Ideal and Real Gases

Why it matters

Understanding the behavior of ideal and real gases is crucial in thermodynamics as it helps in designing and analyzing systems like engines, refrigerators, and air conditioners. These concepts are fundamental in predicting how gases will react under different conditions, which is essential for efficient energy use and system optimization.

Key ideas

• Ideal Gas: An ideal gas is a hypothetical gas that perfectly follows the ideal gas law, where interactions between molecules are negligible, and the volume of the molecules themselves is much smaller than the volume the gas occupies.
• Real Gas: Real gases deviate from ideal behavior due to intermolecular forces and the finite volume of molecules, especially at high pressures and low temperatures.
• Ideal Gas Law: The equation PV = nRT describes the state of an ideal gas, where P is pressure, V is volume, n is the number of moles, R is the universal gas constant, and T is temperature.
• Van der Waals Equation: This equation accounts for the behavior of real gases by introducing constants a and b to correct for intermolecular forces and molecular volume, respectively.
• Compressibility Factor (Z): Used to quantify how much a real gas deviates from ideal behavior, defined as Z = PV/nRT.

Use absolute pressure and absolute temperature in equations of state. The mass form is PV = m R_specific T with R_specific = R_universal/M, using compatible molar-mass units. The van der Waals equation is an approximate model; accurate real-fluid calculations may need a validated equation of state. Z near one indicates near-ideal volumetric behavior at the state considered.

Formulas

• Ideal Gas Law: PV = nRT
  • P: Pressure (Pa)
  • V: Volume (m³)
  • n: Number of moles (mol)
  • R: Universal gas constant (8.314 J/(mol·K))
  • T: Temperature (K)
• Van der Waals Equation: (P + a(n/V)²)(V - nb) = nRT
  • a: Measure of attraction between particles (Pa·m⁶/mol²)
  • b: Volume occupied by particles (m³/mol)
• Compressibility Factor: Z = PV/nRT

Worked example

Problem: Calculate the pressure exerted by 2 moles of a gas at 300 K occupying a volume of 0.05 m³ using the ideal gas law.

Given:

• n = 2 mol
• T = 300 K
• V = 0.05 m³
• R = 8.314 J/(mol·K)

1. Use the ideal gas law: PV = nRT
2. Rearrange to find P: P = nRT/V
3. Substitute the values: P = (2 mol)(8.314 J/(mol·K))(300 K) / 0.05 m³
4. Calculate: P = 99768 Pa

Answer: 99768 Pa

Common mistakes

• Confusing the units of the gas constant R.
• Applying the ideal gas law to conditions where real gas behavior is significant.
• Forgetting to convert temperature to Kelvin.

For GATE ME

Questions often involve calculating properties of gases using the ideal gas law or Van der Waals equation. Practice problems involving conversions between different units and conditions where real gas behavior is significant.

Quick check

1. What is the ideal gas law equation?
2. How does the Van der Waals equation modify the ideal gas law?
3. What does the compressibility factor indicate?

Answers: 1. PV = nRT; 2. It introduces constants a and b to account for intermolecular forces and molecular volume; 3. It indicates the deviation of a real gas from ideal behavior.