2.4 Ideal and real gas behavior

Gas behavior is a crucial concept in thermodynamics. Ideal gases follow simple laws, but real gases deviate due to molecular interactions and volume. Understanding these differences helps predict how gases behave under various conditions.

Real gas equations, like Van der Waals and virial equations, account for these deviations. They introduce factors like compressibility and reduced properties, which are essential for accurately describing gas behavior in real-world applications.

Ideal Gas Behavior

Ideal Gas Law and Its Assumptions

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• Ideal gas law relates pressure, volume, temperature, and amount of gas ([PV = nRT](https://www.fiveableKeyTerm:pv_=_nrt))
• Assumes gas molecules have negligible volume compared to the container
• Assumes no intermolecular forces between gas molecules
• Assumes elastic collisions between molecules and container walls
• Assumes average kinetic energy of molecules is proportional to absolute temperature

Deviations from Ideal Behavior

• Real gases deviate from ideal behavior at high pressures and low temperatures
• Deviations occur due to finite volume of gas molecules (affects volume available for motion)
• Deviations also caused by intermolecular forces between gas molecules (affects pressure)
• Critical point is the highest temperature and pressure at which a substance can exist as a liquid and gas in equilibrium
• Compressibility factor (Z) quantifies the deviation of a real gas from ideal behavior ($Z = \frac{PV}{nRT}$)

Real Gas Equations of State

Van der Waals Equation

• Modifies ideal gas law to account for finite molecular volume and intermolecular forces
• Introduces constants a (accounts for intermolecular attraction) and b (accounts for molecular volume)
• Van der Waals equation: $\left(P + \frac{an^2}{V^2}\right)(V - nb) = nRT$
• Predicts the existence of a critical point and liquid-vapor equilibrium
• Provides a more accurate description of real gas behavior than the ideal gas law

Virial Equation

• Expresses the compressibility factor as a power series in terms of pressure or volume
• Virial equation: $Z = 1 + \frac{B(T)}{V} + \frac{C(T)}{V^2} + ...$
• B(T) and C(T) are the second and third virial coefficients, which depend on temperature
• Virial coefficients account for the cumulative effect of intermolecular interactions
• Truncated virial equations (e.g., truncated after the second term) are often used for simplicity

Real Gas Properties

Compressibility Factor and Its Applications

• Compressibility factor (Z) is the ratio of the actual volume of a gas to the volume predicted by the ideal gas law at the same pressure and temperature
• Z = 1 for an ideal gas, Z < 1 for a gas with dominant intermolecular attractions, and Z > 1 for a gas with dominant repulsive interactions
• Compressibility factor is used to correct the ideal gas law for real gas behavior ($PV = ZnRT$)
• Z can be determined experimentally or estimated using generalized compressibility charts (Nelson-Obert charts)

Reduced Properties and the Corresponding States Principle

• Reduced properties are dimensionless quantities that relate the actual properties of a gas to its critical properties
• Reduced pressure: $P_r = \frac{P}{P_c}$, reduced temperature: $T_r = \frac{T}{T_c}$, reduced volume: $V_r = \frac{V}{V_c}$
• The corresponding states principle states that all gases, when compared at the same reduced conditions, have approximately the same compressibility factor
• This principle allows the estimation of real gas properties using generalized charts or equations with reduced properties
• The law of corresponding states is useful for predicting the behavior of gases when limited experimental data is available