Unit 2: Gaseous State II (Real Gases)

Behaviour of Real Gases
Van der Waals Equation of State
Isotherms and Critical State
Law of Corresponding States and Liquefaction

Behaviour of Real Gases

Deviations from Ideal Gas Behaviour

An ideal gas obeys the equation PV = nRT at all conditions. Real gases only obey this law at low pressure and high temperature. At high pressure or low temperature, they show significant deviations.

Compressibility Factor (Z)

The extent of deviation is expressed by the compressibility factor, Z.

Formula: Z = (PV_real) / (nRT)

For an Ideal Gas, Z = 1 always.
For a Real Gas, Z ≠ 1.

Variation of Z with Pressure

At very low pressures: All gases approach Z = 1 (behave ideally).
At moderate pressures (Negative Deviation): For most gases (like N₂, CO₂), Z < 1. This means the gas is more compressible than ideal.
Reason: Intermolecular forces of attraction are dominant, "pulling" the molecules together and reducing the volume.
At high pressures (Positive Deviation): For all gases, Z > 1. This means the gas is less compressible than ideal.
Reason: Molecular volume (repulsive forces) is dominant. The molecules themselves occupy space, so the "free volume" is less than the container volume.
Gases like H₂ and He: These small molecules have very weak attractive forces, so they show Z > 1 (positive deviation) at almost all pressures (except at very low T).

Causes of Deviation from Ideal Behaviour

Deviations arise because two postulates of the Kinetic Theory of Gases are incorrect for real gases:

Faulty Postulate 1: "The volume of gas molecules is negligible."
Correction: This is false at high pressure, where molecules are crowded and their own volume becomes significant. (Leads to Z > 1).
Faulty Postulate 2: "There are no intermolecular forces of attraction."
Correction: This is false at low temperature, where molecules move slowly and attractive forces become significant. (Leads to Z < 1).

Van der Waals Equation of State

Van der Waals modified the ideal gas equation (P_idealV_ideal = nRT) by introducing two correction terms, 'a' and 'b', to account for the faulty postulates.

Derivation of the Equation

Volume Correction:
- The molecules themselves occupy a finite volume, so the "free space" available for movement is not V, but (V - nb).
- V_ideal = V_container - nb
- b is the "excluded volume" per mole, related to the actual molecular size.
Pressure Correction:
- Molecules at the wall experience an inward pull from molecules behind them, reducing the force of their impact.
- This "pressure deficit" is proportional to the number of attracting molecules and the number of attracted molecules, both of which are proportional to density (n/V).
- Pressure correction term \propto (n/V) × (n/V) = an²/V².
- P_ideal = P_observed + (an²) / (V²)
- a is a constant that measures the strength of intermolecular attraction.

Substituting these corrections into P_idealV_ideal = nRT gives:

Van der Waals Equation: ( P + (an²) / (V²) ) (V - nb) = nRT
For 1 mole: ( P + (a) / (V_m²) ) (V_m - b) = RT

Application in Explaining Real Gas Behaviour

At Low Pressure: V is large, so b is negligible compared to V. The term a/V² is small but significant.
( P + (a) / (V_m²) ) V_m ≈ RT ⇒ PV_m + (a) / (V_m) ≈ RT
PV_m ≈ RT - (a) / (V_m)
Z = (PV_m) / (RT) ≈ 1 - (a) / (RTV_m). This correctly predicts Z < 1 (negative deviation) due to attractions.
At High Pressure: P is large, so a/V_m² is negligible compared to P. The term b is significant.
P(V_m - b) ≈ RT ⇒ PV_m - Pb ≈ RT
PV_m ≈ RT + Pb
Z = (PV_m) / (RT) ≈ 1 + (Pb) / (RT). This correctly predicts Z > 1 (positive deviation) due to molecular volume.

Isotherms and Critical State

PV Isotherm of Carbon Dioxide

Andrew's experiments on CO₂ showed P-V relationships at constant temperature (isotherms).

Above T_c (e.g., 50°C): The curve looks like an ideal gas (Boyle's Law).
Below T_c (e.g., 21°C):
- At high volume (gas phase), pressure increases as V decreases.
- At a certain pressure (vapour pressure), the gas begins to liquefy (horizontal line). Here, P is constant as V decreases (liquid and gas coexist).
- Once all gas is liquid, the curve becomes very steep (liquids are incompressible).
At T_c (31.1°C for CO₂): The horizontal liquefaction part reduces to a single point, the critical point.

Critical State

The critical point is a unique state for every substance, defined by three critical constants:

Critical Temperature (T_c): The temperature above which a gas cannot be liquefied, no matter how much pressure is applied.
Critical Pressure (P_c): The minimum pressure required to cause liquefaction at the critical temperature.
Critical Volume (V_c): The volume occupied by one mole of the substance at T_c and P_c.

Relation between Critical Constants and van der Waals Constants

The van der Waals equation can be used to derive mathematical relationships for the critical constants. (Derivation not required by syllabus, but results are important).

Formulas:

V_c = 3b

P_c = (a) / (27b²)

T_c = (8a) / (27Rb)

These can be used to calculate the vdW constants 'a' and 'b' from experimentally measured critical constants.

Law of Corresponding States and Liquefaction

Law of Corresponding States

This law states that if two or more gases are at the same reduced temperature and reduced pressure, they will have the same reduced volume.

Reduced Variables are dimensionless quantities found by dividing the actual variable by its critical constant:

Reduced Pressure: P_r = P / P_c
Reduced Volume: V_r = V_m / V_c
Reduced Temperature: T_r = T / T_c

When the vdW equation is rewritten in these terms, it becomes:

Reduced Equation of State: ( P_r + (3) / (V_r²) ) (3V_r - 1) = 8T_r

Significance: This new equation contains no 'a' or 'b' constants. It is a universal equation that applies to all real gases. It implies that all gases behave identically when in their "corresponding states".

Liquefaction of Gas

To liquefy a gas, it must be cooled below its critical temperature (T_c) and then compressed. Methods of cooling include:

Joule-Thomson Effect: The cooling of a real gas (except H₂, He at room temp) when it expands from a region of high pressure to low pressure through a porous plug or valve. This is the principle of the Linde process.
Adiabatic Expansion: Cooling a gas by making it do external work (e.g., pushing a piston). This is the principle of the Claude process.

Inversion Temperature (T_i)

Definition: Every gas has an inversion temperature.

If T_gas < T_i, the gas cools upon Joule-Thomson expansion (e.g., N₂, O₂ at room temp).

If T_gas > T_i, the gas heats up upon Joule-Thomson expansion (e.g., H₂, He at room temp).

This is why H₂ and He must be pre-cooled below their T_i (which are very low) before they can be liquefied by the Joule-Thomson effect.

Formula: T_i = (2a) / (Rb) (This is (27) / (4) times T_c)