What is the fugacity coefficient?

The fugacity coefficient φ is a dimensionless ratio defined as φ = f / P, where f is fugacity and P is pressure. It measures how far a real gas deviates from ideal behavior. For an ideal gas, φ = 1. Values of φ less than 1 usually indicate attractive forces dominating, while φ greater than 1 indicates repulsive forces dominating.

When can I approximate fugacity as equal to pressure?

At low to moderate pressures, typically below a few bar and far from the critical point or strong associating effects, most gases behave nearly ideally and the fugacity coefficient φ is close to 1. In that regime, it is common to approximate fugacity f ≈ P without introducing significant error.

How do I get the compressibility factor Z for fugacity calculations?

The compressibility factor Z can be obtained from an equation of state such as Peng–Robinson or Soave–Redlich–Kwong, from generalized compressibility charts, or from experimental PVT data. Once Z is known as a function of pressure at fixed temperature, it can be used to compute the fugacity coefficient via ln φ = ∫0^P (Z − 1)/P dP.

Why is fugacity important in chemical engineering?

Fugacity is essential for phase equilibrium and reaction equilibrium calculations in non-ideal systems. It allows you to write equilibrium conditions such as equality of fugacity in coexisting phases, or equilibrium constants in terms of fugacity or fugacity coefficients, which is more accurate than using raw pressures at high pressure or in strongly non-ideal mixtures.

Fugacity & Fugacity Coefficient Calculator

Q: What is fugacity in thermodynamics?

Fugacity is an effective pressure that replaces the real pressure in thermodynamic equations for non-ideal gases and mixtures. It has the same units as pressure, but it accounts for intermolecular interactions and deviations from ideal-gas behavior. For an ideal gas, fugacity equals the pressure.

Compute fugacity f and fugacity coefficient φ of a real gas from pressure, temperature, and compressibility factor Z. Includes ideal-gas limit, quick approximation, and full integral form.

Fugacity calculator

Pressure P (bar)

Temperature T (K, optional)

T is not required for the basic Z-based calculation, but is useful for documentation and context.

Compressibility factor Z

Z = PV / (RT). Z = 1 for an ideal gas; use EOS or charts for real gases.

Reference Z at low pressure Z_ref (optional)

If provided, used for a simple trapezoidal integration between P_ref and P.

Reference pressure P_ref (bar, optional)

Common choice: 1 bar or a low-pressure state where gas is nearly ideal.

Reference fugacity coefficient φ_ref (optional)

If omitted, the tool assumes φ_ref = 1 at P_ref.

Calculation mode

Quick approximation — uses ln φ ≈ Z − 1 at the specified P. Two-point integral — uses Z and Z_ref to approximate the integral of (Z − 1)/P between P_ref and P.

What is fugacity?

In thermodynamics, fugacity is an effective pressure that replaces the real pressure in equilibrium and chemical potential expressions for non-ideal gases and mixtures. It has the same units as pressure, but it incorporates intermolecular interactions and deviations from ideal-gas behavior.

For an ideal gas, fugacity equals pressure:

\( f = P \quad \text{(ideal gas)} \)

For a real gas, we define the fugacity coefficient \(\phi\) as:

\( \phi = \dfrac{f}{P} \quad\Rightarrow\quad f = \phi P \)

When \(\phi = 1\) the gas behaves ideally. Values of \(\phi \neq 1\) quantify non-ideality:

\(\phi < 1\): attractive forces dominate (effective pressure lower than actual).
\(\phi > 1\): repulsive forces dominate (effective pressure higher than actual).

Relationship between fugacity coefficient and compressibility factor Z

For a pure real gas at constant temperature, the fugacity coefficient is related to the compressibility factor \(Z = \dfrac{PV}{RT}\) by:

\[ \ln \phi(T,P) = \int_{0}^{P} \frac{Z(T,P') - 1}{P'}\,\mathrm{d}P' \]

In practice, we often integrate from a low reference pressure \(P_\text{ref}\) where the gas is nearly ideal:

\[ \ln \phi(T,P) = \ln \phi(T,P_\text{ref}) + \int_{P_\text{ref}}^{P} \frac{Z(T,P') - 1}{P'}\,\mathrm{d}P' \]

If the gas is nearly ideal at \(P_\text{ref}\), we can set \(\phi(T,P_\text{ref}) \approx 1\), so \(\ln \phi(T,P_\text{ref}) \approx 0\).

Quick approximation used in this calculator

When you only know a single value of \(Z\) at the pressure of interest, a common engineering approximation is:

\[ \ln \phi \approx Z - 1 \quad\Rightarrow\quad \phi \approx \exp(Z - 1) \]

This is accurate when \(Z\) does not change rapidly with pressure over the range from low pressure to \(P\), which is often true at moderate pressures and away from the critical region.

Two-point integral approximation

If you know \(Z\) at both a reference pressure \(P_\text{ref}\) and the target pressure \(P\), you can approximate the integral by assuming \(Z - 1\) varies linearly with \(\ln P\) between the two points. A simple trapezoidal approximation in \(\ln P\) gives:

\[ \ln \phi(T,P) \approx \ln \phi(T,P_\text{ref}) + \frac{(Z - 1) + (Z_\text{ref} - 1)}{2} \, \ln\left(\frac{P}{P_\text{ref}}\right) \]

With \(\phi(T,P_\text{ref}) \approx 1\), this reduces to:

\[ \ln \phi(T,P) \approx \frac{(Z - 1) + (Z_\text{ref} - 1)}{2} \, \ln\left(\frac{P}{P_\text{ref}}\right) \]

Worked example

Given:

Gas: CO₂ at 298 K
Pressure: \(P = 50\ \text{bar}\)
Compressibility factor at 50 bar: \(Z = 0.85\)

Approximate fugacity coefficient:

\[ \ln \phi \approx Z - 1 = 0.85 - 1 = -0.15 \] \[ \phi \approx e^{-0.15} \approx 0.86 \]

Fugacity:

\[ f = \phi P \approx 0.86 \times 50\ \text{bar} \approx 43\ \text{bar} \]

So at 50 bar and 298 K, CO₂ behaves as if its “effective pressure” in equilibrium expressions were about 43 bar instead of 50 bar.

Limitations and good practices

At very high pressures or near the critical point, use a full equation of state and numerical integration for accurate fugacity.
For mixtures, fugacity becomes component fugacity \(f_i = y_i \phi_i P\); this tool focuses on pure-component behavior.
Always keep track of units: fugacity has the same units as pressure (bar, Pa, atm, etc.).