Calculator
Formula used
The equipartition theorem assigns an average energy of ½ kBT to each independent quadratic degree of freedom.
If a system has f such degrees per particle, then:
- Per particle:
Ē = (f/2) kB T - For N particles:
Ētotal = (f/2) N kB T - For n moles:
Ētotal = (f/2) n R T, sinceR = NAkB
Use integer or fractional f when an effective model is intended.
How to use this calculator
- Enter the temperature and select its unit.
- Provide degrees of freedom f for your model.
- Select quantity type: moles or particle count.
- Enter the amount and choose the energy unit.
- Press Calculate to view results above.
To export, calculate first, then use the CSV or PDF buttons.
Example data table
| Temperature | f | Amount | Type | Energy unit | Total energy (avg) |
|---|---|---|---|---|---|
| 300 K | 3 | 1 | mol | kJ | 3.7415 kJ |
| 500 K | 5 | 2 | mol | kJ | 20.7862 kJ |
| 298 K | 3 | 1.0e20 | particles | J | 0.6175 J |
Equipartition energy guide
1) What the theorem states
In the classical limit, each independent quadratic term in the energy contributes an average of ½ kBT. This calculator applies that rule to estimate the mean thermal energy for a chosen temperature, degrees of freedom, and amount of substance.
2) Typical degrees of freedom values
For a monatomic ideal gas, translational motion gives f = 3. A rigid diatomic gas often has f ≈ 5 (3 translation + 2 rotation) at moderate temperatures. Vibrational modes can add 2 per active vibration (kinetic + potential) when they are thermally excited.
3) Per-particle and per-mole energy scale
At 300 K, the average energy per quadratic degree of freedom is about ½ kBT ≈ 2.07×10−21 J (≈ 0.0129 eV). For f = 3, the mean energy per particle becomes 3.11×10−21 J. For 1 mol with f = 3, the total is (3/2)RT ≈ 3.74 kJ.
4) When moles are more convenient
Many laboratory calculations use moles, so the calculator offers (f/2)nRT. This avoids extremely large particle counts and matches common thermodynamics tables where energy is reported per mole or per kilogram.
5) Output units and conversions
Energy can be displayed in J, kJ, eV, or erg. Electron-volts are useful at the microscopic scale, while kJ is practical for molar totals. The tool converts after computing in joules to keep numerical consistency.
6) Validity limits and caveats
Equipartition is not universal. At low temperatures, quantum level spacing can “freeze out” rotations or vibrations, reducing the effective f. Strong interactions, constraints, or non-quadratic potentials can also shift averages away from ½ kBT.
7) Using effective degrees of freedom
Real systems may not activate modes fully. An effective f (including fractional values) can approximate partial activation over a temperature range. This is common in fitting heat-capacity data or estimating energy budgets without a full quantum model.
8) Practical workflow for analysis
Start with a physically motivated f, choose temperature and quantity, then compare “per particle” and “total” results. Export CSV for record-keeping or plotting, and use PDF export to share a clean summary with reports or lab notes.
FAQs
1) What does f represent in this calculator?
It is the number of independent quadratic degrees of freedom per particle used by your model, such as translations, rotations, and active vibrational contributions.
2) Why is the energy per DOF equal to ½kBT?
In the classical limit, equipartition assigns the same mean energy to each quadratic term in the Hamiltonian, giving an average contribution of one half kBT.
3) Should I use particles or moles?
Use particles for microscopic counts in simulations. Use moles for laboratory-scale problems, since results align with thermodynamic relations using the gas constant R.
4) Can I enter fractional degrees of freedom?
Yes. Fractional f can represent partially activated modes or an effective model that fits measurements across a temperature range.
5) Why does the calculator warn about very low temperatures?
At low temperatures, quantum effects can suppress rotations or vibrations, so classical equipartition may overestimate energy unless you adjust f accordingly.
6) What is the relationship between kB and R?
They are connected by Avogadro’s number: R = NA × kB. That is why molar totals use (f/2)nRT while particle totals use (f/2)NkBT.
7) How does the PDF download work?
The PDF option opens the browser print dialog and formats the page to keep only the results section. You can then choose “Save as PDF” on most systems.