Two nondegenerate levels have energies 0 and ε with exp(−βε)=0.25. Find Z.
Answer and explanation
1.25
Add weights 1+0.25.
Subjects
Statistical mechanics
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Sample exercises
For a nondegenerate two-level canonical model with energies 0 and ε and exp(−βε)=0.25, find the excited-state probability.
Answer and explanation
0.2
Divide the excited weight by total weight: 0.25/1.25.
What extra factor enters a level's weight when it has degeneracy g?
- g² always
- g
- No factor in any case
- 1/g always
Answer and explanation
g
There are g microstates at that energy, each contributing a Boltzmann weight.
A ground level has degeneracy 2 and excited level degeneracy 3. With excited weight per microstate 0.5, find Z.
Answer and explanation
3.5
Z=2×1+3×0.5.
What quantity is −∂ln Z/∂β for a temperature-independent spectrum?
- Mean position always
- Volume always
- Mean energy
- Zero in all ensembles
Answer and explanation
Mean energy
Differentiating the canonical partition sum produces the weighted energy mean.
Why should probabilities be normalized after assigning Boltzmann factors?
Answer and explanation
Individual weights need not sum to one. Dividing by Z ensures a valid probability distribution across all included microstates.
Credit Z as the normalization sum.
Study notes
Canonical ensembles and partition functions
For a system at inverse temperature β=1/(kBT), the canonical probability of a microstate of energy E is exp(−βE)/Z.
Advanced universityCanonical ensembles and partition functions · revision notes
For a system at inverse temperature β=1/(kBT), the canonical probability of a microstate of energy E is exp(−βE)/Z.
Advanced universityPartition function of a two-level system
Ground energy is zero with degeneracy 1; the excited level has degeneracy g. Energies are measured from the ground state.
Advanced universityPartition function of a two-level system · formula reference
Ground energy is zero with degeneracy 1; the excited level has degeneracy g. Energies are measured from the ground state.
Advanced universityExcited-state population with degeneracy
The probability refers to the whole excited level, including all its degenerate states.
Advanced universityExcited-state population with degeneracy · formula reference
The probability refers to the whole excited level, including all its degenerate states.
Advanced universityEntropy difference from multiplicity ratio
Use the microcanonical equal-a-priori-probability model and a ratio of multiplicities.
Advanced universityEntropy difference from multiplicity ratio · formula reference
Use the microcanonical equal-a-priori-probability model and a ratio of multiplicities.
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