FYUGP B.Sc Chemistry Semester 1: Unit 7 - Gaseous State

This unit explores the behavior of real gases, deviations from ideal gas behavior, and the equations that describe real gases. Topics include the van der Waals equation, critical phenomena, and the virial equation. These notes are designed to help students understand the complexities of gas behavior and their thermodynamic implications.

1. Causes of Deviation from Ideal Gas Behavior

Real gases deviate from ideal gas behavior due to:

• Intermolecular Forces: Attractive and repulsive forces between gas molecules.
• Finite Molecular Volume: Gas molecules occupy space, unlike ideal gas molecules.

2. Compressibility Factor (Z)

The compressibility factor (Z) measures the deviation of a real gas from ideal gas behavior:

Z = PV / nRT

• For an ideal gas, Z = 1.
• For real gases, Z ≠ 1 (Z < 1 at low pressure, Z > 1 at high pressure).

Fig 1: Variation of Compressibility Factor with Pressure

3. State Variables and Equation of State for Real Gases

State variables (P, V, T) describe the state of a gas. The van der Waals equation is a modified equation of state for real gases:

(P + a(n/V)²)(V - nb) = nRT

• a: Corrects for intermolecular forces.
• b: Corrects for finite molecular volume.

4. Derivation and Application of van der Waals Equation

The van der Waals equation is derived by modifying the ideal gas equation to account for real gas behavior:

• Pressure Correction: Accounts for intermolecular attractions (a(n/V)²).
• Volume Correction: Accounts for the volume occupied by gas molecules (nb).

5. Failure of van der Waals Equation

The van der Waals equation fails under certain conditions:

• High Pressure: Intermolecular forces become negligible, and the volume correction dominates.
• Low Temperature: Gas molecules condense into liquids, violating the assumptions of the equation.

6. Interpretation of van der Waals Pressure-Volume Isotherm

The van der Waals isotherm shows the relationship between pressure and volume at constant temperature:

• Critical Point: The point where the gas and liquid phases become indistinguishable.
• Subcritical Isotherms: Show regions of gas, liquid, and coexistence.

Fig 2: van der Waals Pressure-Volume Isotherm

7. Critical State and Phenomena

The critical state is characterized by:

• Critical Temperature (T_c): The temperature above which a gas cannot be liquefied.
• Critical Pressure (P_c): The pressure required to liquefy a gas at T_c.
• Critical Volume (V_c): The volume occupied by one mole of gas at T_c and P_c.

8. Relation Between Critical Constants and van der Waals Constants

The critical constants are related to the van der Waals constants (a and b) as follows:

T_c = 8a / 27Rb

P_c = a / 27b²

V_c = 3b

9. Introduction to Virial Equation

The virial equation describes real gas behavior using a power series expansion:

PV = nRT [1 + B(T)/V + C(T)/V² + ...]

• B(T), C(T): Virial coefficients that depend on temperature.

10. Derivation of Boyle Temperature

The Boyle temperature (T_B) is the temperature at which a real gas behaves like an ideal gas over a range of pressures:

T_B = a / Rb

Practical Applications

• Gas Liquefaction: Understanding critical phenomena is essential for gas liquefaction processes.
• Industrial Applications: Real gas behavior is crucial in industries like petroleum and chemical engineering.
• Thermodynamic Calculations: The van der Waals and virial equations are used in thermodynamic modeling.

These notes provide a comprehensive understanding of the gaseous state and real gas behavior. Practice problems and thermodynamic calculations will help reinforce these concepts.