   |   |   |   |   |   |
|   |   |   |   |   |   |   |   |
|   |   |   |   |   |   |   |   |
|   |   |   |   |   |   |   |   |

## Analyze

1. Where is population competition strongest?

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2. Why does high degeneracy matter even at fixed energy?

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3. Which run shows a heat-capacity peak?

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4. What physical assumption justifies the canonical ensemble?

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## Evidence-bounded conclusion

At gap ___, raising temperature from ___ to ___ changed excited population from ___ to ___; degeneracy altered the balance because ___.

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