Fugacity


Fugacity (f) is a chemical term with units of pressure that is intended to better describe a gas' real world pressure than the ideal pressure "P" used in the ideal gas law. Fugacities are determined experimentally or estimated for various models such as "van der Waals gas" that are closer to reality than an "ideal gas". The ideal gas pressure and fugacity are related through the dimensionless fugacity coefficient \phi \,.[1]

\phi = f/P \,

Since partial pressures are the most common means describe a gases or vapors concentration fugacity is extremely useful in determining real world gas concentrations. Thus just as activities are used to modify concentrations in condensed states the fugacity coefficient is used to modify the effective concentration of gas. Yet fugacity has a relationship to the condensed state. Fugacity can play significant role under any conditions where pressure is a significant factor, including condensed states.

The word "fugacity" is derived from the Latin for "fleetness" which is often interpreted as “the tendency to flee or escape”. The concept of fugacity was introduced by American chemist Gilbert N. Lewis.[2]

Contents

Technical detail

Fugacity is used to better approximate the pressure of real world gases than estimations using the ideal gas law. Yet fugacity allows the use of many of the relationships developed for an idealized system.

In the real world, gases approach ideal gas behavior at low pressures and high temperatures; under such conditions the value of fugacity approaches the value of pressure. Yet no substance is truly ideal. At moderate pressures real gases have attractive interactions and at high pressures intermolecular repulsions become important. Both interactions result in a deviation from "ideal" behavior for which interactions between gas atoms or molecules are ignored.

Derivations

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{{RT}}{P}

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

Reordering, we get

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

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} {{f^\circ }}

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\bar V - RT}} {P}

We can obtain values for Φ experimentally easily by measuring V, T and P. (note that for an ideal gas, Φ = 1)

From the expression above we have

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

We can then write

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

Where

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

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

\mu ^\circ + RT\ln \frac{f} {{f^\circ }} = \mu ^\circ + RT\ln \frac{P} {{P^\circ }} + \int_{P^\circ }^P {\Phi dP}
\Rightarrow RT\ln \frac{f} {{f^\circ }} - RT\ln \frac{P} {{P^\circ }} = \int_{P^\circ }^P {\Phi dP}
RT\ln \frac{{fP^\circ }} {{Pf^\circ }} = \int_{P^\circ }^P {\Phi dP}

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

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

The fugacity coefficient will then verify

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

The integral can be evaluated via graphical integration if we measure experimentally values for Φ 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 temperature 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} {{P^\circ }}

See also

References

  1. ^ Atkins, Peter; John W. Locke (2002-01). Atkins' Physical Chemistry (7th ed.). Oxford University Press. ISBN 0198792859. 
  2. ^ Lewis, Gilbert Newton (1908-05-01). "The Osmotic Pressure of Concentrated Solutions, and the Laws of Perfect Solution.". Journal of the American Chemical Society 30 (5): 668-683. doi:10.1021/ja01947a002. http://dx.doi.org/10.1021/ja01947a002. 






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