Counting states to get chemistry

Science · Chemistry · ages 20-21

Name ______________________ Date ____________

A two-level molecule has ground energy zero and an excited Boltzmann factor of 0.3. What is q?
Answer: ______________
What does the molecular partition function sum over?
- Only the single ground level of the molecule
- Every level, weighted by its Boltzmann factor times degeneracy
- Only the levels measured in one experiment
How do you sum the vibrational ladder of equally spaced levels?
- As a geometric series giving 1 over 1 minus the factor
- By counting only the ground level and ignoring the rest
- By adding the spacings instead of the factors
Why may you multiply translational, rotational, vibrational, and electronic parts?
- Because temperature affects every motion identically
- Because the motions are nearly independent of each other
- Because degeneracy is always exactly one
Temperature drops near zero. What happens to q?
- It explodes toward infinity as levels multiply
- It turns negative since energies invert
- It sinks toward 1 as only the ground level counts
A vibrational factor e to minus spacing over kT equals 0.2. Using q equal to 1 over 1 minus the factor, what is q?
Answer: ______________
At very high temperature, only the ground level contributes to q.
Circle one: True False
How does q connect to bulk free energy?
- The logarithm of q times thermal energy gives the free energy
- q equals the free energy directly with no further math
- Free energy adds q values together without logarithms

Answer key

For grown-ups. Fold this page away before handing over the rest.

1.3 · Add the ground term of one to the excited factor. Two levels means two terms.
Every level, weighted by its Boltzmann factor times degeneracy · Every state contributes its thermal weight. Accessible states dominate and remote ones fade.
As a geometric series giving 1 over 1 minus the factor · Equal spacing makes each term a fixed multiple of the last. The infinite series closes neatly.
Because the motions are nearly independent of each other · Independence lets the sum split into a product. Each motion contributes its own factor.
It sinks toward 1 as only the ground level counts · Cold closes every door but the ground floor. One surviving term means q near one.
1.25 · Subtract the factor from one, then divide one by the result. The closed form covers all levels.
False · Heat opens every door: countless levels contribute. The ground only rule belongs to the cold limit.
The logarithm of q times thermal energy gives the free energy · The log route turns the microscopic sum into the working energy. That bridge is the payoff of the whole sum.