10.3 Real gas behavior and equations of state

Real gases don't always play by the rules of ideal gas behavior. As pressure rises and temperature drops, intermolecular forces come into play, causing deviations from the ideal gas law.

To accurately model real gases, we need more complex equations. The van der Waals and virial equations account for molecular interactions and size, giving us a better picture of how gases behave in extreme conditions.

Real Gas Behavior

Ideal vs real gas behavior

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• Ideal gas behavior assumes no intermolecular forces between gas molecules (helium at room temperature) and follows the ideal gas law: $PV = nRT$, which is accurate at low pressures (< 1 atm) and high temperatures (> 300 K)
• Real gas behavior considers intermolecular forces between gas molecules (nitrogen at high pressure), deviates from the ideal gas law, especially at high pressures (> 10 atm) and low temperatures (< 100 K), and requires more complex equations of state to accurately model behavior (van der Waals equation)

Intermolecular forces in gases

• Intermolecular forces include attractive and repulsive forces between gas molecules, such as dipole-dipole interactions (HCl), London dispersion forces (CH4), and hydrogen bonding (H2O)
• Strength of intermolecular forces depends on molecular size (larger molecules have stronger forces) and polarity (polar molecules have stronger forces)
• Effects on real gas behavior include causing gas molecules to occupy a non-zero volume, reducing the pressure exerted by the gas compared to ideal gas predictions, and leading to condensation and liquefaction at high pressures and low temperatures (propane in a tank)

Equations of State for Real Gases

Van der Waals equation of state

• van der Waals equation of state modifies the ideal gas law to account for intermolecular forces and molecular size using the equation: $(P + \frac{an^2}{V^2})(V - nb) = nRT$
• Parameters $a$ (attraction) and $b$ (repulsion) are specific to each gas and determined experimentally (a = 1.39 L2atm/mol2 and b = 0.0391 L/mol for CO2)
• Improvements over the ideal gas law include predicting the existence of a critical point and phase transitions and being more accurate at high pressures and low temperatures (predicting liquid-vapor equilibrium)

Virial equation for gases

• Virial equation of state expresses the compressibility factor ($Z$) as a power series in pressure ($Z = 1 + B(T)\frac{P}{RT} + C(T)(\frac{P}{RT})^2 + ...$) or density ($Z = 1 + B(T)\rho + C(T)\rho^2 + ...$)
• Virial coefficients $B(T)$ (second) and $C(T)$ (third) account for two-molecule and three-molecule interactions, respectively, and are determined experimentally or theoretically for each gas (B(300 K) = -123.2 cm3/mol for N2)
• Application involves truncating the virial equation to the desired level of accuracy, using experimental data to determine virial coefficients, and calculating the compressibility factor and other properties using the virial equation (predicting the behavior of natural gas in pipelines)