Theoretical Background

This section provides the theoretical foundation for cubic equations of state and their implementation in sandlercubics. The presentation follows closely the treatment found in Chemical, Biochemical, and Engineering Thermodynamics by Stan Sandler [1].

Cubic Equations of State

Cubic equations of state are mathematical models that relate pressure, temperature, and molar volume of a pure substance or mixture. They are called “cubic” because they result in a cubic equation expressed in terms of compressibility factor \(Z\):

\[Z \equiv \frac{Pv}{RT}\]

Each cubic equation can be rewritten in terms of \(Z\):

\[Z^3 + pZ^2 + qZ + r = 0\]

where \(p, q, r\) are equation-specific functions of temperature and pressure. The three possible classes of solutions to the cubic equation depend on the thermodynamic state:

  1. Single real root (supercritical or single-phase states)

  2. Three real roots (two-phase region below critical point)

When three real roots exist, they correspond to:

  1. Vapor phase (largest Z)

  2. Liquid phase (smallest positive Z)

  3. Non-physical root (intermediate Z, discarded)

Van der Waals Equation

The van der Waals equation (1873) is the original and simplest cubic equation of state:

\[P = \frac{RT}{v-b} - \frac{a}{v^2}\]

Parameters

\[a = \frac{27R^2T_c^2}{64P_c}\]
\[b = \frac{RT_c}{8P_c}\]

where \(T_c\) and \(P_c\) are the critical temperature and pressure. When cast as a cubic in \(Z\), the van der Waals equation becomes:

\[Z^3 - (1+B)Z^2 + AZ - AB = 0\]

where:

\[A = \frac{aP}{R^2T^2}\]

and

\[B = \frac{bP}{RT}\]

By convention, all cubics use the same formulations of \(A\) and \(B\) for consistency.

Peng-Robinson Equation

The Peng-Robinson equation (1976) is widely used in the petroleum and gas industries:

\[P = \frac{RT}{v-b} - \frac{a\alpha(T)}{v(v+b) + b(v-b)}\]

Parameters

\[a = 0.45724\frac{R^2T_c^2}{P_c}\]
\[b = 0.07780\frac{RT_c}{P_c}\]
\[\alpha(T) = \left[1 + \kappa\left(1 - \sqrt{T_r}\right)\right]^2\]
\[\kappa = 0.37464 + 1.54226\omega - 0.26992\omega^2\]

where:

  • \(T_r = T/T_c\) is the reduced temperature

  • \(\omega\) is the acentric factor

When expressed as a cubic in \(Z\), the Peng-Robinson equation becomes:

\[Z^3 - (1-B)Z^2 + (A - 3B^2 - 2B)Z - (AB - B^2 - B^3) = 0\]

Soave-Redlich-Kwong Equation

The Soave-Redlich-Kwong (SRK) equation (1972) modifies the original Redlich-Kwong equation:

\[P = \frac{RT}{v-b} - \frac{a\alpha(T)}{v(v+b)}\]

Parameters

\[a = 0.42748\frac{R^2T_c^2}{P_c}\]
\[b = 0.08664\frac{RT_c}{P_c}\]
\[\alpha(T) = \left[1 + m\left(1 - \sqrt{T_r}\right)\right]^2\]
\[m = 0.480 + 1.574\omega - 0.176\omega^2\]

When cast as a cubic in \(Z\), the SRK equation becomes:

\[Z^3 - (1-B)Z^2 + (A - B - B^2)Z - AB = 0\]

Departure Functions

Departure functions quantify the difference between real and ideal gas properties.

Enthalpy Departure

\[H^{dep} = H^{real} - H^{ig} = \int_\infty^v \left[T\left(\frac{\partial P}{\partial T}\right)_v - P\right] dv\]

This integral can be evaluated analytically for each cubic equation.

Entropy Departure

\[S^{dep} = S^{real} - S^{ig} = \int_\infty^v \left[\left(\frac{\partial P}{\partial T}\right)_v - \frac{R}{v}\right] dv\]

Phase Equilibrium

At vapor-liquid equilibrium, the cubic equation has three real roots. The equilibrium condition is:

\[f^V = f^L\]

where \(f\) is the fugacity. For cubic equations:

\[\ln \phi = \frac{1}{RT}\int_\infty^v \left[P - \frac{RT}{v}\right] dv - \ln Z\]

where \(\phi = f/P\) is the fugacity coefficient.

Implementation Notes

Numerical Methods

sandlercubics uses two main numerical techniques:

  1. Root finding: Analytical solution of cubic equation or iterative methods

  2. Saturation calculations: Iterative solution of fugacity equality as explained in Chapter 7 of Sandler [1]

Convergence

Near the critical point, cubic equations become numerically challenging:

  • Multiple roots converge

  • Small changes in T or P cause large changes in properties

  • Derivatives become very large

Limitations

Users should be aware of fundamental limitations:

  1. Polar compounds: Cubic equations are less accurate for highly polar substances (water, alcohols, acids)

  2. Associating fluids: Hydrogen bonding effects are not captured

  3. Critical region: Accuracy decreases near the critical point

  4. Quantum effects: Not applicable to cryogenic helium or hydrogen at very low temperatures

  5. Mixtures: Current implementation is for pure substances only

References

  1. Sandler, S. I. (2017). Chemical, Biochemical, and Engineering Thermodynamics (5th ed.). Wiley.

  2. van der Waals, J. D. (1873). “Over de Continuiteit van den Gas- en Vloeistoftoestand”. PhD thesis, Leiden University.

  3. Peng, D. Y., & Robinson, D. B. (1976). “A New Two-Constant Equation of State”. Industrial & Engineering Chemistry Fundamentals, 15(1), 59-64.

  4. Soave, G. (1972). “Equilibrium constants from a modified Redlich-Kwong equation of state”. Chemical Engineering Science, 27(6), 1197-1203.

  5. Redlich, O., & Kwong, J. N. S. (1949). “On the Thermodynamics of Solutions. V. An Equation of State. Fugacities of Gaseous Solutions”. Chemical Reviews, 44(1), 233-244.

Further Reading

For deeper understanding of thermodynamic theory and cubic equations:

  • Reid, R. C., Prausnitz, J. M., & Poling, B. E. (1987). The Properties of Gases and Liquids (4th ed.). McGraw-Hill.

  • Elliott, J. R., & Lira, C. T. (2012). Introductory Chemical Engineering Thermodynamics (2nd ed.). Prentice Hall.

  • Poling, B. E., Prausnitz, J. M., & O’Connell, J. P. (2001). The Properties of Gases and Liquids (5th ed.). McGraw-Hill.