Van der Waals Real Gas Solver
Solve the Van der Waals equation of state, (P + an²/V²)(V − nb) = nRT, for pressure, volume, or temperature. Pick a built-in gas or enter your own a and b constants, and see the result compared against the ideal gas law with the compressibility factor Z.
Real Gas vs. Ideal Gas Comparison
How far the Van der Waals result for Carbon dioxide (CO2) drifts from the ideal gas law, plus the compressibility factor Z (Z = 1 means perfectly ideal).
Step-by-Step Solution
Here's exactly how this answer was calculated, one step at a time.
Given: P = 1 atm, V = 22.4 L, T = 273.15 K, n = 1 mol, gas = Carbon dioxide (CO2)
Step 1: Convert temperature to kelvin
As with the ideal gas law, the Van der Waals equation needs absolute temperature in kelvin.
T = 0 °C → 273.15 KStep 2: Write the Van der Waals equation
The a term corrects pressure for attraction between gas particles; the b term corrects volume for the space the particles themselves take up.
(P + an²/V²)(V − nb) = nRTStep 3: Use this gas's constants
a = 3.59 L²·atm/mol², b = 0.0427 L/mol (Carbon dioxide (CO2))Step 4: Rearrange and solve
P = nRT/(V − nb) − an²/V² = 0.99538 atmStep 5: Convert to your chosen unit
Result = 0.99538 atmStep 6: Compare with the ideal gas law
This shows exactly how far the ideal gas law's assumption-free answer drifts from the more realistic Van der Waals result for this gas and these conditions.
Ideal gas law gives 1.00062 atm for the same values — a difference of -0.524%
The Van der Waals result is:
0.99538 atm
Free Van der Waals Real Gas Solver
The ideal gas law, PV = nRT, works well for most classroom problems, but it quietly assumes gas particles have no size and never attract each other. Real gases don't quite follow that rule, especially at high pressure or low temperature. The Van der Waals equation fixes both assumptions with two correction terms, and this solver works it out for you — pick pressure, volume, or temperature to solve for, choose a real gas from the built-in list (or type your own constants), and get a full step-by-step answer.
Alongside the Van der Waals result, this solver always shows what the plain ideal gas law would have predicted for the same numbers, so you can see exactly how much the real-gas correction changes the answer, and by what percentage.
The Van der Waals Equation Explained
The Van der Waals equation is (P + an²/V²)(V − nb) = nRT. It starts from the same PV = nRT and adds two correction terms. The an²/V² term is added to the measured pressure because real gas particles attract each other slightly, which pulls them together and makes the measured pressure a little lower than it would be for particles with zero attraction — so the equation adds back what's 'missing'. The nb term is subtracted from the volume because real gas particles take up actual physical space, so the space they can freely move around in is a little less than the full container volume.
Each gas has its own pair of constants, a and b, worked out experimentally. A large a value means the gas molecules attract each other more strongly — polar or easily-condensed gases like water vapour, ammonia, and sulfur dioxide sit high on this scale. A large b value means the molecules themselves are bulkier — larger molecules like chlorine and sulfur dioxide again sit high here too. Small, light, weakly-attracting gases like helium and hydrogen have small a and b values and behave closest to an ideal gas.
How to Use This Solver
Pick what you're solving for — pressure, volume, or temperature — then choose the gas you're working with from the dropdown. Built-in constants are included for fourteen common gases, from helium and hydrogen (nearly ideal) to carbon dioxide, ammonia, and water vapour (noticeably non-ideal). If your gas isn't listed, or your textbook gives you different a and b values, choose 'Custom' and type them in directly.
Fill in the remaining known values — pressure, volume, temperature, and the amount of gas in moles — using whatever units are convenient, and the solver converts everything internally. The result shows both the Van der Waals answer and, right next to it, what the ideal gas law alone would have predicted, along with the percentage difference between them.
Why Solving for Volume Needs a Different Method
Solving the Van der Waals equation for pressure or temperature is simple algebra — you can rearrange the equation directly and plug numbers in. Solving for volume is not so simple, because once you multiply everything out, volume appears three separate times, including a V³ term. There's no clean rearrangement that isolates V on one side the way there is for P or T.
Instead, this solver uses a numerical method (Newton-Raphson) to home in on the correct volume, starting from the ideal gas volume as a first guess and refining it step by step until the equation balances. This is exactly what happens 'under the hood' in most real chemistry and engineering software when volume is the unknown, and it's why a plain rearrange-and-substitute approach doesn't work for this particular case.
Worked Example: Carbon Dioxide at High Pressure
Take 1 mole of carbon dioxide gas at 0°C, compressed into a volume of 0.5 litres — a fairly tight squeeze compared to the roughly 22.4 litres an ideal gas would occupy at STP. Using CO2's Van der Waals constants (a = 3.59 L²·atm/mol², b = 0.0427 L/mol), the solver works out the real pressure needed to hold that gas in that volume at that temperature, and compares it to what the ideal gas law alone would predict.
At this kind of tight volume, the two answers noticeably diverge — the ideal gas law overestimates the pressure needed, because it doesn't account for the fact that the CO2 molecules attract each other and effectively pull the gas 'inward', lowering the pressure it actually exerts on the container walls compared to a hypothetical gas with zero attraction between particles.
The Compressibility Factor, Z
The compressibility factor Z = PV/(nRT) is a quick way to describe how far any gas has drifted from ideal behaviour under a given set of conditions. For a perfectly ideal gas, Z always equals exactly 1. This solver calculates Z automatically from the Van der Waals result.
When Z is greater than 1, the gas is harder to compress than an ideal gas would be — this usually happens at very high pressure, where the physical size of the molecules (the b term) starts to matter more than their attraction to each other. When Z is less than 1, attractive forces between molecules (the a term) are winning out, pulling the gas into a smaller volume than the ideal gas law would predict — this is common at moderate pressure and lower temperature, and gets more pronounced as a gas approaches the conditions where it would condense into a liquid.
Which Gases Deviate the Most from Ideal Behaviour?
Small, non-polar, weakly-interacting gases stay closest to ideal — helium, hydrogen, and neon are the classic examples, and their small a and b constants reflect that. They stay close to ideal behaviour across a wide range of everyday pressures and temperatures, which is part of why the ideal gas law works so well for so many textbook problems.
Gases that are polar, easily liquefied, or made of larger, bulkier molecules deviate more. Water vapour, ammonia, sulfur dioxide, and chlorine all have comparatively large Van der Waals constants, because their molecules either attract each other strongly (polar molecules) or take up more physical space (larger, heavier molecules). Carbon dioxide sits in between — light enough to often be treated as roughly ideal at everyday pressure, but showing real, measurable deviation once it's compressed or cooled.
When Should You Actually Use the Van der Waals Equation?
For most everyday chemistry problems — a reaction at room temperature and atmospheric pressure, a typical lab gas sample, a classroom stoichiometry question — the ideal gas law is close enough, and the extra complexity of the Van der Waals equation usually isn't worth it. The two answers only start to meaningfully diverge at high pressure, low temperature, or when the gas in question is one of the more strongly-interacting ones like ammonia, water vapour, or a gas close to its condensation point.
Use this solver when you specifically need to see or demonstrate that deviation — comparing real versus ideal predictions for an assignment, checking how far off the ideal gas law is for a particular gas and condition, or working through an engineering-style problem where the extra accuracy actually matters, such as high-pressure gas storage or industrial gas compression.
Limits of This Model
The Van der Waals equation is itself an approximation, not a perfect description of real gas behaviour — it's simply a large step up in accuracy from the ideal gas law for most everyday conditions. At extremely high pressures, or very close to a gas's critical point, even the Van der Waals equation starts to lose accuracy, and more advanced equations of state (like Redlich-Kwong or Peng-Robinson) are used instead in serious engineering work.
This solver is intended for educational use — comparing real and ideal gas behaviour, understanding what the a and b constants represent, and working through textbook-style Van der Waals problems. For safety-critical or regulatory work involving compressed or liquefied gases, always use validated, industry-standard software and follow your organisation's engineering procedures.
Van der Waals Solver: Quick Reference
The equation is (P + an²/V²)(V − nb) = nRT. Rearranged for pressure: P = nRT/(V − nb) − an²/V². Rearranged for temperature: T = [(P + an²/V²)(V − nb)] / (nR). Volume has no simple rearrangement and is solved numerically.
a corrects for attraction between molecules (units of L²·atm/mol²) and b corrects for the physical volume the molecules themselves occupy (units of L/mol). Both constants are specific to each gas and are usually looked up from a reference table rather than calculated from first principles.
Frequently Asked Questions
What is the Van der Waals equation?
It's a corrected version of the ideal gas law: (P + an²/V²)(V − nb) = nRT, where a accounts for attraction between gas molecules and b accounts for the physical space the molecules take up.
What are typical values for a and b?
They vary by gas — small, weakly-interacting gases like helium have small values (a ≈ 0.03, b ≈ 0.02), while larger or more polar gases like sulfur dioxide or ammonia have much bigger values. This solver includes built-in constants for fourteen common gases.
Why can't volume be solved directly like pressure or temperature?
Volume appears three separate times in the equation once it's expanded, including a cubed term, so there's no simple algebraic rearrangement. This solver uses a numerical method (Newton-Raphson) to find volume instead.
What does the compressibility factor Z mean?
Z = PV/(nRT). Z = 1 means the gas is behaving exactly like an ideal gas. Z below 1 means attractive forces are dominating; Z above 1 means the physical size of the molecules is dominating, usually at high pressure.
Which gases behave closest to ideal?
Small, non-polar, weakly-interacting gases like helium, hydrogen, and neon stay closest to ideal behaviour across a wide range of conditions.
When does the ideal gas law stop being accurate?
It becomes noticeably less accurate at high pressure, low temperature, or for gases that are close to condensing — that's exactly when the Van der Waals correction starts to matter.
Is the Van der Waals equation perfectly accurate?
No — it's a big improvement over the ideal gas law for most conditions, but it's still an approximation. At extreme pressures or near a gas's critical point, more advanced equations of state are used instead.