Van der Waals Equation Calculator: Real Gas Pressure or Volume, Compressibility Factor and Critical Point

Real gas behaviour compared with the ideal gas law.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

Choose the gas and whether you want the pressure or the volume.

Enter the amount of gas, the temperature, and the volume or the pressure.

The calculator gives the real answer, the ideal gas answer, the compressibility factor and the critical point.

Formulas used

  • \left(P + \frac{a n^2}{V^2}\right)(V - nb) = nRT
  • Z = \frac{PV}{nRT}
  • T_c = \frac{8a}{27Rb},\ P_c = \frac{a}{27b^2}

Worked example

With the default numbers in the calculator above:

  1. Real gas pressure from the van der Waals equation: 4907.2 kPa
  2. Ideal gas pressure for comparison: 5820.1 kPa
  3. Compressibility factor: 0.84314
  4. How far the ideal gas law is out: 18.6 %
  5. Critical temperature of this gas: 304 K
  6. Critical pressure of this gas: 7.4044 MPa

Change the numbers in the calculator to see the full working, every step, for your own case.

A second worked example

A different case, with these numbers: What do you want to find? Volume, from the pressure and temperature; Pressure (used when you find the volume) 5000 kPa.

  1. Real gas volume from the van der Waals equation: 0.48873 L
  2. Ideal gas volume for comparison: 0.58201 L
  3. Compressibility factor: 0.83972
  4. How far the ideal gas law is out: 19.09 %
  5. Critical temperature of this gas: 304 K
  6. Critical pressure of this gas: 7.4044 MPa

Common questions

What do a and b mean in the van der Waals equation?

The constant a allows for the attraction between molecules and b for the volume the molecules take up themselves.

What is the compressibility factor?

Z is the real gas volume divided by the ideal gas volume. It is 1 for an ideal gas, below 1 when attraction dominates and above 1 when molecule size dominates.

When does the ideal gas law fail?

At high pressure and low temperature, close to the point where the gas condenses.

Related calculators

Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Any chemistry or thermodynamics text (real gases, the van der Waals equation).

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