1.0041
atm
1.0048
atm
0.07
%
0.9993
1.0041
atm
1.0048
atm
0.07
%
0.9993
The Van der Waals Equation Calculator computes the pressure of a real gas using the van der Waals equation, comparing it to the ideal gas prediction. Johannes Diderik van der Waals derived this equation in his 1873 doctoral thesis (Nobel Prize in Physics 1910) by introducing two corrections to the ideal gas law that account for the finite size of molecules and the attractive interactions between them.
The van der Waals equation is (P + a*n^2/V^2)(V - n*b) = n*R*T, where a accounts for intermolecular attractions (which effectively reduce the pressure compared to ideal) and b accounts for the finite volume of molecules (which reduces the available volume compared to ideal). Equivalently, P = nRT/(V-nb) - an^2/V^2.
The parameter a is larger for gases with stronger intermolecular forces (polar molecules, easily polarizable molecules): CO2 has a = 3.640, H2O has a = 5.536, while H2 has a = 0.244 and He has a = 0.034 (in L^2*atm/mol^2). The parameter b is related to the physical size of molecules: larger molecules have larger b.
The compression factor Z = PV/(nRT) measures deviation from ideality: Z = 1 for an ideal gas. At high pressures Z > 1 (repulsion dominates, hard-core effect of b). At moderate pressures Z < 1 (attraction dominates, Boyle temperature where Z = 1 momentarily). Real gas behavior is essential for understanding high-pressure industrial processes, gas pipelines, liquefied natural gas, and supercritical fluid extraction.
The van der Waals equation, despite its simplicity, correctly predicts gas-liquid phase transitions, critical phenomena (critical temperature, pressure, and molar volume), and the general shape of isotherms. It remains foundational in physical chemistry.
Van der Waals pressure: P = nRT/(V-nb) - an^2/V^2. Ideal gas pressure: P_ideal = nRT/V. R = 0.08206 L*atm/(mol*K). Deviation = |P_vdw - P_ideal| / P_ideal * 100%. Compression factor: Z = P_vdw*V/(nRT). A = a correction reduces pressure; b correction increases pressure.
At low pressures and high temperatures (large V per mole), P_vdw approaches P_ideal. Deviations are large near the critical point and at high pressures. If P_vdw < P_ideal, attractions dominate (low T, moderate P). If P_vdw > P_ideal, repulsions dominate (very high P). Typical a and b values: N2: a=1.408, b=0.0391. CO2: a=3.640, b=0.0427. H2O: a=5.536, b=0.0305.
Inputs
Results
At 0.5 L/mol and 300 K, CO2 gives P_vdw = 33.5 atm vs ideal 49.2 atm — a 32% deviation with Z = 0.679. The strong a value for CO2 shows large attractive interactions at this density.
Inputs
Results
Nitrogen near STP conditions (22.4 L/mol, 273 K) deviates from ideal by only 0.28%. Z = 0.997, confirming N2 is nearly ideal at standard conditions.
(P + an^2/V^2)(V - nb) = nRT. It modifies the ideal gas law with two parameters: a correcting for attractive intermolecular forces (reducing effective pressure) and b correcting for molecular volume (reducing effective volume).
The constant a (L^2*atm/mol^2) quantifies intermolecular attractions. Larger a means stronger attractive forces (polar molecules, easily polarizable gases). It reduces the pressure because attracted molecules hit the walls with less force.
The constant b (L/mol) is approximately 4 times the volume of one mole of molecules. It represents the excluded volume — the space unavailable to molecular motion because of the finite size of molecules. It increases the effective pressure at high densities.
Z = PV/(nRT). For ideal gas Z = 1. Z < 1 means attractions are dominant. Z > 1 means repulsions (excluded volume) dominate. At the Boyle temperature, attractive and repulsive corrections cancel and Z = 1 over a range of pressures.
The state point above which liquid and gas phases are indistinguishable (T_c, P_c, V_c). The van der Waals equation predicts T_c = 8a/(27Rb), P_c = a/(27b^2), V_c = 3nb per mole. Critical constants can be used to calculate a and b from experimental data.
At high pressures (high density — molecules interact frequently), low temperatures (thermal energy is comparable to interaction energy), and near the critical point. For most gases below 10 atm and above 200 K, the ideal gas law is accurate within a few percent.
Yes. The Redlich-Kwong, Soave-Redlich-Kwong, and Peng-Robinson equations are more accurate for engineering calculations. The virial equation of state gives a systematic expansion in 1/V. But the van der Waals equation retains pedagogical value for explaining real gas phenomena.
Above both T_c and P_c, a substance is a supercritical fluid — neither liquid nor gas but with properties of both. Supercritical CO2 (T_c = 304 K, P_c = 72.8 atm) is widely used as a solvent in extraction, chromatography, and decaffeination.
Intermolecular attractive forces: London dispersion forces (induced dipole-induced dipole, proportional to polarizability), Keesom forces (permanent dipole-dipole), and Debye forces (permanent-induced dipole). The parameter a in the equation of state reflects their total effect.
Below T_c, the van der Waals isotherm has three P-V branches for a given T, with a region of negative slope (unphysical — Maxwell construction replaces it with a flat portion representing the liquid-vapor coexistence region and the phase transition pressure).
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
How helpful was this calculator?
Be the first to rate!
Electron Configuration Calculator
Atomic & Molecular Physics Calculators
Ionization Energy Calculator
Atomic & Molecular Physics Calculators
Electron Affinity Calculator
Atomic & Molecular Physics Calculators
Electronegativity Calculator
Atomic & Molecular Physics Calculators
De Broglie Wavelength Calculator
Atomic & Molecular Physics Calculators
Heisenberg Uncertainty Calculator
Atomic & Molecular Physics Calculators
