Questions: Partition Function: Definition and Properties
3 questions to test your understanding
Score: 0 / 3
Question 1 Multiple Choice
For a two-level system with energies 0 and ε, the partition function is Z = 1 + exp(−ε/kT). As temperature T → ∞, what value does Z approach?
A0
B1
C2
D∞
As T → ∞, ε/kT → 0, so exp(−ε/kT) → exp(0) = 1, giving Z → 1 + 1 = 2. Physically, at very high temperature, both states become equally probable (Boltzmann factors equalize) and Z counts the number of accessible microstates. This illustrates that Z is not constant — it depends on temperature.
Question 2 True / False
The partition function Z is just a normalization constant that ensures probabilities sum to one; once the probabilities p_i = exp(−E_i/kT)/Z are known, Z itself has no further physical content.
TTrue
FFalse
Answer: False
Z is the generating function for all equilibrium thermodynamics, not merely a normalization constant. Free energy F = −kT ln Z; average energy U = −∂ln Z/∂β; entropy S = k(ln Z + β ∂ln Z/∂β); pressure P = kT(∂ln Z/∂V). Every macroscopic equilibrium observable is a derivative of ln Z. The probabilities p_i tell you about individual microstates; Z tells you about the macroscopic system.
Question 3 Short Answer
A system has two configurations: one with 1 microstate at energy E, and another with 100 microstates at energy E + δ (slightly higher). At high temperature, which configuration dominates, and why does the partition function capture this?
Think about your answer, then reveal below.
Model answer: At high temperature the 100-microstate configuration dominates because the Boltzmann penalty exp(−δ/kT) ≈ 1 becomes negligible, so entropy (more states) wins. Z = exp(−E/kT) + 100·exp(−(E+δ)/kT) shows the second term dominates when kT ≫ δ, correctly weighting both energy and degeneracy.
This illustrates that Z naturally encodes the competition between energy minimization and entropy maximization that underlies the free energy F = U − TS. The sum in Z weights each state by its Boltzmann factor; highly degenerate energy levels contribute many terms. This is why Z — not just the ground state energy — determines equilibrium behavior.