Partition functions: from molecular levels to thermodynamics
Learn translational, rotational and vibrational partition functions, how they give U, A and S, with two solved practice questions and a study plan.
This post shows how one sum over molecular energy levels gives you energy, entropy and free energy.
You will learn the three molecular partition functions, two solved practice questions and a short study plan.
What the partition function measures
A molecule can sit in many energy levels. At temperature T, lower levels are more crowded than higher ones. The weight of a level of energy ε is the Boltzmann factor, e−ε/kT. Here k is the Boltzmann constant.
The partition function adds these weights over all levels. A common way to write it for a system is Z = Σ e−β × energy, where β stands for 1/kT. If a level has degeneracy g (g states at the same energy), count it g times. The result is dimensionless, a pure number.
Read its size like this. If q is close to 1, almost every molecule is in the lowest level. If q is large, many levels are occupied. So q measures the "accessible states".
The fraction of molecules in a level i is g e−ε/kT divided by q. Remember this ratio. Many exam questions are only this ratio with different numbers.
There are two related symbols. Small q is for one molecule. Capital Q (or Z) is for the whole system, which exchanges heat with its surroundings at fixed temperature, volume and number of particles. This is called the canonical ensemble. For N identical, non-interacting molecules that move freely (a gas), Q = qN/N!. The N! corrects for the fact that identical molecules cannot be told apart. For localised particles, as in a crystal, there is no N!.
Translational, rotational and vibrational functions
Different motions have different energy scales. So the total q splits into a product, and each factor is evaluated separately.
| Motion | High-temperature form | Depends on | Quick check |
|---|---|---|---|
| Translation (3-D box) | qtrans = (2πmkT/h2)3/2 V | mass, T, volume | Grows as T3/2 and as V |
| Rotation, linear molecule | qrot = T/(σΘrot) | T, moment of inertia, symmetry | σ = 2 for homonuclear, 1 for heteronuclear |
| Vibration, one mode | qvib = 1/(1 − e−Θvib/T) | T, vibrational frequency | Tends to 1 when T is far below Θvib |
Θrot and Θvib are characteristic temperatures. They convert a level spacing into a temperature, so you can compare it directly with T. Θrot = h2/(8π2 × I × k), where I is the moment of inertia. Θvib = hν/k, where ν is the vibrational frequency.
Translation. The levels are so close that the sum becomes an integral. That is why the formula is a smooth T3/2 form. A heavier molecule or a larger volume gives a larger qtrans. Translation is the largest factor, so it dominates the entropy of a gas.
Rotation. The formula above is valid only when T is much larger than Θrot. This holds at room temperature for most molecules. The symmetry number σ is the number of indistinguishable orientations you get by rotating the molecule. Do not forget it. Leaving σ out for a homonuclear diatomic is a very common error.
Vibration. Here the spacing is large, so you must sum the series, not integrate. Measure energy from the lowest vibrational level, so the zero-point energy is taken as zero. Then the sum is a geometric series and gives the form in the table. If a question measures energy from the bottom of the potential well, an extra factor e−Θvib/2T appears. Check which convention the question uses.
Electronic. At ordinary temperatures only the ground state counts. So qelec is usually just its degeneracy, often 1.
From q to thermodynamic properties
Once you have ln Q, everything follows by differentiation. Learn these relations:
- Internal energy: U = kT2 (∂ ln Q/∂T)V
- Free energy (at constant T and V): A = −kT ln Q
- Entropy: S = U/T + k ln Q
- Pressure: p = kT (∂ ln Q/∂V)T
Because the log of a product is a sum, ln q = ln qtrans + ln qrot + ln qvib. So the internal energy, free energy and entropy also split into additive translational, rotational and vibrational parts. This is why you can treat each motion on its own.
Two quick results save time. Since qrot is proportional to T, the rotational energy of a linear molecule is RT per mole. Since qtrans is proportional to T3/2, the translational energy is (3/2)RT per mole. Both match the equipartition idea from your earlier course. Also, qtrans is proportional to V, so p = kT (∂ ln Q/∂V) gives pV = NkT, the ideal gas law. Derive this once by hand. It shows the framework is consistent.
How examiners ask this topic
Questions in this area usually fall into a few types. These are patterns, not a prediction of any paper.
- Scaling questions: how does q change if T doubles, or the mass or volume changes?
- Population questions: what fraction of molecules is in a given level?
- Property questions: find energy, entropy or free energy from a given q.
- Limit questions: what happens to qvib as T tends to 0 or to infinity?
- Symmetry questions: what is σ, and how does it change the rotational entropy?
Most of these need only the table above and a calculator. Marks are lost on units, on σ and on the zero of energy.
Two solved practice questions
These are practice questions written for this post. They are not past-paper questions.
Practice question 1. A diatomic molecule has Θvib equal to T, so Θvib/T = 1. Energy is measured from the ground vibrational level. Find (a) qvib, (b) the fraction of molecules in v = 0, and (c) the fraction in v = 1.
Solution.
- qvib = 1/(1 − e−1). Since e−1 = 0.3679, the denominator is 0.6321. So qvib ≈ 1.58.
- The fraction in v = 0 is 1/qvib = 1 − e−1 ≈ 0.632. About 63% of molecules are in the ground level.
- The fraction in v = 1 is e−1/qvib = 0.3679 × 0.6321 ≈ 0.233.
Check: the v = 1 fraction is smaller than the v = 0 fraction, as it must be. Their ratio is e−1 = 0.368, which matches 0.233/0.632.
Practice question 2. A homonuclear diatomic gas has Θrot = 3.0 K. At 300 K, find qrot and the molar rotational entropy. Use R = 8.314 J K−1 mol−1.
Solution.
- T is much larger than Θrot, so the high-temperature form applies. The molecule is homonuclear, so σ = 2.
- qrot = T/(σΘrot) = 300/(2 × 3.0) = 50.
- For a linear rotor the molar rotational energy is RT. So the entropy is S = U/T + R ln q = R + R ln 50 = R(1 + ln 50).
- ln 50 = 3.912. So the rotational entropy = 8.314 × 4.912 ≈ 40.8 J K−1 mol−1.
Common trap: if you forget σ, you get qrot = 100. The entropy then comes out higher by R ln 2, about 5.8 J K−1 mol−1. Also note that no N! appears here, because that factor is applied once to the whole gas, not to each motion.
A short study plan
- Step 1: Write the definition q = Σ g e−ε/kT from memory. Solve three population-ratio problems.
- Step 2: Learn the three formulas in the table. Note when each is valid, for example T much larger than Θrot.
- Step 3: Derive U, S and A from ln Q. Reproduce pV = NkT from qtrans.
- Step 4: Practise limits, such as T tending to 0 and to infinity, for each q. Then redo the two practice questions above without looking.
- Step 5: Attempt timed statistical thermodynamics questions from the ABC Chemistry question bank. Keep a list of mistakes, especially σ, N! and the zero of energy.
Revisit the mistake list after a week. If you repeat the same error, the concept is not yet clear. Go back to Step 1 for that point.
Key takeaways
- q is a sum of Boltzmann factors and counts the thermally accessible states. It is dimensionless.
- The total q is a product of translational, rotational, vibrational and electronic parts, so ln q is a sum.
- Always check σ for rotation, N! for a gas, and the zero of energy for vibration.
- U = kT2(∂ ln Q/∂T)V, A = −kT ln Q, and S = U/T + k ln Q.
- Practise the same few problem types until the steps are automatic.
Frequently asked questions
Is the partition function the same as the number of states?
No. It is a weighted count. Each state is weighted by its Boltzmann factor, so high-energy states count for less at low temperature.
Why is qvib close to 1 at low temperature?
The vibrational spacing is large compared with kT. Only the lowest level is occupied, and that level contributes 1 when energy is measured from it.
When can I use the high-temperature rotational formula?
When T is much larger than Θrot. If a question gives a very low temperature, sum the levels term by term instead.
What is the difference between q and Q?
Small q belongs to one molecule. Capital Q belongs to the whole system. For a gas of N identical molecules, Q = qN/N!.
Which ensemble does the canonical partition function describe?
A system that can exchange heat with its surroundings at fixed temperature, volume and number of particles.
