The Boltzmann Factor and the Partition Function · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Weighing states: Boltzmann factor and partition function

Science · Thermal & Statistical Physics · ages 20-22
Name ______________________   Date ____________

  1. What is the partition function Z?

    • The sum of Boltzmann factors over all states
    • The product of all energies
    • The single largest factor

  2. A two-level system has an energy gap between lower and upper. What is the population ratio of upper to lower?

    • Gap times kT
    • exp(-gap/kT)
    • 1 minus gap over kT

  3. At very low temperature, nearly all members of a two-level system sit in the lower state.

    Circle one:   True   False

  4. Temperature climbs very high above the gap. How do the populations settle?

    • All below
    • Nearly even
    • All above

  5. A system has Z equal to 4, and one state carries factor 1. What is that state's probability?

    • 4
    • 1
    • 1/4

  6. Two states: E equal to 0 with factor 3, and E equal to 2 with factor 1. What is the mean energy?

    • 0.5
    • 1
    • 2

  7. A student computes Z from the lower state alone, then reports odds. What is the error?

    • One state suffices for Z
    • Z must sum every state, or odds and means skew
    • Z ignores the upper state by rule

  8. Why does heating level the populations instead of emptying the upper state?

    • Heat destroys the states
    • Gaps shrink with temperature
    • kT grows against fixed gaps, so exponentials approach one
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Answer key

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Weighing states: Boltzmann factor and partition function W1-mt_u9MSoTgJkF-s1

  1. The sum of Boltzmann factors over all states · Z is the bookkeeping total from which odds and averages flow.
  2. exp(-gap/kT) · Upper weight over lower weight is the exponential of minus gap over kT.
  3. True · Cold makes the upper weight vanish next to the lower one.
  4. Nearly even · Heat levels the field, so both states fill nearly equally.
  5. 1/4 · Odds are factor over Z, so 1 over 4.
  6. 0.5 · Weight each energy by its odds: (0 times 3 plus 2 times 1) over 4 is 0.5.
  7. Z must sum every state, or odds and means skew · Z is the total over all states. Skipping states inflates every odd.
  8. kT grows against fixed gaps, so exponentials approach one · Fixed gaps matter less as kT grows, evening the weights.
Worksheet · LightMySky