How is the partition function Z defined for a discrete set of microstates?

• Sum the Boltzmann factors over all microstates
• Multiply the Boltzmann factors over all microstates
• Sum the microstate energies without their factors

Two levels each contain one state. Their fixed energy gap is D>0, and they are in thermal equilibrium at T>0. What is upper population divided by lower population?

• exp(D/(k_B T))
• exp(-D/(k_B T))
• exp(-k_B T/D)

For two nondegenerate levels with fixed gap D>0 in thermal equilibrium, nearly all particles occupy the lower level when k_B T is much smaller than D.

Circle one:   True   False

Two fixed levels each contain one state. In thermal equilibrium, k_B T becomes much larger than their positive gap. How do their populations compare?

• Nearly all remain in the lower state
• Nearly equal in the two states
• Nearly all move into the upper state

A system has partition function Z=5. One microstate has Boltzmann factor 1. What is that state's probability?

• 5/1
• 4/5
• 1/5

Two microstates have energies 0 and 3 energy units. Their Boltzmann factors are 1 and 1/2, respectively. What is the mean energy?

• 1 energy unit
• 1.5 energy units
• 2 energy units

Two allowed states have positive weights. Rafi uses only the lower state's weight as Z and reports its full-system probability as 1. What must he correct?

• Replace the sum with a product of the two weights
• Include both weights in the probability denominator
• Replace the lower weight with the energy difference

For two nondegenerate levels with a fixed positive gap D, why does the upper/lower population ratio approach 1 as T grows?

• D/(k_B T) approaches zero, so exp(-D/(k_B T)) approaches 0
• D/(k_B T) approaches infinity, so exp(-D/(k_B T)) approaches 1
• D/(k_B T) approaches zero, so exp(-D/(k_B T)) approaches 1