Real Gases

An ideal gas is a theoretical construct in which gas molecules occupy no volume and exert no intermolecular forces on one another. While this model is mathematically convenient and works reasonably well at low pressures and high temperatures, real gases — every gas that actually exists — deviate from this ideal behavior under many common conditions. These deviations arise because real molecules do have finite size and do attract or repel one another, effects that become increasingly significant as pressure rises or temperature falls. The compressibility factor, Z, is the key quantitative measure of how much a real gas deviates from ideal behavior. Defined as Z = PV/nRT, it equals exactly 1 for a perfect ideal gas. When Z is less than 1, intermolecular attractions dominate and the gas is more compressed than expected. When Z is greater than 1, repulsive interactions and finite molecular volume dominate, making the gas less compressible than an ideal gas would be. Plotting Z against pressure at constant temperature produces characteristic curves that are unique fingerprints of each real gas. The van der Waals equation, proposed by Dutch physicist Johannes Diderik van der Waals in 1873, was the first successful attempt to modify the ideal gas law to account for real-gas behavior. It introduces two empirical constants — a, which corrects for intermolecular attractions, and b, which corrects for the excluded volume occupied by the molecules themselves. Understanding real gases is essential not only for theoretical chemistry but also for industrial processes such as the liquefaction of gases, the design of high-pressure reactors, and the storage of compressed fuels such as natural gas.

Key takeaways

The Ideal Gas Assumptions and Their Limits
Effect of Intermolecular Attractions (Low to Moderate Pressure)
Effect of Finite Molecular Volume (High Pressure)
The van der Waals Equation
Boyle Temperature

The Ideal Gas Assumptions and Their Limits

The ideal gas law, PV = nRT, rests on two central assumptions: (1) gas molecules are point masses with zero volume, and (2) there are no intermolecular forces between molecules. At very low pressures (molecules far apart) and very high temperatures (high kinetic energy overwhelming any attractions), these assumptions hold reasonably well. However, as pressure increases to hundreds of atmospheres or temperature drops toward the boiling point, both assumptions break down noticeably.

Effect of Intermolecular Attractions (Low to Moderate Pressure)

At moderate pressures, the dominant deviation comes from intermolecular attractions — London dispersion forces, dipole–dipole interactions, and hydrogen bonding. These attractive forces pull molecules toward one another, reducing the effective pressure the gas exerts on the container walls. As a result, the measured volume is smaller than the ideal gas law predicts, giving Z < 1. This is why gases like CO₂ and NH₃, which have significant intermolecular attractions, show pronounced negative deviations from ideality even at modest pressures.

Effect of Finite Molecular Volume (High Pressure)

At very high pressures, molecules are forced so close together that their own physical size becomes significant. The actual free space available for molecular motion is less than the total container volume V, because each molecule excludes a certain volume (the co-volume, related to van der Waals constant b) to all other molecules. This makes the actual volume larger than predicted by the ideal gas law, pushing Z above 1. For gases with large molecules, this positive deviation dominates even at relatively modest pressures.

The van der Waals Equation

Van der Waals modified the ideal gas law with two correction terms. The pressure correction term, a/V_m², accounts for the reduction in effective pressure due to intermolecular attractions (where V_m is the molar volume). The volume correction term, b, subtracts the excluded volume per mole of gas. The corrected equation is: (P + a/V_m²)(V_m − b) = RT. Values of a and b are determined experimentally and tabulated for each gas. For example, for CO₂: a = 3.640 L²·atm·mol⁻², b = 0.04267 L·mol⁻¹; for H₂: a = 0.2444 L²·atm·mol⁻², b = 0.02661 L·mol⁻¹. The small a value for H₂ reflects its weak dispersion forces, explaining why hydrogen shows positive deviations (Z > 1) even at relatively low pressures.

Boyle Temperature

At a special temperature called the Boyle temperature (T_B = a/Rb), the repulsive and attractive deviations cancel each other out over a range of moderate pressures, and the gas behaves nearly ideally over that range. For nitrogen, T_B ≈ 327 K; for CO₂, T_B ≈ 710 K. Above the Boyle temperature, Z > 1 across most pressures; below it, Z dips below 1 at moderate pressures before rising again at very high pressures.

The Key Equation

\left(P + \frac{a}{V_m^2}\right)\left(V_m - b\right) = RT

P — Pressure (atm or Pa): The measured pressure of the real gas.
a — Van der Waals attraction constant (L²·atm·mol⁻²): Empirical constant accounting for intermolecular attractions; larger a means stronger attractions.
V_m — Molar volume (L·mol⁻¹): Volume occupied by one mole of the gas.
b — Van der Waals volume constant (L·mol⁻¹): Empirical constant representing the excluded volume per mole due to finite molecular size.
R — Universal gas constant (L·atm·mol⁻¹·K⁻¹): R = 0.08206 L·atm·mol⁻¹·K⁻¹.
T — Absolute temperature (K): Temperature in Kelvin.

Worked Example

Problem. Calculate the compressibility factor Z for 1 mole of CO₂ at T = 300 K and P = 10.0 atm, using the van der Waals equation (a = 3.640 L²·atm·mol⁻², b = 0.04267 L·mol⁻¹, R = 0.08206 L·atm·mol⁻¹·K⁻¹). Compare this to ideal behavior.

Z = \frac{PV_m}{RT} = \frac{10.0 \times 2.353}{0.08206 \times 300} \approx 0.956

Answer. Z ≈ 0.956, indicating CO₂ behaves non-ideally with attractive forces dominating (Z < 1).

Figure

Diagram

Origins · 1 minJohannes Diderik van der Waals and how this was discovered

Johannes Diderik van der Waals

1837–1923 · Dutch

Van der Waals derived the equation of state for real gases in his 1873 doctoral thesis, introducing the correction terms for intermolecular attractions (constant a) and finite molecular volumes (constant b). This was the first molecular-scale theoretical explanation for the liquefaction of gases and the existence of a critical point. He was awarded the Nobel Prize in Physics in 1910 for this work. Van der Waals had no formal education until his mid-twenties and taught himself physics largely through self-study. He only obtained his PhD at age 36, yet his thesis immediately impressed James Clerk Maxwell, who praised it enthusiastically in the scientific press.

Ideal Gas Law — The ideal gas law PV = nRT is the baseline from which real gas deviations are measured; understanding it is prerequisite to understanding Z.
Kinetic Molecular Theory — KMT provides the molecular-level assumptions (point masses, no forces) that define ideal behavior; real gases violate these assumptions.
Intermolecular Forces — London dispersion, dipole–dipole, and hydrogen bonding are the intermolecular forces responsible for negative deviations (Z < 1) in real gases.
Liquefaction of Gases — Real gas behavior, especially the role of intermolecular attractions at high pressure, directly explains the conditions under which gases can be liquefied.
Critical Point and States of Matter — The critical temperature and pressure, predicted by van der Waals theory, mark the boundary above which a gas cannot be liquefied regardless of pressure.