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The van der Waals equation corrects the ideal-gas model for intermolecular attraction and finite molecular volume.
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Formula
p in Pa · V in m3 · n in mol · T in K · a in Pa·m6/mol2 · b in m3/mol
| Symbol | Meaning | Unit |
|---|---|---|
| Gas pressure | Pa | |
| Gas volume | m3 | |
| Amount of gas | mol | |
| Absolute temperature | K | |
| Substance-specific attraction parameter | Pa·m6/mol2 | |
| Substance-specific co-volume parameter | m3/mol |
\left(p+a\frac{n^2}{V^2}\right)(V-nb)=nRTThe a n2/V2 term corrects pressure for attractions that reduce wall impacts; V−nb reduces the free volume for molecular motion. Parameters are substance-specific. This is a useful simple real-gas model, not a high-accuracy equation near every phase boundary or critical region. Keep pressure, volume and parameters in a consistent unit system.
A simple real-gas model; more accurate equations of state may be needed near critical and phase boundaries.
The van der Waals equation corrects the ideal-gas model for intermolecular attraction and finite molecular volume.
Rearrange for the target quantity
Check units and model limits before substitution.
Calculate p for 1.00 mol at V=0.0240 m3 and T=300 K using the stated a and b values.
For n=1.00 mol, V=0.0240 m3, T=300 K, a=0.137 Pa·m6/mol2 and b=3.86×10−5 m3/mol, p=nRT/(V−nb)−a(n/V)2≈103.8 kPa.
Check: setting a=b=0 returns the ideal-gas value, about 103.9 kPa.
Applying the equation outside its reference conditions or with mixed units.
Check units and model assumptions before substitution; the worked example states the relevant limit.
Formula cluster
The relation (p+an2/V2)(V−nb)=nRT complements related quantities but does not replace unit checks or model assumptions.
For n=1.00 mol, V=0.0240 m3, T=300 K, a=0.137 Pa·m6/mol2 and b=3.86×10−5 m3/mol, p=nRT/(V−nb)−a(n/V)2≈103.8 kPa. Check: setting a=b=0 returns the ideal-gas value, about 103.9 kPa.
Estimate real-gas states and compare them with the ideal-gas approximation
Here is how to work through a typical van der Waals Equation ((p+an²/V²)(V−nb)=nRT) task step by step:
Task
Calculate p for 1.00 mol at V=0.0240 m3 and T=300 K using the stated a and b values.
Solution path
For n=1.00 mol, V=0.0240 m3, T=300 K, a=0.137 Pa·m6/mol2 and b=3.86×10−5 m3/mol, p=nRT/(V−nb)−a(n/V)2≈103.8 kPa.
Check: setting a=b=0 returns the ideal-gas value, about 103.9 kPa.
The van der Waals equation corrects the ideal-gas model for intermolecular attraction and finite molecular volume. The relation van der Waals Equation answers a specific question about Estimate real-gas states and compare them with the ideal-gas approximation. For n=1.00 mol, V=0.0240 m3, T=300 K, a=0.137 Pa·m6/mol2 and b=3.86×10−5 m3/mol, p=nRT/(V−nb)−a(n/V)2≈103.8 kPa. Check: setting a=b=0 returns the ideal-gas value, about 103.9 kPa. Its check verifies the calculation and the expected balance or sign. Map each symbol to the actual sample before substituting values. A plausible number alone does not prove that the model assumptions are satisfied in an experiment.
The quantities mean: p: Gas pressure (Pa); V: Gas volume (m3); n: Amount of gas (mol); T: Absolute temperature (K); a: Substance-specific attraction parameter (Pa·m6/mol2); b: Substance-specific co-volume parameter (m3/mol). The unit rule is p in Pa · V in m3 · n in mol · T in K · a in Pa·m6/mol2 · b in m3/mol. Numerator and denominator must describe the same sample and compatible measurement scales. For solutions, distinguish final volume from solvent mass; for gases, pressure and volume units must match the gas constant. Check units after full substitution. This exposes a missing conversion factor before it disappears inside a rounded result. Keep each symbol tied to its definition.
A simple real-gas model; more accurate equations of state may be needed near critical and phase boundaries. The a n2/V2 term corrects pressure for attractions that reduce wall impacts; V−nb reduces the free volume for molecular motion. Parameters are substance-specific. This is a useful simple real-gas model, not a high-accuracy equation near every phase boundary or critical region. Keep pressure, volume and parameters in a consistent unit system. Check this scope before calculating: temperature, composition, phase and concentration scale can affect the result. If the sample violates an assumption, extra decimal places will not fix the model. State the mismatch or choose a better description, such as activities for nonideal solutions or a real-gas model at high pressure.
Recheck the example independently: For n=1.00 mol, V=0.0240 m3, T=300 K, a=0.137 Pa·m6/mol2 and b=3.86×10−5 m3/mol, p=nRT/(V−nb)−a(n/V)2≈103.8 kPa. Check: setting a=b=0 returns the ideal-gas value, about 103.9 kPa. Then verify the unit (p in Pa · V in m3 · n in mol · T in K · a in Pa·m6/mol2 · b in m3/mol) and substitute the result back into the relation. A material balance must remain consistent; a pressure or temperature relation must change in the physically expected direction. A failed check often points to mixed reference quantities, an unconverted unit, or an assumption that does not fit the sample.
van der Waals Equation sits beside Ideales Gasgesetz and Compressibility Factor for Real Gases but serves a distinct purpose: Estimate real-gas states and compare them with the ideal-gas approximation. Linked relations may supply inputs, but they do not replace this equation. Check standard state, ideal-mixture assumptions and reaction step separately for each relation. In a multistep problem, first establish the balance or measured quantity, then apply the equilibrium model, and finally verify units. Distinguish them by naming the quantity each predicts and the measurement each requires.
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Curated exam set for "van der Waals Equation":
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Which formula describes van der Waals Equation?
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How do you rearrange van der Waals Equation for Rearrange for the target quantity?
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Which common mistake happens with van der Waals Equation?
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