# Dictionary Definition

fugacity

### Noun

1 the tendency of a gas to expand or escape

2 the lack of enduring qualities (used chiefly of
plant parts) [syn: fugaciousness]

# User Contributed Dictionary

## English

### Noun

- A measure of the tendency of a fluid to expand or escape
- A measure of the relative stability of different phases of a substance under the same conditions
- Transience

#### Translations

- Italian: fugacità (1,2)

# Extensive Definition

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(f, V, T) \, is defined as

- \mathcal(f, 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) \, . 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 \ln \mathcal(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 = = \,

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:

- = \exp \left ( \int_^ dG \right) = \exp \left ( \int_^ \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 = \int_^P

Now, for the ideal gas \bar V = \frac

\mu - \mu ^\circ = \int_^P = RT\ln \frac
Reordering, we get

\mu = \mu ^\circ + RT\ln \frac 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
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 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 _ \frac = 1

We determine f by defining a function

\Phi = \frac 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 + \Phi

We can then write

\int_^\mu = \int_^P = \int_^P + \int_^P

Where

\mu = \mu ^\circ + RT\ln \frac + \int_^P

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

\mu ^\circ + RT\ln \frac = \mu ^\circ + RT\ln
\frac + \int_^P

\Rightarrow RT\ln \frac - RT\ln \frac =
\int_^P

RT\ln \frac = \int_^P

Suppose we choose P \to 0. Since \mathop _ f = P,
we obtain

RT\ln \frac = \int_0^P

The fugacity coefficient will then verify

\ln \phi = \frac \int_0^P

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

### 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 = e^

Expanding in Taylor
series about 0,

\frac \approx 1 + \frac = 1 + \frac \left(
\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.

## References

## External links

- Lewis' 1908 paper (subscription required to view paper)

fugacity in German: Fugazität

fugacity in Spanish: Fugacidad

fugacity in Korean: 퓨개서티

fugacity in Italian: Fugacità

fugacity in Dutch: Fugaciteit

fugacity in Japanese: フガシティー

fugacity in Polish: Lotność (gaz)

fugacity in Slovak: Fugacita

fugacity in Swedish: Fugacitet