2.6 Statistical Mechanics of Real Gases

Real gases deviate from ideal behavior due to intermolecular forces and molecule size. These effects become more noticeable at high pressures and low temperatures, impacting gas properties and behavior in ways the ideal gas law can't predict.

Statistical mechanics helps us understand these deviations by tweaking the partition function. This accounts for molecule interactions and excluded volume, leading to more accurate equations of state like the van der Waals equation for real gases.

Real Gas Deviations from Ideal Behavior

Intermolecular Interactions and Molecular Size Effects

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• Real gases deviate from ideal gas behavior due to intermolecular interactions (dispersion forces, dipole-dipole interactions, hydrogen bonding) and the finite size of molecules
• These deviations become more significant at high pressures and low temperatures where intermolecular forces and molecular size effects are more pronounced
• The non-zero volume occupied by the molecules themselves contributes to the deviations from ideal gas behavior
• The compressibility factor, $Z = PV/nRT$, measures the deviation of a real gas from ideal gas behavior
  • For an ideal gas, $Z = 1$, while for real gases, $Z$ can be greater or less than 1, depending on the pressure and temperature

Quantifying Deviations using Statistical Mechanics

• Statistical mechanics quantifies these deviations by modifying the partition function to account for intermolecular interactions and the excluded volume of the molecules
• The virial equation of state is a power series expansion in terms of the molar density that accounts for deviations from ideal gas behavior
  • The coefficients of the expansion, called virial coefficients, are related to the intermolecular interactions between molecules
• The modified partition function can be expressed as a product of the ideal gas partition function and a correction factor that depends on the intermolecular potential energy
  • The correction factor is often approximated using a mean-field approach (van der Waals approximation) which assumes each molecule experiences an average potential energy due to its interactions with all other molecules in the system

Intermolecular Interactions in Real Gases

Types of Intermolecular Interactions

• Intermolecular interactions in real gases include:
  • Dispersion forces (London forces): arise from temporary fluctuations in the electron distribution of molecules, creating instantaneous dipoles
  • Dipole-dipole interactions: occur between molecules with permanent dipole moments (polar molecules like HCl)
  • Hydrogen bonding: a strong type of dipole-dipole interaction involving hydrogen atoms bonded to highly electronegative elements (O, N, F)
• These interactions arise from the electrostatic forces between molecules and can be attractive or repulsive, depending on the distance between the molecules and their relative orientations

Effects on the Partition Function

• The presence of intermolecular interactions modifies the partition function of real gases by introducing additional terms that account for the potential energy of interaction between molecules
• The modified partition function can be expressed as:
  • $Q_\text{real} = Q_\text{ideal} \times Q_\text{interaction}$
  • $Q_\text{ideal}$: partition function for an ideal gas
  • $Q_\text{interaction}$: correction factor accounting for intermolecular interactions
• The correction factor depends on the intermolecular potential energy, which is often modeled using pairwise potentials (Lennard-Jones potential)

Statistical Mechanics for Real Gas Equations of State

Van der Waals Equation of State

• The van der Waals equation is a modified equation of state that accounts for the finite size of molecules and the attractive intermolecular interactions in real gases
• The equation is derived by considering the excluded volume and the mean-field potential energy of interaction between molecules in the partition function
  • Excluded volume: accounted for by subtracting a term proportional to the square of the molar density from the molar volume
  • Attractive interactions: represented by a term that is inversely proportional to the square of the molar volume
• The van der Waals equation is given by:
  • $(P + a/V^2)(V - b) = nRT$
  • $P$: pressure, $V$: volume, $n$: number of moles, $R$: gas constant, $T$: temperature
  • $a$: van der Waals constant measuring the strength of attractive interactions between molecules
  • $b$: van der Waals constant representing the excluded volume per mole of the gas

Limitations and Applications

• The van der Waals equation predicts the behavior of real gases more accurately than the ideal gas equation, particularly at high pressures and low temperatures
  • Accounts for the condensation of gases into liquids and the existence of a critical point (temperature and pressure above which a substance cannot exist as a liquid)
• However, the van der Waals equation still has limitations and may not accurately describe the behavior of gases near the critical point or in the liquid state
• Other equations of state have been developed to improve upon the van der Waals equation, such as the Redlich-Kwong and Peng-Robinson equations, which are widely used in the chemical industry for process design and optimization