Van der Waals Equation Calculator
Estimate the pressure of a real gas, compare it with the ideal-gas result, and derive substance-specific van der Waals constants from critical-point data.
Gas and critical point
Selecting a preset loads critical pressure and temperature, then derives a and b.
Positive decimal using a dot; example 7.3773.
Absolute temperature must be greater than 0 K.
Derived from the critical-point relation Vc = 3RTc/(8Pc).
Gas parameters
Required positive amount; scientific notation is not accepted.
Must exceed the excluded molecular volume n × b.
Required absolute temperature; convert Celsius by adding 273.15.
Van der Waals constants a and b
Higher a increases the attractive-pressure correction.
Higher b reduces the free volume available to molecules.
Live results
Van der Waals pressure
246.851 kPa
The real-gas estimate is 1.036% below the ideal-gas pressure for this state.
- Ideal-gas pressure
- 249.434 kPa PV = nRT benchmark
- Attraction correction
- 3.657 kPa a(n/V)²
- Free volume
- 9.95715 L V – nb
- Compressibility factor
- 0.989644 Z = PV/(nRT)
- Ideal deviation
- – 1.03563% Relative pressure difference
- Excluded volume
- 0.0428453 L n × b
Van der Waals pressure 246.851 kilopascals. Compressibility factor 0.989644.
Calculation breakdown
| Quantity | Current value | Role in the model |
|---|---|---|
| Repulsion-adjusted pressure term | 250.507 kPa | nRT/(V – nb), before subtracting attraction |
| Attraction correction | 3.657 kPa | a(n/V)², subtracted from the repulsion-adjusted term |
| Van der Waals pressure | 246.851 kPa | Final real-gas pressure estimate |
| Ideal-gas pressure | 249.434 kPa | Reference result from nRT/V |
| Free-volume ratio | 99.5715% | (V – nb)/V; must remain positive |
All rows use the same canonical calculation as the result cards and Excel workbook. The model is invalid when V ≤ nb because no positive free volume remains.
How to use the Van der Waals equation calculator
What this calculator does
This calculator estimates the pressure of a real gas from its amount, container volume, absolute temperature, and substance-specific van der Waals constants. It also derives the constants from critical pressure and critical temperature when you choose a gas preset. The calculation corrects the ideal-gas law for two effects: molecules occupy finite space, and molecules attract one another. It is a compact equation-of-state estimate, not a substitute for a high-accuracy property database, phase-equilibrium package, or safety-rated process simulation.
When to use it
Use it to compare a classroom ideal-gas result with a real-gas correction, check how pressure changes as a vessel becomes more crowded, reproduce textbook exercises involving a and b, or make an initial thermodynamics estimate before consulting a more detailed equation of state. It is especially instructive when pressure is elevated, temperature is comparatively low, or the gas has stronger intermolecular attraction. For engineering-grade property work, compare the result with the NIST thermophysical fluid-property system.
How to calculate
- The calculator opens with a complete carbon-dioxide demonstration: critical pressure 7.3773 MPa, critical temperature 304.1282 K, 1 mol, 10 L, and 300 K. The first result and a validated example Excel workbook are available immediately.
- Choose a Gas preset or select Custom. A preset replaces Critical pressure and Critical temperature and recalculates Attraction constant a, Repulsion constant b, and Critical molar volume.
- Enter Amount of substance in moles, Volume in liters, and Temperature in kelvins. Values use a decimal point; commas and scientific notation are rejected to avoid ambiguous interpretation.
- Review Van der Waals pressure, the ideal comparison, correction terms, free volume, compressibility factor, and the breakdown table. Results update without a Calculate button.
- Select Download Excel to export the current typed inputs and outputs. Reset clears the demonstration values, results, validation residue, and workbook state; Excel export remains unavailable until a complete valid state is entered again.
Input guide
Gas preset is a required selection that loads a named gas or Custom mode. Critical pressure is a positive decimal in MPa, such as 7.3773; increasing it while critical temperature is fixed generally lowers derived a and b. Critical temperature is a positive absolute temperature in K, such as 304.1282; it affects both derived constants. Critical molar volume is a read-only derived output in L/mol, calculated as 3RTc/(8Pc). The exact molar gas constant used is the NIST CODATA value of R.
Amount of substance is required and must be greater than zero; 1 mol is a typical instructional value. More moles raise both the repulsive and attractive terms. Volume is required in liters; 10 L is the startup example. It must be greater than n × b, because V – nb is the volume available for molecular motion. Temperature is required in kelvins; 300 K is the example. Do not enter Celsius directly. Higher temperature raises the kinetic pressure term. Attraction constant a is a nonnegative value in Pa·m⁶/mol²; a larger value subtracts more pressure. Repulsion constant b is nonnegative in m³/mol; a larger value reduces free volume. Editing a or b switches the preset to Custom so the manual constants remain authoritative. Reference values for many gases are available in the LibreTexts van der Waals constants table.
Output guide
Van der Waals pressure is the primary estimate in kPa. Ideal-gas pressure is nRT/V in kPa. Attraction correction is a(n/V)² in kPa and is subtracted. Free volume is V – nb in liters; zero or a negative value invalidates the state. Excluded volume is n × b. Compressibility factor is Z = PV/(nRT): Z = 1 matches the ideal relation, while values below or above 1 indicate net departure. Ideal deviation is the percentage difference between the real-gas and ideal-gas pressures. The breakdown table also reports the repulsion-adjusted pressure term and free-volume ratio. These are model estimates or exact identities within the selected equation, not experimental guarantees.
Worked example
For the startup carbon-dioxide case, the critical data produce a = 0.365652123 Pa·m⁶/mol² and b = 0.0000428453254 m³/mol. With n = 1 mol, V = 10 L = 0.010 m³, and T = 300 K, the available volume is 0.010 – 1 × 0.0000428453254 = 0.00995715467 m³. The repulsion-adjusted term is 250.507 kPa. The attraction correction is 3.657 kPa. Subtracting gives a Van der Waals pressure of 246.851 kPa. The ideal result is 249.434 kPa, so the real-gas estimate is 1.03563% lower and Z = 0.989644. The conceptual derivation is explained in the LibreTexts treatment of the van der Waals equation.
How the model works
The first term raises pressure relative to an ideal gas because molecules cannot use the excluded volume nb. The second term lowers pressure because attractive forces reduce the momentum transferred to a container wall. Either correction can dominate depending on density, temperature, and the selected gas. At low density, nb is small relative to V and a(n/V)² becomes small, so the equation approaches PV = nRT.
The preset constants are derived from the van der Waals critical-point identities b = RTc/(8Pc), a = 27R²Tc²/(64Pc), and Vc = 3b. These identities force the model's critical compressibility factor to 3/8, whereas measured fluids do not all share that value. That is one reason the classic equation is educational and qualitative rather than universally precise.
Interpretation and limitations
A compressibility factor near 1 means the current state is close to ideal behavior under this model. A value below 1 indicates that the attractive correction is dominant; a value above 1 indicates that excluded-volume repulsion is dominant. Negative model pressure can occur for mathematically permitted but physically unstable combinations, particularly at low temperature and high density. Treat that as a warning that a simple single-phase calculation is outside its useful region.