# Fugacity

In chemical thermodynamics , fugacity is an effective partial pressure for a real gas of a gas which converts the mechanical partial pressure into an accurate calculation of the chemical equilibrium constant. It is equal to the pressure of an ideal gas having the same temperature and molar Gibbs free energy as a real gas.

Fugitiveness is determined experimentally or estimated from various models such as van der Waals gas which are closer to reality than an ideal gas. The pressure and fugacity of the real gas are related to the dimensionless medium by the fugacity coefficient .

{\displaystyle \varphi ={\frac {f}{P}}\,}

For an ideal gas, fugacity and pressure are equal and so = 1 . Taken at the same temperature and pressure, the difference between the molar Gibbs free energy of a real gas and that of the corresponding ideal gas is equal to r t ln .

Fugacity is closely related to thermodynamic activity . For a gas, activity is simply the runoff divided by a reference pressure to give a dimensionless quantity. This reference pressure is called the standard state and is usually chosen as 1 atmosphere or 1 bar .

For real gases, fugacity should be used instead of pressure in accurate calculations of chemical equilibrium . The thermodynamic condition for chemical equilibrium is that the total chemical potential of the reactants is equal to that of the products. If the chemical potential of each gas is expressed as a function of fugacity, the equilibrium state can be converted into the familiar reaction quotient form (or the law of mass action ), except that the pressures are expressed by fugacity. is replaced.

For a condensed phase (liquid or solid) in equilibrium with its vapor phase, the chemical potential is equal to the vapor potential, and hence the fugacity is equal to the vapor’s fugitiveness. When the vapor pressure is not very high, it is approximately equal to the stampede vapor pressure .

## Pure Substance

The fugacity is closely related to the chemical potential μ . In a pure substance, μ is equal to the Gibbs energy g m for a mole , of the substance and

{\displaystyle d\mu =dG_{\mathrm {m} }=-S_{\mathrm {m} }dT+V_{\mathrm {m} }dP},

where T and P are temperature and pressure, Vm is the volume per mole and Sm is the entropy per mole.

### gas

The equation of state for an ideal gas can be written as:

{\displaystyle V_{\mathrm {m} }^{\mathrm {ideal} }={\frac {RT}{P}}},

where r is the ideal gas constant . The difference in chemical potential between two different pressures but the same temperature (i.e., dT = 0 ) is given by the change

{\displaystyle d\mu =V_{\mathrm {m} }dP=RT\,{\frac {dP}{P}}=RT\,d\ln P.}

The equation of state for real gases will diverge from the simpler one, and for an ideal gas the above result will be a good approximation only provided that (a) the specific size of the molecule is negligible compared to the average distance between individual molecules, and ( b) The short-range behavior of the intermolecular potential can be neglected, that is, when the molecules can be assumed to rebound elastically from each other during a molecular collision. In other words, real gases behave like ideal gases at low pressure and high temperature. [3] At moderately high pressures, the attractive interactions between molecules reduce the pressure compared to the ideal gas law; And at very high pressures, the sizes of the molecules are no longer negligible and the repulsive forces between the molecules increase the pressure. At lower temperatures, molecules are more likely to stick together rather than rebound elastically. [4]

The ideal gas law can still be used to describe the behavior of a real gas if the pressure is replaced by a fugacity f , so as to be defined as

{\displaystyle d\mu =RT\,d\ln f}

And

{\displaystyle \lim _{P\to 0}{\frac {f}{P}}=1.}

That is, at low pressure f is proportional to pressure, so it has the same units as pressure. Ratio

{\displaystyle \varphi ={\frac {f}{P}}}

It is called the fugacity coefficient .

If a reference state is represented by a zero superscript, then integrating the equation for the chemical potential is obtained by

{\displaystyle \mu - \mu ^ {0} = RT \, \ln a,}

where a , a dimensionless quantity, is called activity .

Numerical example: Nitrogen gas (N2 ) at 0 °C and a pressure of p = 100 atmospheres (atm) has a fugacity of f = 97.03 atm . [1] This means that the molar Gibbs energy of real nitrogen at a pressure of 100 atm is equal to the molar Gibbs energy of nitrogen as an ideal gas at 97.03 atm . runaway coefficient

The contribution of nonideality to the molar Gibbs energy of a real gas is equal to t ln . For nitrogen at 100 atm, m = m, id + rt .9703 ln , which is less than the ideal value m, id because of molecular attractive forces. In the end, the activity is just 97.03 without the units .

### Condensed phase

The opacity of a condensed phase (liquid or solid) is defined in the same way as for a gas:

{\displaystyle d\mu _{\mathrm {c} }=RT\,d\ln f_{\mathrm {c} }}

And

{\displaystyle \lim _{P\to 0}{\frac {f_{\mathrm {c} }}{P}}=1.}

It is difficult to directly measure fugitiveness in a condensed phase; But if the condensed phase is saturated (in equilibrium with the steam phase) the full chemical potentials in the two phases are equal ( μc = μg ) . With the above definition, it implies that

{\displaystyle f_{\mathrm {c} }=f_{\mathrm {g} }.}

When computing the opacity of the compressed phase, one can usually assume that the volume is constant. At constant temperature, the stampede changes as the pressure from the saturation presses sets to P

{\displaystyle \ln {\frac {f}{f_{\mathrm {sat} }}}={\frac {V_{\mathrm {m} }}{RT}}\int _{P_{\mathrm {sat} }}^{P}dp={\frac {V\left(P-P_{\mathrm {sat} }\right)}{RT}}.}

This fraction is known as the Poynting factor . fset = set using the p set, where set is the runaway coefficient ,

{\displaystyle f=\varphi _{\mathrm {sat} }P_{\mathrm {sat} }\exp \left({\frac {V\left(P-P_{\mathrm {sat} }\right)}{RT}}\right).}

This equation allows the fugacity to be calculated using the tabular values ​​for the saturated vapor pressure. The pressure is often low enough for the vapor phase to be considered an ideal gas, so the fugacity coefficient is approximately equal to 1.

Unless the pressure is very high, the Poynting factor is usually small and the exponential term is close to 1. Often, the opacity of a pure liquid is used as a reference condition when defining and using the mixing activity coefficient.

### Mix

Fugacity is most useful in mixtures. It doesn’t add any new information compared to chemical potential, but it has computational advantages. As soon as the molar fraction of a component becomes zero, the chemical potential dissipates but the stampede becomes zero. In addition, there are natural reference states for runaway (for example, an ideal gas forms a natural reference state for a gas mixture because runoff and pressure converge at low pressures).
gases

In a mixture of gases, the fugacity of each component I is with a similar definition partial molar volume instead of molar volume (e.g., GI instead of Gm and VI instead Vm 🙂

{\displaystyle dG_{i}=RT\,d\ln f_{i}}

And

{\displaystyle \lim _{P\to 0}{\frac {f_{i}}{P_{i}}}=1,}

where P i is the partial pressure component of i . Partial pressures obey Dalton’s law :

{\displaystyle P_{i}=y_{i}P,}

where P is the total pressure and y is the mole fraction of the i component (so the partial pressure gets added to the total pressure). Fugitives generally follow a similar law called the Lewis and Randall Rule:

{\displaystyle f_{i}=y_{i}f_{i}^{*},}

where f*IIf the whole gas had that composition at the same temperature and pressure I would have that component. Both laws are expressions of the assumption that gases behave independently.

Liquid substance

In a liquid mixture, the volatility of each component is equal to that of the vapor component in equilibrium with the liquid. In an ideal solution , the fugitives follow the Lewis–Randel rule:

{\displaystyle f_{i}=x_{i}f_{i}^{*},}

where x i is the mole fraction of the liquid and f*
Iis the opacity of the pure liquid phase. This is a good approximation when the constituent molecules have the same size, shape and polarity. [ 2 ] : 264,269 -270

In a dilute solution with two components, the component with the larger molar fraction ( solvent ) may still obey Raoult’s law even though the other component ( solute ) has different properties. This is because its molecules experience essentially the same environment they would in the absence of a solute. In contrast, each solute molecule is surrounded by solvent molecules, so it obeys a different law known as Henry’s law . [9] : 171 According to Henry’s law, the melting point of a solute is proportional to its concentration. The constant of proportionality (a measured Henry’s constant) depends on whether the concentration is represented by a mole fraction, molality, or molarity.

## Temperature and Pressure Dependence

Pressure dependence of runaway (at constant temperature) [2] : 260. given by

{\displaystyle \left({\frac {\partial \ln f}{\partial P}}\right)_{T}={\frac {V_{\mathrm {m} }}{RT}}}

And always positive.

The dependence of temperature on constant pressure is

{\displaystyle \left({\frac {\partial \ln f}{\partial T}}\right)_{P}={\frac {\Delta H_{\mathrm {m} }}{RT^{2}}},}

where h m is the change in molar enthalpy as the gas expands, the liquid evaporates, or the solid turns into a vacuum. [2] : 262 Further, if the pressure p is 0 , then

{\displaystyle \left({\frac {\partial \left(T\ln {\frac {f}{p^{0}}}\right)}{\partial T}}\right)_{P}=-{\frac {S_{\mathrm {m} }}{R}}<0.}

Since temperature and entropy are positive, ln F / Po decreases with increasing temperature.

## Measurement

Stampede can be estimated from the measurement of volume as a function of pressure at constant temperature. in that case,

{\displaystyle \ln \varphi ={\frac {1}{RT}}\int _{0}^{P}\left(V_{m}-V_{\mathrm {m} }^{\mathrm {ideal} }\right)dP.}

This integral can also be calculated using the equation of state. [2] : 251-252

The integral can be re-formulated into an alternative form using the compressibility factor

{\displaystyle Z={\frac {PV_{\mathrm {m} }}{RT}}.}

Then

{\displaystyle \ln \varphi =\int _{0}^{P}\left({\frac {Z-1}{P}}\right)dP.}

This is useful because of the theorem of corresponding states: If the pressure and temperature at the critical point of the gas are c and c , then we can define the reduced properties as Pr = P / Pc and Tr = Tea / Tc ,

Then, to a good approximation, most gases have the same value of Z for the same low temperature and pressure . However, in geochemical applications, this principle ceases to be accurate at the pressures where metamorphosis occurs. [11] : 247

For a gas obeying the van der Waals equation, the obvious formula for the fugacity coefficient is

{\displaystyle RT\ln \varphi ={\frac {RTb}{V_{\mathrm {m} }-b}}-{\frac {2a}{V_{\mathrm {m} }}}-RT\ln \left(1-{\frac {a(V_{\mathrm {m} }-b)}{RTV_{\mathrm {m} }^{2}}}\right)}

This formula is difficult to use, because the pressure depends on the molar volume via the equation of state; So one must choose a volume, calculate the pressure, and then use these two values ​​on the right side of the equation.

## History

The word fugitive is derived from Latin fugere , to run away. In the sense of an “avoidance”, it was introduced to thermodynamics in 1901 by the American chemist Gilbert N. Lewis and popularized in 1923 by Lewis and Merle Randall, an influential textbook on thermodynamics and the free energy of chemical substances . [13] The “tendency to escape” referred to the flow of matter between phases, and temperature played a similar role in the flow of heat.