Calculator
Pick a mode, enter values, then compute. Optional higher-order terms improve accuracy at higher densities.
Formula used
Virial equation (density form): Z = 1 + Bρ + Cρ² + Dρ³ + …
Here, Z is the compressibility factor, ρ is molar density (mol/m³), and B, C, D are virial coefficients.
Pressure from Z: P = ρRTZ, where R is the gas constant and T is temperature (K).
Second virial from van der Waals: B₂(T) = b − a/(RT), linking molecular attraction and excluded volume.
How to use this calculator
- Select a mode: virial series or van der Waals B₂.
- Enter temperature and choose your units.
- For the virial series, provide ρ or Vₘ and B.
- Enable C and D if you have higher-order data.
- Press Calculate to view Z and pressure above.
- Use Download CSV or Download PDF for exporting.
Example data table
| Mode | T (K) | ρ (mol/m³) | B (m³/mol) | C (m⁶/mol²) | Z | P (kPa) |
|---|---|---|---|---|---|---|
| Virial series | 300 | 40 | -1.60×10⁻⁴ | 2.00×10⁻⁷ | 0.99392 | 99.1 |
| Virial series | 450 | 20 | -9.00×10⁻⁵ | (not used) | 0.99820 | 74.8 |
| van der Waals B₂ | 300 | a = 3.60 bar·L²/mol² | b = 0.0427 L/mol | B₂ ≈ -0.00100 L/mol | ||
Values are illustrative only. Use published coefficients for your gas and temperature range.
Virial coefficients guide
1) Why virial coefficients matter
Real gases deviate from ideal behavior when intermolecular forces and finite molecular size become important. Virial coefficients quantify those departures using a controlled expansion. The second coefficient B(T) captures the leading correction, while higher terms add refinement at increasing density. In practice, virial data helps engineers model moderate-pressure gases, validate molecular potentials, and compare experimental datasets.
2) Interpreting the sign of B(T)
The sign of B(T) is a quick diagnostic. Negative B typically indicates net attractive interactions dominate at that temperature, pulling Z below 1 at low density. Positive B suggests repulsive or excluded-volume effects dominate, pushing Z above 1. Near the Boyle temperature, B(T) approaches zero and the gas behaves almost ideally over a broader density range.
3) Density-form virial series used here
This calculator uses the density form Z = 1 + Bρ + Cρ² + Dρ³. It is convenient when you know molar density ρ (mol/m³) or molar volume Vm = 1/ρ. Units must be consistent: B has m³/mol, C has m⁶/mol², and D has m⁹/mol³. Proper unit handling prevents large scaling errors.
4) Pressure prediction from Z
Once Z is computed, pressure follows from P = ρRTZ. For example, at T = 300 K and ρ = 40 mol/m³, an ideal gas gives P ≈ 99.8 kPa. If B = −1.60×10−4 m³/mol, then Bρ ≈ −0.0064, giving Z ≈ 0.9936 before higher terms. That small change shifts pressure by roughly 0.6%.
5) When to include C and D
Including C and D improves accuracy as density rises, but only if reliable coefficients are available for the same temperature. A useful rule is that each additional term should be smaller than the previous one: |Bρ| > |Cρ²| > |Dρ³|. If higher terms are comparable, the expansion may not converge well and a different equation of state may be preferable.
6) van der Waals link to B₂(T)
The calculator also provides B₂(T) from van der Waals parameters using B₂(T) = b − a/(RT). Here, b represents excluded volume and a represents attractions. Because a/(RT) decreases with temperature, B₂ increases as T rises. This simplified estimate is useful for quick checks, but experimental virial tables are more accurate for real substances.
7) Data sources and typical ranges
Virial coefficients are commonly obtained from precise PVT measurements, sound-velocity data, or fitted intermolecular potentials. Many gases have B values on the order of 10−4 to 10−3 m³/mol in common temperature ranges, while C is often around 10−7 to 10−5 m⁶/mol², depending on the substance. Always match coefficients to the exact temperature.
8) Practical workflow for best results
Start with temperature and the best available B(T). Compute Z and pressure at your density, then enable C and D if trustworthy values exist. Compare the incremental change in Z to judge sensitivity. For reporting, export CSV and archive the coefficient source and units. For design work, validate results against at least one independent reference model or dataset.
FAQs
1) What is the compressibility factor Z?
Z equals PVm/(RT). It measures how much a gas deviates from ideal behavior. Z = 1 is ideal, Z < 1 indicates net attractions, and Z > 1 indicates net repulsions at the chosen state.
2) Can I use molar volume instead of density?
Yes. Choose the molar volume option and enter Vm in m³/mol, L/mol, or cm³/mol. The calculator converts it to density using ρ = 1/Vm before evaluating the virial series.
3) Why does my pressure change when I enable C or D?
C and D add higher-order density corrections to Z. At higher densities, these terms can noticeably shift Z and therefore P. If the added term is not small compared to previous terms, the expansion may be outside its reliable range.
4) What does a negative B mean physically?
A negative B typically means attractive interactions dominate at low density, lowering Z below 1. As temperature increases, attractions become less important relative to thermal energy, and B often moves toward zero and may become positive.
5) What is the Boyle temperature?
The Boyle temperature is the temperature where B(T) ≈ 0. Near this point, the leading non-ideal correction vanishes, so the gas can behave almost ideally over a wider density range than usual.
6) Is van der Waals B₂ accurate for real gases?
It is a rough estimate useful for sanity checks. Real gases often require experimentally fitted virial coefficients or more advanced equations of state for accurate work, especially near saturation, high pressure, or low temperature.
7) Which units should I use for B, C, and D?
Use units consistent with molar density in mol/m³: B in m³/mol, C in m⁶/mol², and D in m⁹/mol³. If you enter L-based units, the calculator converts them internally to maintain consistency.