Fugacity

Fugacity is a measure of chemical potential in the form of 'adjusted pressure.' It directly relates to the tendency of a substance to prefer one phase (liquid, solid, or gas) over another, and the word can be literally defined as "the tendency to flee or escape." This applies in that fugacity describes a substance's tendency to "flee" from one particular state to another, and at a fixed temperature and pressure, a homogeneous substance will have a different fugacity for each phase. The phase with the lowest fugacity will be the most favorable as the substance will have the lowest tendency to leave or "flee" this state, and the substance will have a minimized Gibbs free energy as a result. The concept of fugacity was introduced by American chemist Gilbert N. Lewis in his paper "The osmotic pressure of concentrated solutions, and the laws of the perfect solution."

Applications
As well as predicting the preferred phase of a single substance, fugacity is also useful for multi-component equilibrium involving any combination of solid, liquid and gas equilibria. It is useful as an engineering tool for predicting the final phase and reaction state of multi-component mixtures at various temperatures and pressures without doing the actual lab test.

Fugacity is not a physical property of a substance; rather it is a calculated property which is intrinsically related to chemical potential. When a system approaches the ideal gaseous state (very low pressure), chemical potential approaches negative infinity, which for the purposes of mathematical modeling is undesirable. Under the same conditions, fugacity approaches zero and the fugacity coefficient (defined below) approaches 1. Thus, fugacity is much easier to manipulate mathematically.

Definition from Statistical Mechanics
In statistical mechanics, the fugacity is one of the parameters that define the grand canonical ensemble (a system that may exchange particles with the environment). It represents the effort of adding an additional particle to the system. Its logarithm, multiplied by $$k_B T \,$$, is the chemical potential, $$\mu\,$$


 * $$ \mu = k_B T \log f \, $$

where, $$k_B \,$$ is the Boltzmann constant, and $$T\,$$ is the temperature. (More commonly, the fugacity is denoted by symbol $$z\,$$ instead of $$f\,$$ used here. ) In other words, fugacity


 * $$ f = \exp ( \mu / k_B T ) . \,$$

The grand canonical ensemble is a weighted sum over systems with different numbers of particles. Its partition function, $$ \mathcal{Z}(f, V, T) \, $$ is defined as


 * $$ \mathcal{Z}(f, V, T) =

\sum_{N=0}^{\infty} f^N \, Z(N, V, T) \, $$

where $$N\,$$ is the number of particles of the system, and the canonical partition function is defined for a system with a fixed number of particles $$N\,$$, at temperature $$T\,$$, of volume $$V\,$$ as, $$ Z(N, V, T) = \sum \exp(-\frac{E}{k_B T}) \, $$. Here the summation is performed over all microscopic states, and $$E \,$$ is the energy of each microscopic state. The position of fugacity in grand canonical ensemble is similar to that of temperature in the canonical ensemble as a weighting factor.

Many physically important quantities can be obtained by differentiating the partition function. A most important relation is about the average number of particles of the grand canonical ensemble,


 * $$ \langle N \rangle = f\frac{\partial} {\partial f} \ln \mathcal{Z}(f, V, T) $$,

while the partition function is related to the pressure $$P\,$$ of the system as $$ P V = k_B T  \ln \mathcal{Z}(f, V, T) $$.

Technical detail
Fugacity is a state function of matter at fixed temperature. It only becomes useful when dealing with substances other than an ideal gas. For an ideal gas, fugacity is equal to pressure. In the real world, though under low pressures and high temperatures some substances approach ideal behavior, no substance is truly ideal, so we use fugacity not only to describe non-ideal gases, but liquids and solids as well.

The fugacity coefficient is defined as the ratio fugacity/pressure. For an ideal gas (which is a good approximation for any gas at sufficiently low pressure), fugacity is equal to pressure. Thus, for an ideal gas, the ratio $$ \phi = f/P \,$$ between fugacity $$f\,$$ and pressure $$P\,$$ (the fugacity coefficient) is equal to 1. This ratio can be thought of as 'how closely the substance behaves like an ideal gas,' based on how far it is from 1.

For a given temperature $$ T\,$$, the fugacity $$ f\,$$ satisfies the following differential relation:


 * $$ d \ln {f \over f_0} = {dG \over RT} = {{\bar V dP} \over RT} \,$$

where $$G\,$$ is the Gibbs free energy, $$R\,$$ is the gas constant, $$\bar V\,$$ is the fluid's molar volume, and $$f_0\,$$ is a reference fugacity which is generally taken as that of an ideal gas at 1 bar. For an ideal gas, when $$f=P$$, this equation reduces to the ideal gas law.

Thus, for any two mutually-isothermal physical states, represented by subscripts 1 and 2, the ratio of the two fugacities is as follows:


 * $$ {f_2 \over f_1} = \exp \left ({1 \over RT} \int_{G_1}^{G_2} dG \right) = \exp \left ({1 \over RT} \int_{P_1}^{P_2} \bar V\,dP \right) \,$$

Fugacity and chemical potential
For every pure substance, we have the relation $$dG = -SdT + VdP$$ for Gibbs free energy and we can integrate this expression remembering the chemical potential is a function of $$T$$ and $$P$$. We must also set a reference state. In this case, for an ideal gas the only reference state will be the pressure, and we set $$P$$ = 1 bar.

$$\int_{\mu^\circ }^\mu {d\mu }  = \int_{P^\circ }^P {\bar VdP}$$

Now, for the ideal gas $$\bar V = \frac{P}$$

$$\mu - \mu ^\circ  = \int_{P^\circ }^P {\frac {P}dP} = RT\ln \frac{P} $$

Reordering, we get

$$\mu = \mu ^\circ  + RT\ln \frac{P} $$

Which gives the chemical potential for an ideal gas in an isothermal process, where the reference state is $$P$$=1 bar.

For a real gas, we cannot calculate $$\int_{P^\circ }^P {\bar VdP}$$ because we do not have a simple expression for a real gas’ molar volume. On the other hand, even if we did have one expression for it (we could use the Van der Waals equation, Redlich-Kwong or any other equation of state), it would depend on the substance being studied and would be therefore of a very limited usability.

We would like the expression for a real gas’ chemical potential to be similar to the one for an ideal gas.

We can define a magnitude, called fugacity, so that the chemical potential for a real gas becomes

$$\mu = \mu ^\circ  + RT\ln \frac{f} $$

with a given reference state (discussed later).

We can see that for an ideal gas, it must be $$f=P$$

But for $$P \to 0$$, every gas is an ideal gas. Therefore, fugacity must obey the limit equation

$$\mathop {\lim }_{P \to 0} \frac{f} {P} = 1$$

We determine $$f$$ by defining a function

$$\Phi = \frac {P}$$

We can obtain values for $$\Phi$$ experimentally easily by measuring $$V$$, $$T$$ and $$P$$. (note that for an ideal gas, $$\Phi$$ = 0)

From the expression above we have

$$\bar V = \frac {P} + \Phi$$

We can then write

$$\int_{\mu ^\circ }^\mu {d\mu }  = \int_{P^\circ }^P {\bar VdP}  = \int_{P^\circ }^P {\frac {P}dP} + \int_{P^\circ }^P {\Phi dP}$$

Where

$$\mu = \mu ^\circ  + RT\ln \frac{P} + \int_{P^\circ }^P {\Phi dP}$$

Since the expression for an ideal gas was chosen to be $$\mu = \mu ^\circ  + RT\ln \frac{f} $$,we must have

$$\mu ^\circ + RT\ln \frac{f} = \mu ^\circ + RT\ln \frac{P} + \int_{P^\circ }^P {\Phi dP}$$

$$ \Rightarrow RT\ln \frac{f} - RT\ln \frac{P} = \int_{P^\circ }^P {\Phi dP}$$

$$RT\ln \frac = \int_{P^\circ }^P {\Phi dP}$$

Suppose we choose $$P \to 0$$. Since $$\mathop {\lim }_{P \to 0} f = P$$, we obtain

$$RT\ln \frac{f} {P} = \int_0^P {\Phi dP}$$

The fugacity coefficient will then verify

$$\ln \phi = \frac{1} \int_0^P {\Phi dP}$$

The integral can be evaluated via graphical integration if we measure experimentally values for $$\Phi$$ while varying $$P$$.

We can then find the fugacity coefficient of a gas at a given pressure $$P$$ and calculate

$$f = \phi P\,$$

The reference state for the expression of a real gas’ chemical potential is taken to be “ideal gas, at $$P$$ = 1 bar and work $$T$$”. Since in the reference state the gas is considered to be ideal (it is an hypothetical reference state), we can write that for the real gas

$$\mu = \mu ^\circ  + RT\ln \frac{f} $$

Alternative methods for calculating fugacity
If we suppose that $$\Phi$$ is constant between 0 and $$P$$ (assuming it is possible to do this approximation), we have

$$\frac{f} {P} = e^{\frac }$$

Expanding in Taylor series about 0,

$$\frac{f} {P} \approx 1 + \frac = 1 + \frac{1} \left( {\frac {P}} \right)P = 1 + \frac - 1 = \frac $$

Finally, we get

$$ f \approx \frac $$

This formula allows us to calculate quickly the fugacity of a real gas at $$P$$,$$T$$, given a value for V (which could be determined using any equation of state), if we suppose is constant between 0 and $$P$$.

We can also use generalized charts for gases in order to find the fugacity coefficient for a given reduced temperature.

Fugacity could be considered a “corrected pressure” for the real gas, but should never be used to replace pressure in equations of state (or any other equations for that matter). That is, it is false to write expressions such as

$$fV = nRT$$

Fugacity is strictly a tool, conveniently defined so that the chemical potential equation for a real gas turns out to be similar to the equation for an ideal gas.