Real Gases, van der Waals Equation & Deviations

Real Gases & Deviations | Advanced Thermal Physics

Advanced Physics → Advanced Thermal Physics → Real Gases & Deviations

Comparison between ideal gas and real gas molecular behavior — Ideal gases assume point particles with no interaction, while real gases consist of finite molecules with intermolecular forces.

Quick reference
• Ideal gas assumptions break down at high pressure and low temperature
• Real gases exhibit intermolecular attraction and finite molecular volume
• van der Waals equation corrects the ideal gas law

1. Why Ideal Gas Theory Fails

The kinetic theory of gases treats molecules as point particles with no intermolecular forces except during elastic collisions.

This approximation works well only at low pressure and high temperature.

Real gases deviate from ideal behavior when molecular interactions can no longer be ignored.

These deviations arise from two fundamental microscopic effects.

2. Two Fundamental Causes of Deviation

(a) Intermolecular Forces

At moderate separations, gas molecules attract each other due to van der Waals forces.

This attraction reduces the effective pressure exerted on the container walls.

Attractive forces between gas molecules at short distances — Weak intermolecular attractions become significant at low temperatures and high pressures.

(b) Finite Molecular Size

Molecules occupy finite volume. At high pressure, the free volume available for motion is significantly reduced.

Gas molecules occupying finite volume at high density — At high densities, the finite volume of molecules reduces the free space available for motion.

3. van der Waals Corrections

Pressure Correction

$P_{\text{corrected}} = P + \dfrac{a}{V^2}$

Volume Correction

$V_{\text{corrected}} = V - b$

4. van der Waals Equation of State

$\left(P + \dfrac{a}{V^2}\right)(V - b) = RT$

a measures intermolecular attraction
b accounts for finite molecular volume

Deviation of real gas pressure volume curve from ideal gas behavior — Real gases deviate from ideal behavior at high pressure and low temperature.

5. Physical Interpretation

At low density, correction terms become negligible and the ideal gas law is recovered.

At high density, deviations become significant and ideal gas theory fails completely.

Gas molecules clustering due to intermolecular attraction leading to liquefaction — Intermolecular attraction allows real gases to liquefy under suitable conditions.

6. Critical Phenomena

Density fluctuations in a gas near the critical point — Near the critical point, large density fluctuations appear and gas–liquid distinction vanishes.

When a real gas is compressed and cooled, the distinction between gas and liquid gradually disappears. The specific temperature and pressure at which this happens are called the critical temperature and critical pressure.

At the critical point, the system undergoes a profound change: large-scale density fluctuations appear, and the concept of a sharp gas–liquid boundary loses meaning.

Near this point, the gas becomes highly compressible. Small changes in pressure or temperature produce unusually large changes in density. This behavior cannot be explained using the ideal gas model.

At the critical point, the correlation length of density fluctuations diverges — microscopic interactions influence macroscopic behavior.

From the perspective of van der Waals theory, the critical point corresponds to an inflection point on the pressure–volume isotherm:

$\left(\dfrac{\partial P}{\partial V}\right)_T = 0, \quad \left(\dfrac{\partial^2 P}{\partial V^2}\right)_T = 0$

Physically, this means that the restoring tendency of pressure against volume change vanishes. As a result, the system becomes unstable against density fluctuations.

Critical phenomena reveal a deep limitation of classical thermodynamics. Near the critical point, mean-field models like the van der Waals equation fail to predict experimental behavior accurately. A full understanding requires ideas from statistical mechanics and scaling theory.

7. Where Real Gas Theory Appears Next

Liquefaction of gases
Critical phenomena
Statistical mechanics
Phase transitions

Practice Problems

Level 1 — Conceptual

Why do real gases behave ideally at low pressure?

Solution

Molecules are far apart, so interactions and volume effects are negligible.

What physical effect does parameter a represent?

Solution

Intermolecular attractive forces.

Level 2 — Analytical

What happens to van der Waals equation when $a=0$ and $b=0$ ?

Solution

It reduces to the ideal gas equation $PV = RT$ .

Why does pressure correction scale as $1/V^2$ ?

Solution

Because attraction depends on pairwise molecular interactions.

For CO₂, why is $Z < 1$ at moderate pressures?

Solution

Intermolecular attraction reduces effective pressure on container walls.

Level 3 — Advanced (Physical Reasoning)

Why do real gases liquefy while ideal gases cannot?

Solution

Liquefaction requires attractive forces, absent in ideal gases.

Why does ideal gas theory fail near critical points?

Solution

Density fluctuations and interactions dominate near phase transitions.

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