7.3 Equations of State for Real Gases

Ideal Gas vs Real Gas Behavior

Assumptions and Limitations of the Ideal Gas Equation

The ideal gas equation $PV = nRT$ rests on two assumptions: gas molecules have zero volume and exert no forces on each other. Neither is true for real gases, and the errors grow as conditions push molecules closer together.

Deviations from ideal behavior become significant when:

• Pressure is high — molecules are forced close enough for repulsive and attractive forces to matter.
• Temperature is low — molecules move slowly enough that attractive forces can "pull" them together.
• The gas is near its critical point — the distinction between liquid and gas phases vanishes, and intermolecular effects are at their strongest.

The type of molecule also matters. Larger molecules (long-chain hydrocarbons), polar molecules (water vapor), and molecules capable of hydrogen bonding (alcohols) all show bigger deviations because their intermolecular interactions are stronger.

Quantifying Deviations: The Compressibility Factor

The compressibility factor $Z$ gives you a single number that captures how far a gas is from ideal behavior:

$Z = \frac{PV}{RT}$

(written here on a per-mole basis, so $V$ is molar volume)

• $Z = 1$: the gas behaves ideally.
• $Z < 1$: attractive forces dominate, pulling molecules together and reducing the volume below what the ideal gas law predicts. This is typical at moderate pressures and low temperatures.
• $Z > 1$: repulsive forces dominate, meaning the finite size of molecules forces the gas to occupy more volume than predicted. This is typical at very high pressures.

At low pressures and high temperatures, molecules are far apart and moving fast, so $Z \to 1$ and the ideal gas law works well. You can visualize these trends on generalized compressibility charts (Z-charts), which plot $Z$ against reduced pressure $P_r = P/P_c$ at various reduced temperatures $T_r = T/T_c$.

Equations of State for Real Gases

Van der Waals Equation of State

The van der Waals equation was the first widely used correction to the ideal gas law. It modifies both the pressure and volume terms:

$\left(P + \frac{a}{V^2}\right)(V - b) = RT$

where $V$ is molar volume, and $a$ and $b$ are substance-specific constants.

The two corrections work as follows:

1. Pressure correction $\left(\frac{a}{V^2}\right)$: Attractive intermolecular forces reduce the pressure a gas exerts on its container walls. The parameter $a$ reflects the strength of those attractions. Adding $a/V^2$ to the measured pressure recovers what the pressure would be without attractions.
2. Volume correction $(b)$: Real molecules occupy space. The parameter $b$ approximates the volume excluded by the molecules themselves, so $V - b$ is the "free" volume available for molecular motion.

The van der Waals equation is a good conceptual model, and it qualitatively predicts liquid-vapor phase transitions. However, its quantitative accuracy is limited, especially for polar gases and near the critical point, because the $a/V^2$ term oversimplifies how attractive forces depend on density and temperature.

Redlich-Kwong Equation of State

The Redlich-Kwong (RK) equation improves on van der Waals by making the attractive term temperature-dependent:

$P = \frac{RT}{V - b} - \frac{a}{T^{1/2}\, V(V + b)}$

The key improvement is the $T^{1/2}$ factor in the denominator of the attraction term. Attractive forces effectively weaken at higher temperatures (molecules have enough kinetic energy to overcome them), and the RK equation captures this. The denominator $V(V+b)$ also provides a better description of how the attractive correction varies with volume compared to the simpler $V^2$ in van der Waals.

Parameters $a$ and $b$ are determined from the critical temperature $T_c$ and critical pressure $P_c$ of the substance:

$a = 0.42748\,\frac{R^2 T_c^{5/2}}{P_c}, \qquad b = 0.08664\,\frac{RT_c}{P_c}$

The RK equation gives noticeably better results than van der Waals for non-polar and slightly polar gases across a wider range of pressures.

Peng-Robinson Equation of State

The Peng-Robinson (PR) equation further refines the attractive term and is one of the most widely used cubic equations of state in engineering practice:

$P = \frac{RT}{V - b} - \frac{a\,\alpha(T)}{V(V + b) + b(V - b)}$

The function $\alpha(T)$ depends on temperature and the acentric factor $\omega$, which characterizes how non-spherical a molecule's intermolecular force field is. A common form is:

$\alpha(T) = \left[1 + \kappa\left(1 - \sqrt{T/T_c}\right)\right]^2$

where $\kappa = 0.37464 + 1.54226\,\omega - 0.26992\,\omega^2$.

The PR equation's modified denominator $V(V+b) + b(V-b)$ improves liquid density predictions compared to RK, which is why it's the standard in the oil and gas industry. It performs especially well for hydrocarbons and other non-polar gases near the critical point and at high pressures.

Both the RK and PR equations require only $T_c$, $P_c$, and $\omega$ as inputs. These values are tabulated for hundreds of substances.

Choosing the Right Equation of State

The right equation depends on three things:

1. The gas itself — polarity, molecular size, and strength of intermolecular forces.
2. Operating conditions — temperature and pressure range.
3. Required accuracy — a quick estimate vs. a rigorous process simulation.

Here's a practical breakdown:

• Soave-Redlich-Kwong (SRK): Replaces the $T^{1/2}$ dependence in RK with an $\alpha(T)$ function similar to PR. Widely used alongside PR in process simulators.
• Benedict-Webb-Rubin (BWR): A multi-parameter equation with eight constants per substance. More accurate over wide ranges but computationally heavier and requires more data.
• Virial equation: Derived from statistical mechanics, expressed as a power series in $1/V$ (or $P$). Truncated to two or three terms, it's simple and theoretically grounded, but only reliable at low-to-moderate pressures.
• Lee-Kesler correlation: A generalized corresponding-states method that uses $T_r$, $P_r$, and $\omega$ to estimate $Z$ and other properties without solving a cubic. Convenient for quick calculations.

When choosing, also consider computational cost (cubic equations are fast; multi-parameter equations are slower), availability of parameters for your substance, and compatibility with the rest of your thermodynamic model (mixing rules for mixtures, activity coefficient models for liquid phases, etc.).

Thermodynamic Properties from Equations of State

Calculating Key Properties

Once you have an equation of state that accurately represents $P$-$V$-$T$ behavior, you can derive a wide range of thermodynamic properties:

• Density $\rho$: Solve the equation of state for molar volume $V$ at given $T$ and $P$, then $\rho = M/V$ (where $M$ is molar mass).
• Compressibility factor: $Z = PV/(RT)$, calculated directly from the solved molar volume.
• Fugacity $f$: The "effective pressure" of a real gas that replaces $P$ in equilibrium calculations. It's related to pressure through the fugacity coefficient $\phi$:

$f = \phi\, P$

The fugacity coefficient is obtained by integrating the equation of state. For a cubic equation of state, $\ln \phi$ can be expressed analytically in terms of $Z$, $a$, $b$, and $T$.

• Enthalpy, entropy, and Gibbs free energy: These require departure functions (see below).

Departure Functions and Residual Properties

You can't measure absolute values of $H$ or $S$ directly, but you can calculate how much a real gas property departs from the ideal gas value at the same conditions. That's what departure functions do.

A departure function is defined as:

$\Delta M^{dep} = M(T, P) - M^{ig}(T, P)$

where $M$ is any molar property ($H$, $S$, $G$, etc.), and the superscript $ig$ denotes the ideal gas value. The departure is computed from the equation of state using integrals involving $P$-$V$-$T$ derivatives. For example, the enthalpy departure at constant $T$ is:

$H(T,P) - H^{ig}(T,P) = \int_{\infty}^{V} \left[T\left(\frac{\partial P}{\partial T}\right)_V - P\right] dV + PV - RT$

The practical workflow for calculating a real gas property change between two states is:

1. Calculate the departure function at state 1 to "convert" from real gas to ideal gas.
2. Calculate the ideal gas property change from state 1 to state 2 (using heat capacity data, for instance).
3. Calculate the departure function at state 2 to "convert" back from ideal gas to real gas.

This three-step path works because ideal gas property changes are straightforward to compute, and the departure functions handle all the non-ideal corrections.

The Joule-Thomson coefficient $\mu_{JT}$ is another property accessible from an equation of state. It describes the temperature change during a throttling (isenthalpic) process:

$\mu_{JT} = \left(\frac{\partial T}{\partial P}\right)_H$

For an ideal gas, $\mu_{JT} = 0$ (no temperature change on throttling). For real gases, $\mu_{JT}$ can be positive (cooling) or negative (heating), and it's calculated from the equation of state using:

$\mu_{JT} = \frac{1}{C_P}\left[T\left(\frac{\partial V}{\partial T}\right)_P - V\right]$

This is critical for designing expansion valves in refrigeration and natural gas processing.

Why This Matters in Practice

The deviations from ideal gas behavior are largest at high pressures, low temperatures, and near the critical point. In engineering applications like natural gas processing, refrigeration system design, and supercritical fluid extraction, using the ideal gas law can produce errors of 20% or more in property calculations. Choosing an appropriate equation of state and correctly applying departure functions is what separates a rough estimate from a reliable design calculation.