Fugacity

Fugacity is a measure of 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 calculated fugacity for each phase. The phase with the lowest fugacity will thermodynamically be the most favorable: the one that minimizes Gibbs free energy. Fugacity, therefore, is a useful engineering tool for predicting the phase state of multi-component mixtures at various temperatures and pressures without doing the actual lab test. And besides predicting the preferred solid, liquid, or vapor phase, fugacity also applies to solid-solution equilibria.

Fugacity is also a useful way to explain otherwise obscure behaviors of substances. For example: Most people are at a loss to describe why liquid water in a dish on a typical day eventually dries up. In the absence of heat energy to convert the water to a vapor, what causes water to spontaneously change phases from liquid to gas? Fugacity provides an answer: When the temperature and pressure is such that the relative humidity in the room is less than 100%, the calculated fugacity of water vapor will be lower than either liquid or solid water. Therefore, liquid water has a lower Gibbs free energy in the vapor phase and will evaporate. Fundamentally, water in a puddle is more ordered, or has less entropy than water dispersed all over the room as individual molecules. As dictated by the second law of thermodynamics all processes are headed towards maximum disorder, or increasing entropy, and water in a puddle is no exception.

Technical Detail

In thermodynamics, the fugacity is a state function of matter at fixed temperature. Fugacity is the change in pressure requred to make a real gas behave like an ideal gas. The fugacity, which has units of pressure, represents the tendency of a fluid to escape or expand isothermally. For gases at low pressures where the ideal gas law is a good approximation, fugacity is nearly equal to pressure. The ratio $\phi =f/P\,$ between fugacity $f\,$ and pressure $P\,$ is called the fugacity coefficient. For an ideal gas, $\phi =1\,$.

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

$d\ln {f \over f_{0}}={dG \over RT}={{{\tilde {V}}dP} \over RT}\,$

where $G\,$ is the Gibbs free energy, $R\,$ is the gas constant, ${\tilde {V}}\,$ 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:

$f_{2}/f_{1}=exp\left({1 \over RT}\int _{G_{1}}^{G_{2}}dG\right)=exp\left({1 \over RT}\int _{P_{1}}^{P_{2}}{\tilde {V}}\,dP\right)\,$

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).

Fugacity and Chemical Potential

For ideal gases, we have the relation

$dG=-SdT+VdP\,$

for Gibbs free energy

and

$\mu _{i}=\left({\frac {\partial G}{\partial n_{i}}}\right)_{T,P,n_{j\neq i}}$

for the chemical potential of the gas $i$

If we choose to study only one pure ideal gas, we can divide both members of the equation by $n$ and we’ll have

$d\mu =-{\bar {S}}dT+{\bar {V}}dP$

where ${\bar {S}}$ and ${\bar {V}}$ are the molar entropy and molar volume for that gas

We choose to study a process where $T$ remains constant.

Therefore,

$d\mu ={\bar {V}}dP$

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.

$\int _{\mu ^{\circ }}^{\mu }{d\mu }=\int _{P^{\circ }}^{P}{{\bar {V}}dP}$

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 $\int _{P^{\circ }}^{P}{{\bar {V}}dP}$ 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 $\Phi$ experimentally easily by measuring $V$, $T$ and $P$. (note that for an ideal gas, $\Phi$ = 1)

From the expression above we have

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

We can then write

$\int _{\mu ^{\circ }}^{\mu }{d\mu }=\int _{P^{\circ }}^{P}{{\bar {V}}dP}=\int _{P^{\circ }}^{P}{{\frac {RT}{P}}dP}+\int _{P^{\circ }}^{P}{\Phi dP}$

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 $P\to 0$. 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

$\ln \phi ={\frac {1}{RT}}\int _{0}^{P}{\Phi dP}$

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

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

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

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

Expanding in Taylor series about 0,

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

Finally, we get

$f\approx {\frac {P^{2}{\bar {V}}}{RT}}$

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.

External links

• Lewis' 1908 paper