Real Gases vs Ideal Gases

The ideal gas model — with its assumptions of point-like particles, no intermolecular forces, and perfectly elastic collisions — is remarkably successful at predicting gas behaviour under everyday conditions. But no real gas is truly ideal. Under certain conditions, real gases deviate significantly from the predictions of pV = nRT. Understanding when and why this happens is important for both exams and practical applications.

Ideal Gas Assumptions — A Reminder

graph LR
    A["Ideal Gas\nAssumptions"] --> B["Negligible\nmolecular volume"]
    A --> C["No intermolecular\nforces"]
    A --> D["Elastic\ncollisions"]
    A --> E["Random\nmotion"]
    B -->|"Breaks down at"| F["HIGH PRESSURE\n(molecules close together)"]
    C -->|"Breaks down at"| G["LOW TEMPERATURE\n(molecules move slowly)"]
    F --> H["Real gas behaviour"]
    G --> H

When Do Ideal Gas Assumptions Break Down?

The two most significant assumptions of the ideal gas model are:

The volume of the molecules is negligible compared to the volume of the container.
There are no intermolecular forces between molecules.

These assumptions are reasonable when the molecules are far apart (low pressure) and moving fast (high temperature). They break down in two key situations:

High Pressure

At high pressures, the gas is compressed into a smaller volume. The molecules are pushed closer together, and:

The volume of the molecules themselves becomes a significant fraction of the container volume. The molecules are no longer negligible point particles — there is a minimum volume below which the gas cannot be compressed, no matter how high the pressure.
The intermolecular forces become significant because the molecules are close enough to attract each other. These attractive forces reduce the pressure below what the ideal gas equation predicts, because molecules approaching the walls are pulled back slightly by attraction from neighbouring molecules.

Low Temperature

At low temperatures, the molecules move slowly. Because they have less kinetic energy:

Intermolecular attractive forces become more significant relative to the kinetic energy. Molecules that approach each other are more likely to be captured by these forces, which reduces the effective collision rate with the walls and hence reduces the pressure.
At sufficiently low temperatures, the attractive forces can cause the gas to condense into a liquid — a phase transition that ideal gas theory cannot predict.

Quantifying Deviations: The Compressibility Factor

The compressibility factor Z provides a quantitative measure of how much a real gas deviates from ideal behaviour:

Z = pV / (nRT)

For an ideal gas, Z = 1 at all conditions. For real gases:

Condition	Z value	Dominant effect
Low pressure, high temperature	Z ≈ 1	Ideal behaviour
Moderate pressure	Z < 1	Attractive forces reduce pressure
Very high pressure	Z > 1	Molecular volume increases effective volume
Near critical point	Z deviates significantly	Both effects compete

For example, nitrogen at 300 K: at 1 atm, Z = 0.9998 (essentially ideal); at 100 atm, Z = 0.984; at 500 atm, Z = 1.24 (molecular volume dominates).

Intermolecular Forces in Real Gases

Real molecules interact through van der Waals forces — weak, short-range attractive forces that arise from temporary dipoles in electron clouds. These forces:

Are attractive at moderate distances (pulling molecules toward each other).
Become strongly repulsive at very short distances (when electron clouds overlap).
Are responsible for the ability of real gases to condense into liquids and solidify into solids.

In an ideal gas, these forces are assumed to be zero. In a real gas, they reduce the pressure (because molecules near the walls are pulled back by neighbours) and reduce the available volume (because each molecule excludes a small region of space around itself).

The strength of intermolecular forces varies with molecular properties:

Factor	Effect on intermolecular forces
More electrons (larger molecules)	Stronger London dispersion forces
Permanent dipole	Additional dipole-dipole forces
Hydrogen bonding (N-H, O-H, F-H)	Much stronger attractive forces
Monatomic noble gases	Weakest forces (London only)

The Van der Waals Equation (Concept Only)

The Dutch physicist Johannes van der Waals proposed a modified equation of state for real gases:

(p + a/V²)(V − b) = nRT

(for one mole of gas, with appropriate modifications for n moles)

where:

a/V² is a correction for intermolecular attractive forces — it increases the effective pressure.
b is a correction for the finite volume of the molecules — it reduces the available volume.
a and b are constants that depend on the specific gas.

Gas	a / Pa m⁶ mol⁻²	b / m³ mol⁻¹	Interpretation
He	0.00346	2.38 × 10⁻⁵	Very weak forces, tiny atoms
H₂	0.0248	2.66 × 10⁻⁵	Weak forces, small molecules
N₂	0.141	3.91 × 10⁻⁵	Moderate forces
CO₂	0.366	4.27 × 10⁻⁵	Stronger forces (polar)
H₂O	0.554	3.05 × 10⁻⁵	Strong hydrogen bonding

You do not need to perform calculations with this equation, but you should understand the concept: the ideal gas equation is modified to account for (1) attractive forces between molecules and (2) the actual volume occupied by the molecules.

Liquefaction and Critical Temperature

When a real gas is cooled sufficiently at high enough pressure, the intermolecular attractive forces overwhelm the kinetic energy of the molecules, and the gas liquefies (condenses).

Every gas has a critical temperature above which it cannot be liquefied by pressure alone, no matter how high the pressure. Below the critical temperature, applying sufficient pressure will cause the gas to condense.

Gas	Critical temperature	Critical pressure / MPa	Ease of liquefaction
Water (H₂O)	647 K (374 °C)	22.1	Easy at room T
Carbon dioxide (CO₂)	304 K (31 °C)	7.38	Possible near room T
Oxygen (O₂)	155 K (−118 °C)	5.04	Requires cooling
Nitrogen (N₂)	126 K (−147 °C)	3.39	Requires significant cooling
Hydrogen (H₂)	33 K (−240 °C)	1.30	Very difficult
Helium (He)	5.2 K (−268 °C)	0.23	Extremely difficult

Helium has an extremely low critical temperature, which is why it is so difficult to liquefy and was the last element to be liquefied (in 1908, by Heike Kamerlingh Onnes).

Ideal gas theory predicts that pV = nRT for all temperatures and pressures, with no possibility of liquefaction. The fact that gases do condense is direct evidence that the ideal model breaks down.

Comparing Real and Ideal Behaviour on pV vs p Diagrams

A useful way to visualise the differences is through a pV vs p diagram.

For an ideal gas: pV = nRT, so pV is constant at constant T. A graph of pV against p is a horizontal line.

For a real gas at moderate temperatures:

At low pressures, pV is slightly less than the ideal value (attractive forces dominate — molecules being pulled together reduces the effective pressure).
At high pressures, pV is greater than the ideal value (molecular volume dominates — the finite size of molecules means the gas cannot be compressed as much as predicted).

At very high temperatures, real gases approximate ideal behaviour more closely because the high kinetic energy makes the intermolecular forces relatively insignificant.

Pressure range	pV compared to ideal	Dominant deviation
Very low	≈ ideal	Neither effect significant
Low to moderate	pV < ideal	Intermolecular attraction
High	pV = ideal (crossover)	Effects balance
Very high	pV > ideal	Molecular volume

Real-World Applications

Understanding real gas behaviour is not just academic:

Industrial gas storage: Gases like oxygen, nitrogen, and CO₂ are stored at high pressures (typically 150–200 atm) where ideal gas calculations would give inaccurate predictions for the amount of gas in a cylinder. The deviation can be 10–20%.
Liquefaction and cryogenics: Producing liquid oxygen (for medical and industrial use), liquid nitrogen (for cryogenics and food freezing), and liquefied natural gas (LNG for transport) all depend on understanding real gas behaviour. The Linde process exploits the Joule-Thomson effect — a real gas effect where gas cools upon expansion through a valve.
Refrigeration and heat pumps: The phase transitions of real refrigerant gases are the basis of all refrigeration and air conditioning systems. These rely on the gas absorbing latent heat during evaporation and releasing it during condensation — behaviour that ideal gas theory cannot describe.
Atmospheric science: Water vapour in the atmosphere condenses to form clouds — behaviour that ideal gas theory cannot explain. Weather prediction models must account for real gas properties of water.
Supercritical fluids: Above the critical temperature and pressure, a substance becomes a supercritical fluid with properties intermediate between liquid and gas. Supercritical CO₂ is used as a solvent in decaffeination of coffee and dry cleaning.

Common Exam Mistakes

Mistake	Correction
Saying "real gases don't obey any gas laws"	Real gases approximate ideal behaviour at low p, high T
Confusing which deviation dominates when	Low/moderate p: attractions dominate (pV < ideal); very high p: volume dominates (pV > ideal)
Saying ideal gases can be liquefied	Ideal gases have no intermolecular forces, so no liquefaction is possible
Forgetting that critical temperature is a maximum	Above T_c, no amount of pressure can liquefy the gas

Summary

Ideal gas assumptions break down at high pressure (molecular volume significant, intermolecular forces matter) and low temperature (intermolecular forces dominate, leading to liquefaction).
Real gas molecules experience attractive van der Waals forces, which reduce the pressure compared to ideal predictions.
The van der Waals equation adds corrections for intermolecular attraction (a/V²) and molecular volume (b).
The compressibility factor Z = pV/(nRT) quantifies deviations: Z < 1 means attractions dominate; Z > 1 means volume dominates.
Every gas has a critical temperature above which it cannot be liquefied by pressure alone.
Ideal gas behaviour is best approximated at low pressure and high temperature.

A-Level Deep Dive: Real vs Ideal Gases

Spec mapping

Edexcel 9PH0 specification Topic 9 — Thermodynamics covers the kinetic theory of gases, the ideal gas equation $pV = nRT$ , the assumptions underlying ideal-gas behaviour, and the conditions under which real gases deviate from those assumptions (refer to the official specification document for exact wording). The same topic requires candidates to relate macroscopic observables — pressure, volume, temperature — to the microscopic motion of molecules, and to interpret deviations from ideality in terms of intermolecular forces and finite molecular volume. Although this lesson sits inside Topic 9, it is examined synoptically: deviations from ideality connect to Topic 4 (Materials, density and molecular structure), Topic 13 (Oscillations, where intermolecular potentials are modelled as quadratic near equilibrium) and the kinetic-theory derivation of $pV = \tfrac{1}{3} N m \overline{c^2}$ . The Edexcel formula booklet provides $pV = NkT$ and $pV = nRT$ , but candidates must memorise the kinetic-model assumptions and the qualitative form of the van der Waals correction.

Worked example with full mark scheme

Question (8 marks):

A sample of carbon dioxide is held at $T = 300\ \text{K}$ and $p = 2.0 \times 10^7\ \text{Pa}$ . Measurements give a molar volume of $V_m = 9.0 \times 10^{-5}\ \text{m}^3\ \text{mol}^{-1}$ .

(a) Calculate the compressibility factor $Z = pV_m / (RT)$ for this sample. Take $R = 8.31\ \text{J}\ \text{mol}^{-1}\ \text{K}^{-1}$ . (3)

(b) State, with reasoning, whether attractive intermolecular forces or the finite volume of the molecules dominates the deviation from ideal behaviour at these conditions. (3)

(c) Suggest one change in conditions under which the same sample of CO $_2$ would behave more closely to an ideal gas, and justify your choice in terms of the kinetic-theory assumptions. (2)

Solution with mark scheme:

(a) Step 1 — substitute into the compressibility expression.

$Z = \dfrac{pV_m}{RT} = \dfrac{(2.0 \times 10^7)(9.0 \times 10^{-5})}{(8.31)(300)}$

M1 — correct substitution of $p$ , $V_m$ , $R$ and $T$ into $Z = pV_m / (RT)$ . Common slip: using $V$ rather than the molar volume $V_m$ , which generates a value that is wrong by a factor of $n$ .

Step 2 — evaluate numerator and denominator.

Numerator: $(2.0 \times 10^7)(9.0 \times 10^{-5}) = 1800\ \text{Pa}\ \text{m}^3\ \text{mol}^{-1}$ . Denominator: $(8.31)(300) = 2493\ \text{J}\ \text{mol}^{-1}$ .

M1 — correct numerical evaluation, with units consistent ( $\text{Pa}\ \text{m}^3 = \text{J}$ ).

Step 3 — final value.

$Z = 1800 / 2493 \approx 0.72$ .

A1 — $Z \approx 0.72$ (accept 0.70–0.73 to 2 s.f.).

(b) Step 1 — interpret $Z < 1$ .

For an ideal gas $Z = 1$ exactly. Here $Z = 0.72 < 1$ , so the measured $pV_m$ product is less than the ideal-gas prediction.

M1 — recognising that $Z < 1$ implies the gas is more compressed than ideal at this $T$ and $p$ .

Step 2 — link to molecular interaction.

A reduction in $pV_m$ relative to ideal indicates that the molecules are pulling on one another — attractive intermolecular forces reduce the rate at which molecules strike the container walls, and so reduce the measured pressure compared with the ideal-gas value.

M1 — attractive forces named as the cause of $Z < 1$ .

A1 — explicit conclusion that attractions dominate over finite-volume effects under these conditions. (Finite-volume effects push $Z > 1$ because excluded volume reduces the space available to the gas; here that effect is being out-competed by attractions.)

Reducing the pressure (e.g. expanding the gas to $p \sim 10^5\ \text{Pa}$ at the same temperature) brings the gas closer to ideal behaviour.

M1 — suggest reducing pressure (or, equivalently, raising $T$ well above the critical temperature).

Step 2 — justify with kinetic-theory assumption.

At low $p$ the mean separation between molecules is much greater than the molecular diameter, so (i) the volume occupied by the molecules themselves is negligible compared with the container volume, and (ii) the time spent within the range of intermolecular forces is small compared with the time spent in free flight. Both ideal-gas assumptions are then well satisfied.

A1 — justification that explicitly references the assumption being restored.

Total: 8 marks (M5 A3, split as shown).

Specimen question modelled on the Edexcel 9PH0 Paper 2 format

Question (6 marks): A student is investigating how a sample of nitrogen behaves as it is compressed isothermally at room temperature.

(a) State two assumptions of the kinetic theory of an ideal gas. (2)

(b) The student finds that, at very high pressures, the product $pV$ for the nitrogen sample increases above the ideal-gas prediction. Explain, in terms of molecular behaviour, why $pV > nRT$ at very high pressures. (4)

Mark scheme decomposition by AO:

(a)

B1 (AO1.1) — any one of: molecules have negligible volume compared with the container; collisions are perfectly elastic; intermolecular forces are negligible except during collisions; molecules are in continuous random motion; time of collision is negligible compared with time between collisions.
B1 (AO1.1) — a second, distinct assumption from the same list.

(b)

M1 (AO1.2) — at very high pressure, molecules are forced close together so the volume occupied by the molecules themselves becomes a significant fraction of the container volume.
M1 (AO2.1) — the available volume for free motion is therefore less than $V$ , so the molecules collide with the walls more frequently than the ideal-gas calculation predicts.
M1 (AO2.1) — this excluded-volume effect raises $p$ above the ideal value, and hence $pV > nRT$ .
A1 (AO2.5) — explicit comparison: at moderate pressures attractive forces would tend to reduce $pV$ below $nRT$ , but at very high pressures the excluded-volume term wins, reversing the sign of the deviation.

Total: 6 marks split AO1 = 3, AO2 = 3. Edexcel uses real-gas questions of this kind to test the synoptic link between assumptions, observations and explanation: AO1 marks reward recall of the assumptions, AO2 marks reward connecting the assumption that has failed to the observed deviation.

Synoptic links

Connects to:

Kinetic theory derivation (Topic 9): the result $pV = \tfrac{1}{3} N m \overline{c^2}$ is derived under the very assumptions that fail for real gases — point particles, no intermolecular forces, elastic collisions only with the walls. Understanding why the derivation works for an ideal gas is exactly what tells you why it must break down when those assumptions fail.
Intermolecular forces (Topic 4 / chemistry crossover): the attractive part of the van der Waals correction is a coarse-grained treatment of dispersion (London) and dipole–dipole forces between molecules. The same forces govern surface tension in liquids, latent heat of vaporisation, and the bulk modulus of a real solid.
Phase transitions and the critical point: when attractive forces are strong enough relative to thermal motion, the gas can condense into a liquid. The critical temperature $T_c$ above which liquefaction by pressure alone is impossible is a direct consequence of the balance between intermolecular attraction and kinetic energy — the very same balance that determines whether a gas behaves ideally.