In the previous modules, we showed that the chemical potential of an ideal gas is given by
\[ \boxed{ \mu = \mu^\circ + RT \ln\left( \frac{p}{p^\circ} \right) } \]
This expression is extremely useful because it relates the chemical potential directly to a measurable quantity: pressure.
Unfortunately, the derivation assumed ideal-gas behavior.
At sufficiently high pressures, real gases deviate from the ideal gas law. Molecular attractions and repulsions become important, and pressure alone no longer provides a complete description of the thermodynamic behavior of the gas.
We therefore need a quantity that plays the role of an effective pressure for real gases.
That quantity is called the fugacity.
Fugacity is defined so that the ideal-gas expression for chemical potential remains valid even when the gas is not ideal.
Specifically, we write
\[ \boxed{ \mu = \mu^\circ + RT \ln\left( \frac{f}{p^\circ} \right) } \]
where \(f\) is the fugacity.
This equation has exactly the same form as the ideal-gas expression, except that fugacity replaces pressure.
For an ideal gas,
\[ \boxed{ f=p } \]
so the familiar ideal-gas expression is recovered.
Since fugacity and pressure have the same units, it is convenient to define the dimensionless fugacity coefficient
\[ \boxed{ \phi = \frac{f}{p} } \]
where the symbol \(\phi\) is commonly used for the fugacity coefficient.
Rearranging,
\[ f=\phi p \]
The fugacity coefficient therefore measures the deviation of a real gas from ideal behavior.
Recall the compression factor introduced in Chapter 2:
\[ \boxed{ Z = \frac{pV_m}{RT} } \]
where \(V_m\) is the molar volume.
For an ideal gas,
\[ Z=1. \]
Deviations from unity indicate non-ideal behavior.
It can be shown that the fugacity coefficient is related to the compression factor through
\[ \boxed{ \ln\phi = \int_0^p \frac{Z-1}{p}\,dp } \]
Thus, experimental measurements of \(Z\) can be used to determine the fugacity coefficient and the fugacity itself.
All gases approach ideal behavior as the pressure approaches zero.
Consequently,
\[ \lim_{p\to0} Z = 1 \]
and therefore
\[ \lim_{p\to0} \phi = 1. \]
In the low-pressure limit,
\[ f \approx p \]
and fugacity becomes indistinguishable from pressure.
Pressure measures the mechanical force exerted by a gas on its container. Fugacity, on the other hand, measures the tendency of a gas to escape, mix, react, or undergo a phase change.
Two gases at the same pressure can have different fugacities if one gas exhibits stronger intermolecular interactions than the other.
Fugacity therefore serves as the thermodynamically correct pressure to use when describing real gases.
Big picture: Fugacity extends the concept of pressure to real gases. By replacing pressure with fugacity, the simple ideal-gas expression for chemical potential can be retained even when gases exhibit substantial deviations from ideal behavior. Fugacity therefore provides the link between measurable pressure data and the thermodynamic behavior of real gases.
The compression factor \(Z\) for \(O_2(g)\) at \(200\ \mathrm{K}\) is measured at several pressures:
| \(p\) (atm) | \(Z\) | \((Z-1)/p\) |
|---|---|---|
| 0.00 | 1.0000 | 0.0000 |
| 1.00 | 0.9970 | -0.00300 |
| 4.00 | 0.9880 | -0.00300 |
| 7.00 | 0.9788 | -0.00303 |
| 10.00 | 0.9700 | -0.00300 |
Estimate the fugacity coefficient \(\phi\) and the fugacity \(f\) at \(10.0\ \mathrm{atm}\).
The fugacity coefficient is related to the compression factor by
\[ \ln\phi = \int_0^p \frac{Z-1}{p}\,dp \]
Since \(Z\rightarrow 1\) as \(p\rightarrow 0\), the integrand approaches zero at very low pressure. We can approximate the integral using the trapezoidal rule.
\[ \ln\phi \approx -0.0286 \]
Therefore,
\[ \phi = e^{-0.0286} = 0.972 \]
Since
\[ \phi = \frac{f}{p} \]
the fugacity is
\[ f = \phi p = (0.972)(10.0\ \mathrm{atm}) \]
\[ \boxed{ f = 9.72\ \mathrm{atm} } \]
Interpretation: Since \(\phi<1\), the fugacity is smaller than the measured pressure. This means the real gas has a lower escaping tendency than an ideal gas at the same pressure, which is consistent with attractive intermolecular forces being important under these conditions.
| Question 1 | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
The compression factor \(Z\) for a gas at \(250\ \mathrm{K}\) is measured
to have the following values:
Use the trapezoidal rule to estimate the fugacity coefficient \(\phi\) and fugacity \(f\) at \(10.0\ \mathrm{atm}\). |
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Show AnswerFirst calculate \((Z-1)/p\):
Using the trapezoidal rule, \[ \ln\phi \approx -0.0205 \] Therefore, \[ \phi = e^{-0.0205} = 0.980 \] and \[ f = \phi p = (0.980)(10.0) = 9.80\ \mathrm{atm} \] |
Big picture: Fugacity extends the concept of pressure to real gases. By replacing pressure with fugacity, the familiar ideal-gas expression for chemical potential remains valid even when gases deviate substantially from ideal behavior. Fugacity therefore provides the essential link between measurable pressure data and the thermodynamics of real gases.