10.4 Compressibility factor and fugacity

Compressibility Factor and Real Gas Behavior

Compressibility factor in real gases

The compressibility factor ($Z$) quantifies how much a real gas deviates from ideal gas behavior. It's defined as the ratio of the actual molar volume of a gas to the molar volume it would occupy if it behaved ideally at the same temperature and pressure:

$Z = \frac{V_{actual}}{V_{ideal}} = \frac{PV_m}{RT}$

How to interpret $Z$:

• $Z = 1$: The gas behaves ideally.
• $Z < 1$: Attractive intermolecular forces dominate, pulling molecules closer together. The gas is more compressible than an ideal gas. This is common at moderate pressures for most gases (e.g., $CO_2$ near its critical point).
• $Z > 1$: Molecular volume (finite size of molecules) dominates, making the gas less compressible than an ideal gas. This occurs at very high pressures or for small, weakly interacting molecules like $H_2$ and $He$.

The physical reasons for deviation come down to two things ideal gas law ignores: intermolecular forces (van der Waals attractions, dipole-dipole interactions) and the fact that molecules take up space. At low pressures and high temperatures, these effects are small and $Z \approx 1$. As pressure increases or temperature drops, deviations grow.

Accurate $Z$ values are essential for designing systems that handle high-pressure gases, such as natural gas pipelines, compressed air storage, and chemical reactors.

Calculation of compressibility factor

There are two main approaches for finding $Z$: generalized charts and equations of state.

Using a compressibility chart (Z-chart)

The principle of corresponding states says that all gases, when compared at the same reduced conditions, show roughly the same deviation from ideality. This lets you use a single generalized chart.

1. Find the critical properties of your gas ($P_c$ and $T_c$). For example, $CO_2$ has $P_c = 7.38$ MPa and $T_c = 304.13$ K.

2. Calculate the reduced pressure and reduced temperature:

   • $P_r = \frac{P}{P_c}$
   • $T_r = \frac{T}{T_c}$

3. On the Z-chart, locate the point corresponding to your $P_r$ and $T_r$ values.

4. Read the $Z$ value from the chart.

This method is quick and reasonably accurate for nonpolar or slightly polar gases. For highly polar gases (like water or ammonia), accuracy drops and you may need corrections.

Using equations of state (EOS)

Equations of state are mathematical models that relate $P$, $V$, and $T$ for a substance. Common ones include van der Waals, Redlich-Kwong, Soave-Redlich-Kwong (SRK), and Peng-Robinson.

The van der Waals equation is the simplest:

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

Here, $V_m$ is the molar volume, $a$ corrects for intermolecular attractions, and $b$ corrects for molecular volume. These are substance-specific constants (for $CO_2$: $a = 0.364$ Pa·m$^6$/mol$^2$, $b = 4.267 \times 10^{-5}$ m$^3$/mol).

To find $Z$ from an EOS:

1. Substitute the known $P$, $T$, and substance-specific parameters ($a$, $b$) into the equation.
2. Solve for $V_m$ (this often requires iterative or cubic equation solving).
3. Calculate $Z = \frac{PV_m}{RT}$.

More advanced EOS like Peng-Robinson give better accuracy, especially near the critical point and for phase equilibrium calculations, but the basic procedure is the same.

Fugacity and Real Gas Behavior

Fugacity and real gas behavior

Fugacity ($f$) is the "effective pressure" of a real gas. It replaces pressure in thermodynamic equations when you need to account for non-ideal behavior.

Why do you need it? The chemical potential of an ideal gas depends on pressure through $\mu = \mu^\circ + RT \ln P$. For a real gas, simply plugging in the actual pressure gives wrong answers because intermolecular forces alter the gas's thermodynamic behavior. Fugacity is defined so that the same functional form holds:

$\mu = \mu^\circ + RT \ln f$

Key properties of fugacity:

• It has the same units as pressure (Pa, atm, bar).
• For an ideal gas, $f = P$ exactly.
• For a real gas, $f$ can be higher or lower than $P$ depending on whether repulsive or attractive interactions dominate.

Fugacity is central to predicting phase equilibria (vapor-liquid equilibrium), chemical reaction equilibria, and mass transfer processes like gas absorption. At phase equilibrium, the fugacity of a component must be equal in all phases.

Determination of gas fugacity

The fugacity coefficient ($\phi$) connects fugacity to the measured pressure:

$\phi = \frac{f}{P} \quad \Rightarrow \quad f = \phi P$

• For an ideal gas, $\phi = 1$.
• For a real gas, $\phi < 1$ when attractive forces dominate (gas is "easier to compress" than ideal), and $\phi > 1$ when repulsive forces dominate.

How to determine $\phi$:

1. From generalized correlations: Using the same reduced properties ($P_r$, $T_r$) as for $Z$, you can read $\phi$ from generalized fugacity coefficient charts. This is the quickest method for pure components.

2. From an equation of state: You can derive $\ln \phi$ analytically from an EOS. The general thermodynamic relation is:

$\ln \phi = \int_0^P \frac{Z - 1}{P} \, dP \quad \text{(at constant } T\text{)}$

This integral captures all the accumulated deviation from ideality as you compress the gas from zero pressure (where all gases are ideal) up to the actual pressure.

1. For mixtures using SRK or Peng-Robinson: The expressions become more involved because you need mixing rules. For example, the SRK form for component $i$ in a mixture is:

$\ln \phi_i = \frac{b_i}{b_m}(Z-1) - \ln(Z-B) - \frac{A}{B}\left(\frac{2\sum_j x_j a_{ij}}{a_m} - \frac{b_i}{b_m}\right)\ln\left(1 + \frac{B}{Z}\right)$

Here $a_m$, $b_m$, $A$, and $B$ are mixture parameters calculated from composition and component properties, and $a_{ij}$ values come from combining rules. You won't typically evaluate this by hand, but understanding the structure matters: each term accounts for a different aspect of non-ideality (size effects, energy interactions, and compressibility).