Model thermal populations across discrete or continuous states. Compare free energy and entropy trends. Export clean reports for faster analysis today.
This tool evaluates the canonical partition function under common modeling choices.
Sample discrete spectrum in eV. You can paste these values into the input box.
| Energy (eV) | Degeneracy g | Notes |
|---|---|---|
| 0.000 | 1 | Ground state |
| 0.025 | 2 | Low excitation |
| 0.080 | 3 | Higher degeneracy |
| 0.150 | 1 | Sparse upper state |
The canonical partition function Z compresses microscopic information into one quantity that predicts macroscopic behavior. Once Z is known, the calculator reports ln(Z), Helmholtz free energy F, internal energy U, and entropy S, letting you move from an energy model to thermodynamic trends quickly.
This tool assumes a fixed temperature T, where probabilities follow exp(−E/kBT). Temperature is entered in kelvin because kB is per kelvin. If your system changes particle number or volume, Z still helps, but interpretation should match the ensemble used.
For quantized systems, paste lines of energy and degeneracy g. Degeneracy multiplies the Boltzmann weight, so a highly degenerate level can contribute strongly to Z even when its energy is higher. The example table uses eV values, but you can enter many levels.
Energy may be entered in joules, electronvolts, or kilojoules per mole. The calculator converts every value to per‑particle joules before computing exp(−βE). This prevents accidental mixing of molar and particle units that would shift Z by huge factors and distort probabilities.
The probability table is a practical diagnostic. Probabilities should sum to 1 within rounding. At low T, weight concentrates in low‑energy states; as T rises, populations spread upward, Z increases, and ln(Z) typically changes smoothly. These patterns help validate your level list.
Free energy is computed from F = −kBT ln(Z), a compact measure of temperature‑dependent “available” energy. Internal energy U is the probability‑weighted mean energy in discrete modes. Entropy follows S = (U − F)/T, combining energetic and multiplicity effects consistently.
The ideal gas option computes translational q and then Z = q^N/N!. Because q^N overflows for large N, the calculator works in ln-space, using ln(N!) via the gamma function. When Z is too large to print, it reports log10(Z) for scale.
Use discrete mode for molecules, spins, defects, or simplified spectra; use the two‑level mode for fast sensitivity checks; and use ideal gas mode for classical translation estimates. Export CSV for calculations and PDF for reporting, then compare how Z, U, F, and S shift with temperature across parameter sweeps with confidence.
Z is the canonical partition function, computed as a weighted sum of Boltzmann factors. It sets the relative statistical weight of accessible states at temperature T and enables the derived outputs ln(Z), F, U, and S.
Degeneracy counts how many distinct microstates share the same energy. The calculator multiplies each Boltzmann factor by g, so larger degeneracy increases that level’s contribution to Z and its probability.
Yes. Select kJ/mol and the calculator converts to per‑particle joules using Avogadro’s number. This keeps the Boltzmann exponent consistent for probabilities and thermodynamic quantities.
Z can become extremely large, especially in ideal gas mode with large N. ln(Z) stays numerically stable and is sufficient to compute F, U, and S. The tool may also show log10(Z) for scale.
Discrete mode accepts any number of levels from your list. Two‑level mode is a fast special case with exactly two states, useful for quick sensitivity checks or teaching demonstrations.
For discrete modes, U is the probability‑weighted mean energy. Free energy is F = −kBT ln(Z). Entropy is S = (U − F)/T. Ideal gas mode uses U = (3/2)NkBT for translation.
Use CSV for further analysis in spreadsheets, plotting, or batch comparisons. Use PDF for a clean, shareable report that summarizes Z, ln(Z), and thermodynamic outputs, plus level probabilities when available.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.