Ideal gas equation, real gases, and Van der Waals equation

Ideal Gas Equation

1. The ideal gas equation is given by PV = nRT, where:
   • P = Pressure
   • V = Volume
   • n = Number of moles
   • R = Universal gas constant (8.314 J/mol·K)
   • T = Temperature in Kelvin
2. This equation is derived by combining Boyle’s law, Charles’s law, and Avogadro’s law.
3. The ideal gas equation assumes that gases behave perfectly under all conditions.
4. The value of R can vary depending on the units of pressure and volume (e.g., 0.0821 L·atm/mol·K).
5. This equation is useful for calculations involving the mole concept, gas density, and molar mass.

Real Gases

1. Real gases deviate from ideal behavior, especially under high pressure and low temperature.
2. The assumptions of the kinetic theory of gases do not hold true for real gases:
   • Gas molecules have a finite volume.
   • There are intermolecular forces between gas molecules.
3. At high pressures, the volume of gas molecules becomes significant compared to the total volume.
4. At low temperatures, intermolecular attractions affect the gas behavior.
5. Deviations from ideal behavior are measured using the compressibility factor (Z), where Z = PV/nRT:
   • Z = 1: Ideal gas behavior.
   • Z > 1: Gas is less compressible due to repulsive forces.
   • Z < 1: Gas is more compressible due to attractive forces.
6. Real gases approximate ideal behavior at low pressures and high temperatures.
7. Examples: N₂, O₂, and CO₂ show deviations from ideality.

Van der Waals Equation

1. The Van der Waals equation modifies the ideal gas equation to account for the behavior of real gases.
2. It is expressed as:
   • [P + a(n/V)²] [V - nb] = nRT
3. The terms a and b are Van der Waals constants, specific to each gas:
   • a: Accounts for intermolecular attractions.
   • b: Represents the finite volume of gas molecules.
4. The correction for pressure (a(n/V)²) adds the effect of attractive forces.
5. The correction for volume (nb) accounts for the finite size of molecules.
6. The Van der Waals equation reduces to the ideal gas equation at high temperatures and low pressures.
7. The constants a and b are determined experimentally.
8. The equation provides better accuracy for predicting the behavior of gases under extreme conditions.

Key Points

1. The ideal gas equation, PV = nRT, assumes no intermolecular forces and negligible molecular volume.
2. Real gases deviate from ideal behavior due to molecular volume and intermolecular attractions.
3. Compressibility factor (Z) is used to measure the deviation from ideality.
4. The Van der Waals equation introduces corrections for pressure and volume to account for real gas behavior.
5. At high pressure, real gases exhibit repulsive forces leading to Z > 1.
6. At low pressure, attractive forces dominate, causing Z < 1.
7. The constants a and b in the Van der Waals equation vary for different gases.
8. Real gases approach ideal behavior at high temperatures and low pressures.
9. Van der Waals forces include dipole-dipole interactions and dispersion forces.
10. Examples of deviations: CO₂ liquefies under high pressure, showing non-ideal behavior.