Sackur–Tetrode Entropy Calculator

Formula used

The Sackur–Tetrode equation gives the absolute entropy of a monatomic ideal gas by including quantum translational states. Two equivalent forms are implemented.

• Temperature form:
  S = N k_B [ ln( (V/N) · (2π m k_B T / h²)^3/2 ) + 5/2 ]
• Energy form:
  S = N k_B [ ln( (V/N) · (4π m U / (3 N h²))^3/2 ) + 5/2 ]

Here N is particle count, m is particle mass, V is volume, T is temperature, U is total internal energy, h is Planck’s constant, and k_B is Boltzmann’s constant.

How to use this calculator

1. Select temperature form if you know T directly.
2. Select energy form if you know U; T is derived when needed.
3. Enter moles n and molar mass M for your monatomic gas.
4. Choose how to supply volume: enter V directly, or compute from pressure.
5. Click Calculate. Results appear above the form with export buttons.

Example data table

Sample outputs below are typical for monatomic ideal gases at moderate conditions.

Notes: The table assumes ideal gas behavior and monatomic structure. Real-gas deviations can change absolute entropy, especially near condensation or at very high pressures.

Technical article

1) What the Sackur–Tetrode equation estimates

The Sackur–Tetrode equation gives the absolute entropy of an ideal, monatomic gas by combining classical thermodynamics with a quantum count of accessible microstates. It is most useful when you want entropy from first principles rather than from tables, and when the gas is dilute enough to behave ideally.

2) Core inputs and what they represent

This calculator uses the amount of substance (n), temperature (T), and either volume (V) or pressure (P). If you provide pressure, it computes volume from the ideal-gas relation V = nRT/P. You also enter molar mass to determine the particle mass, which controls the quantum-scale correction through the thermal wavelength.

3) Thermal de Broglie wavelength and why it matters

A key quantity is the thermal de Broglie wavelength, λ = h/√(2πmk_BT). It measures how “quantum-sized” the particles are compared with their mean spacing. When λ is much smaller than the average spacing, the classical picture is accurate and the entropy is well described by the Sackur–Tetrode form.

4) Typical magnitudes you should expect

Molar entropies for common noble gases at room temperature and 1 atm are usually in the 130–165 J·mol⁻¹·K⁻¹ range. For example, at 300 K and 1 atm, helium is about 135 J·mol⁻¹·K⁻¹, neon about 155 J·mol⁻¹·K⁻¹, and argon about 164 J·mol⁻¹·K⁻¹ (ideal-gas prediction). Values increase with temperature and with increasing specific volume.

5) How entropy responds to changing conditions

At fixed n and T, increasing volume raises entropy logarithmically. Doubling V increases S by nR ln 2, which is about 5.76 J·mol⁻¹·K⁻¹ for one mole. At fixed n and V, increasing temperature also increases entropy because higher T expands the accessible momentum space; the dependence appears through a log(T^3/2) term.

6) Interpreting the “quantum correction”

Classical ideal-gas entropy formulas have an undetermined additive constant. The Sackur–Tetrode derivation fixes that constant by introducing Planck’s constant (h) and correctly counting translational states. That is why absolute values become meaningful and comparable between gases, provided the assumptions (monatomic, dilute, nondegenerate) hold.

7) Validity limits and practical checks

The equation assumes a monatomic ideal gas with negligible interactions and no internal rotational/vibrational modes. It becomes unreliable near condensation, at very high pressures, or at extremely low temperatures where quantum degeneracy appears. A practical check is to ensure the gas is far from saturation and that the computed thermal wavelength is well below the mean particle spacing.

8) Where this calculator is used

Engineers and physicists use Sackur–Tetrode entropy in mixing problems, entropy-balance calculations, and sanity checks against tabulated data. It also supports statistical-mechanics teaching, because it connects macroscopic observables (n, T, P, V) to microscopic constants (k_B, h, particle mass) in a single, testable expression.

FAQs

1) Is this equation valid for diatomic gases like N₂?

Not for full accuracy. Diatomic gases have rotational and vibrational contributions to entropy. Sackur–Tetrode accounts only for translational entropy of monatomic particles, so it underestimates entropy for molecules.

2) Why do I need molar mass?

Molar mass sets the particle mass, which controls the thermal de Broglie wavelength. Heavier particles have a smaller wavelength at the same temperature, shifting the absolute entropy prediction.

3) What happens if I enter pressure instead of volume?

The calculator converts pressure to volume using V = nRT/P (with your chosen units), then evaluates entropy. This is convenient when your state point is defined by n, T, and P.

4) Can I use it near the boiling point or high pressures?

Use caution. Real-gas interactions become important near phase transitions and at high density. In those regimes, the ideal-gas assumption fails and the result can deviate substantially from measured entropy.

5) Why is the entropy sometimes negative for very small volumes?

Extremely small specific volumes can violate the dilute-gas assumptions and push the logarithm term to small values. That signals the model is out of range, not that physical entropy is “truly negative” for a stable gas state.

6) How do I compare the result with tables?

Use molar entropy (J/mol·K) for the same substance and state point. Expect differences if tables include real-gas corrections, reference-state conventions, or internal-mode contributions for non-monatomic species.

7) Which output should I report: total or molar entropy?

Report molar entropy for material-property comparisons, and total entropy when doing system entropy balances. The calculator shows both so you can match the format used in your thermodynamics workflow.