Ideal gas laws simplify gas behavior, assuming no particle interactions and negligible volume. They're great for quick calculations but have limitations. Real gases deviate from ideal behavior due to intermolecular forces and finite particle size, especially at high pressures and low temperatures.

Understanding both ideal and real gas behavior is crucial for thermodynamics. We'll explore how to apply ideal gas laws, their limitations, and methods for modeling real gas behavior. This knowledge is essential for accurately predicting gas properties in various engineering applications.

Ideal Gas Law Applications

Solving Problems with the Ideal Gas Law

• The ideal gas law is PV = nRT, where P is pressure, V is volume, n is number of moles, R is the ideal gas constant, and T is temperature in Kelvin
• The ideal gas law assumes that gas particles have negligible volume, no intermolecular forces, and perfectly elastic collisions
• Apply the ideal gas law to solve problems involving pressure, volume, and temperature by substituting known values and solving for the unknown variable
• Example: Calculate the volume of 2 moles of an ideal gas at 300 K and 1 atm pressure using the ideal gas law

Gas Laws Derived from the Ideal Gas Law

• Boyle's law states that pressure and volume are inversely proportional at constant temperature and amount of gas (P₁V₁ = P₂V₂)
• Charles' law states that volume and temperature are directly proportional at constant pressure and amount of gas (V₁/T₁ = V₂/T₂)
• Gay-Lussac's law states that pressure and temperature are directly proportional at constant volume and amount of gas (P₁/T₁ = P₂/T₂)
• The combined gas law (P₁V₁/T₁ = P₂V₂/T₂) relates pressure, volume, and temperature changes for a fixed amount of gas
• Example: Use Boyle's law to calculate the final volume of a gas initially at 2 atm and 1 L when the pressure is increased to 4 atm at constant temperature

Ideal Gas Law Limitations

Intermolecular Forces and Particle Volume

• The ideal gas law assumes no intermolecular forces, but real gases experience attractive and repulsive forces that affect their behavior, especially at high pressures and low temperatures
• The ideal gas law assumes negligible particle volume, but real gas particles occupy a finite volume, which becomes significant at high pressures
• Example: Hydrogen gas deviates from ideal behavior at high pressures due to significant intermolecular forces and particle volume effects

Phase Transitions and Compressibility Factor

• Real gases may undergo phase transitions (condensation or liquefaction) at certain conditions, which the ideal gas law does not account for
• The compressibility factor (Z) is a measure of the deviation of a real gas from ideal behavior, with Z = 1 for an ideal gas and Z ≠ 1 for real gases
• Example: Carbon dioxide gas can condense to a liquid at high pressures and low temperatures, which the ideal gas law cannot predict

Real Gas Behavior Modeling

Compressibility Factor and Reduced Properties

• The compressibility factor (Z) is defined as Z = PV/(nRT), where Z = 1 for an ideal gas and Z ≠ 1 for real gases
• Compressibility factor charts, such as the generalized compressibility chart, plot Z as a function of reduced pressure (P_r) and reduced temperature (T_r)
• Reduced properties are defined as the ratio of the actual property to the critical property value (e.g., P_r = P/P_c and T_r = T/T_c)
• Example: Use a generalized compressibility chart to determine the compressibility factor of nitrogen gas at a given reduced pressure and temperature

Equations of State for Real Gases

• Equations of state, such as the van der Waals equation and the Redlich-Kwong equation, modify the ideal gas law to account for the effects of intermolecular forces and particle volume
• The van der Waals equation is (P + a(n/V)²)(V - nb) = nRT, where a and b are constants specific to the gas, accounting for intermolecular attractions and particle volume, respectively
• Example: Apply the van der Waals equation to calculate the pressure of carbon dioxide gas at a given volume and temperature, considering intermolecular forces and particle volume

Ideal vs Real Gas Behavior

Pressure and Temperature Effects

• At low pressures and high temperatures, real gases behave more like ideal gases due to reduced intermolecular forces and particle volume effects
• At high pressures and low temperatures, real gases deviate significantly from ideal behavior due to increased intermolecular forces and particle volume effects
• Example: Nitrogen gas behaves more like an ideal gas at room temperature and atmospheric pressure compared to high pressures and cryogenic temperatures

Joule-Thomson Effect and Inversion Temperature

• The Joule-Thomson effect describes the temperature change of a real gas during throttling (adiabatic expansion through a porous plug), which is zero for an ideal gas
• Real gases may exhibit a temperature inversion during the Joule-Thomson process, known as the Joule-Thomson inversion temperature
• Example: Helium gas cools upon throttling at room temperature, while nitrogen gas warms up due to their different Joule-Thomson inversion temperatures

Boyle Temperature and Critical Properties

• The Boyle temperature is the temperature at which a real gas behaves most like an ideal gas over a range of pressures
• Critical properties (critical temperature, pressure, and volume) mark the point beyond which a substance cannot exist as a liquid, regardless of pressure, and the distinction between gas and liquid phases disappears
• Example: Carbon dioxide has a critical temperature of 304.13 K and a critical pressure of 7.38 MPa, above which it exists as a supercritical fluid