Real Gases and the van der Waals Equation

The van der Waals equation, (P + an²/V²)(V − nb) = nRT, is a modification of the ideal gas law that accounts for the finite volume of gas molecules and the attractive forces between them, so that it describes real gases more accurately. In brief: 1.00 mol of CO₂ in 1.00 L at 300 K has an ideal pressure of 24.6 atm, but the van der Waals equation gives 22.1 atm, which is closer to the measured value.

Real gases deviate from ideal behavior at high pressure and low temperature, and eventually they condense. The ideal gas law cannot predict this. The van der Waals equation was the first successful equation that could explain both gas and liquid behavior. This article explains the deviations, the meaning of the constants a and b, a worked example and a comparison table.

Why do real gases deviate from ideal behavior?

The ideal gas law assumes that gas particles have no volume and do not attract each other. Both assumptions are wrong for real gases:

• Molecular volume: the molecules take up some of the container volume, so the free space for movement is smaller than V. This effect raises the pressure above the ideal value, and it is largest at high pressure.
• Intermolecular attraction: molecules near the wall are pulled back by their neighbors, so they hit the wall with less force. This effect lowers the pressure below the ideal value, and it is most noticeable at moderate pressures and low temperatures.

Gases behave most ideally at low pressure and high temperature, when the molecules are far apart and move fast. Small nonpolar molecules such as helium and hydrogen are closest to ideal.

What is the van der Waals equation?

Two corrections to the ideal gas law give:

(P + a n2 / V2) (V − n b) = n R T

Solving for the pressure gives the form that is often most convenient:

P = nRT / (V − nb) − a n2 / V2

The term V − nb is the free volume available to the molecules. The term an²/V² is the reduction of pressure caused by attraction; it grows with the square of the concentration because it depends on pairs of molecules.

What do the constants a and b mean?

The constant a measures the strength of intermolecular attraction. The constant b is the volume excluded per mole of molecules, and is roughly four times the actual volume of one mole of molecules. The constants are found by fitting the equation to experimental data, and they are specific to each gas.

The small value of a for helium reflects its very weak attractions, and the larger values for CO₂ and NH₃ reflect stronger forces. The values are given to a precision of three or four digits, and different reference books list slightly different values.

How do you use the van der Waals equation?

Worked example: carbon dioxide in a small container

Calculate the pressure of 1.00 mol of CO₂ in a 1.00 L container at 300 K, using (a) the ideal gas law and (b) the van der Waals equation.

1. Ideal gas: P = nRT / V = 1.00 × 0.08206 × 300 / 1.00 = 24.6 atm.
2. The van der Waals repulsion term: nRT / (V − nb) = 24.618 / (1.00 − 0.0429) = 24.618 / 0.9571 = 25.72 atm.
3. The van der Waals attraction term: an² / V² = 3.658 × 1.00 / 1.00 = 3.66 atm.
4. P = 25.72 − 3.66 = 22.1 atm.

The ideal gas law overestimates the pressure by about 12 %. Here attraction has the stronger effect, so the pressure is lower than the ideal value. At much higher densities, the excluded-volume term dominates, and the real pressure is above the ideal value.

What is the compressibility factor?

The compressibility factor Z = PV / nRT measures the deviation from ideal behavior. For an ideal gas Z = 1 at all conditions. In the example above, Z = 22.1 / 24.6 = 0.90. A value of Z less than 1 shows that attractive forces dominate, and a value more than 1 shows that repulsion dominates. Typically Z falls below 1 as the pressure rises from zero, passes through a minimum and rises above 1 at very high pressures.

Who was van der Waals?

Johannes Diderik van der Waals (1837–1923), a Dutch physicist, was a schoolteacher until he completed his doctoral thesis at the University of Leiden in 1873. In it he proposed that gases and liquids are different states of one continuous substance, and developed the equation that bears his name. The equation predicted the existence of a critical point, which helped chemists to liquefy gases such as oxygen and hydrogen. He received the Nobel Prize in Physics in 1910. The weak intermolecular forces that explain the constant a were later named van der Waals forces.

Which mistakes are common?

• Mixing the units of a: if you use pressure in atmospheres and volume in liters, the units of a are L²·atm/mol².
• Forgetting the factors of n: the corrections involve n²/V² and nb, not just a and b.
• Thinking the equation is exact: it is an approximation and fails close to the critical point and at very high densities.
• Always using it: at normal pressure the ideal gas law is accurate enough for most gases.
• Assuming that the constants are universal: each gas has its own values of a and b.