3.2 Virial equation of state

The virial equation of state expands on the ideal gas law, accounting for real gas behavior. It uses a power series with virial coefficients to describe how molecules interact at different pressures and temperatures.

Understanding the virial equation helps us grasp how gases deviate from ideal behavior. By looking at the coefficients, we can see how molecular interactions change with temperature and pressure, giving us a more accurate picture of real gases.

Virial Expansion and Coefficients

Virial Expansion Overview

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• Represents a power series expansion of the compressibility factor $Z$ in terms of molar volume $\tilde{V}$ or pressure $P$
• Allows for the calculation of thermodynamic properties of real gases at moderate pressures and densities
• Coefficients in the expansion are called virial coefficients and depend on temperature and the specific gas
• Truncating the series after a certain number of terms provides an approximation of the equation of state

Virial Coefficients

• Second virial coefficient $B(T)$ represents the first correction to ideal gas behavior
  • Accounts for interactions between pairs of molecules
  • Positive values indicate repulsive interactions dominate (Z > 1) while negative values indicate attractive interactions dominate (Z < 1)
• Third virial coefficient $C(T)$ represents the next level of correction
  • Accounts for interactions among three molecules simultaneously
  • Typically has a smaller effect than the second virial coefficient except at high densities
• Higher order coefficients (fourth, fifth, etc.) exist but are often negligible under most conditions

Truncation Considerations

• Truncating the virial expansion after the second term (using only $B(T)$) yields the second virial equation of state
  • Provides a reasonable approximation for many gases at low to moderate pressures
• Including the third virial coefficient $C(T)$ extends the applicable range to higher pressures and densities
• Truncation errors increase at higher pressures and densities where molecular interactions become more significant
• The number of terms needed for accurate results depends on the specific gas and the conditions of interest

Virial Equation Forms

Pressure Explicit Form

• Virial equation written as a series expansion in pressure $P$
  • $Z = 1 + B'P + C'P^2 + D'P^3 + ...$
• Coefficients $B'$, $C'$, $D'$, etc. are related to the virial coefficients $B(T)$, $C(T)$, $D(T)$, etc.
• Useful for describing gas behavior at low to moderate pressures
• Allows for direct calculation of compressibility factor $Z$ from pressure $P$

Volume Explicit Form

• Virial equation written as a series expansion in inverse molar volume $1/\tilde{V}$
  • $Z = 1 + B/\tilde{V} + C/\tilde{V}^2 + D/\tilde{V}^3 + ...$
• Coefficients $B$, $C$, $D$, etc. are the virial coefficients $B(T)$, $C(T)$, $D(T)$, etc.
• Useful for describing gas behavior at low to moderate densities
• Allows for direct calculation of compressibility factor $Z$ from molar volume $\tilde{V}$

Molecular Interactions and Temperature Effects

Molecular Interactions

• Virial coefficients are related to the intermolecular potential energy between molecules
• Second virial coefficient $B(T)$ depends on the pair potential energy $u(r)$ between two molecules
  • Positive values of $B(T)$ indicate predominantly repulsive interactions (hard sphere-like)
  • Negative values of $B(T)$ indicate predominantly attractive interactions (van der Waals-like)
• Third and higher virial coefficients depend on potential energies involving three or more molecules
  • Represent multi-body interactions and are more complex to calculate

Temperature Dependence

• Virial coefficients are functions of temperature $T$
• Second virial coefficient $B(T)$ typically decreases with increasing temperature
  • At high temperatures, kinetic energy dominates and molecular interactions become less significant
  • At low temperatures, attractive interactions become more important and $B(T)$ becomes more negative
• Third and higher virial coefficients also vary with temperature but the dependence is often weaker than for $B(T)$
• The temperature dependence of virial coefficients reflects the changing balance between repulsive and attractive interactions as temperature changes