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Fugacity
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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, gas) over another. At a fixed Temperature and Pressure , water (for example) will have a different fugacity for each phase. The phase with the lowest fugacity will be the most favorable; the substance minimizes Gibbs Free Energy . 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." ''J. Am. Chem. Soc.'' '''30''', 668-683 (1908)


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(- rac{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 angle = f rac{\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 Liquid s and Solid s as well.

The fugacity coefficient is defined as the ratio fugacity/pressure. For Gas es at low pressures (where the Ideal Gas Law is a good approximation), fugacity is roughly 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:


- 1 = rac


Finally, we get


f \approx rac


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.


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