Fugacity coefficient definition

Although the linear mixing rule for b [Eq. (14.42)] has proved generally aceeptable, the quadratie mixing rule for a [Eq. (14.43)] is often unsatisfactory. An alternative is a mixing mle for q that incorporates activity-coefficient data. The comiection between activity coefficients and equation-of-state parameters is provided by activity-coefficient and fugacity-coefficient definitions thus. 

According to the definition of Eq. (4-67), G — G/ is the residual Gibbs energy, The dimensionless ratio is another new property called the fugacity coefficient (j). Thus,... 

According to this definition f is equal to the partial pressure P, in the case of an ideal gas. The fugacity coefficient (j>- is defined by... 

The exact definition of the equilibrium constant given by IUPAC requires it to be defined in terms of fugacity coefficients or activity coefficients, in which case it carries no units. This convention is widely used in popular physical chemistry texts, but it is also common to find the equilibrium constant specified in terms of molar concentrations, pressure or molality, in which cases the equilibrium constant will carry appropriate units. 

Combine definitions of the activity coefficient and the fugacity coefficients ... 

Because of the complex functionality of the K-values, these calculations in general require iterative procedures suited only to computer solution. However, in the case of mixtures of light hydrocarbons, in which the molecular force fields are relatively weak and uncomplicated, we may assume as a reasonable approximation that both the liquid and the vapor phases are ideal solutions. By definition of the fugacity coefficient of a species in solution, =fffxtP. But by Eq. (11.61), f f = xj,. Therefore... 

In this definition, the activity coefficient takes account of nonideal liquid-phase behavior for an ideal liquid solution, the coefficient for each species equals 1. Similarly, the fugacity coefficient represents deviation of the vapor phase from ideal gas behavior and is equal to 1 for each species when the gas obeys the ideal gas law. Finally, the fugacity takes the place of vapor pressure when the pure vapor fails to show ideal gas behavior, either because of high pressure or as a result of vapor-phase association or dissociation. Methods for calculating all three of these follow. 

One of the two ways to describe the fugacity is by using activity coefficients yin the liquid phase and fugacity coefficients tp in the vapour phase. Activity coefficients require the definition of a standard state. In distillation, the two terms are related by... 

The idea here is to exploit the connection between fugacity coefficients and activity coefficients provided by their definitions ... 

If equations of state are available for both the vapor and liquid phases, the above equations may be used to calculate the component fugacity coefficients in both phases by Equation 1.23, and the /f-values by Equation 1.25. Alternatively, the fugacity coefficient of a component in solution may be derived from the total fugacity coefficient expression (Equation 1.21) via the definition of partial properties ... 

It also follows from Equation 1.20 and the definition of fugacity and fugacity coefficient that for ideal solutions 0 = (J) and... 

In the general case, the definitions of fugacity coefficient and activity coefficient... 

From this definition it is clear that the fugacity has units of pressure, and that / —> p as P 0 that is, the fugacity becomes equal to the pressure at pressures low enough that the fluid approaches the ideal gas state. Similarly, the fugacity coefficient = f/P lasP -> 0. Both the fugacity and the fugacity coefficient will be used extensively throughout this book. 

The alternative motivation of definition (4.437) for (real) gas mixtures comes from a statement that a mixture is ideal if Amagat s law (4.440) is valid at any T, P. Indeed, Amagat s laws means Va = v and then by (4.454), (4.458) below, for fugacity coefficients also = v therefore by (4.463), this is an ideal mixture. 

The fugacity coefficient is always positive however, it may be greater or less than unity. The ideal gas has (p,- = 1, but the converse is not true a substance having (pj = 1 is not necessarily an ideal gas. Note that the definition (4.3.18) places no restriction on the kind of phase to which (p,- may be applied it is a legitimate property of solids, liquids, and gases, though it is most often applied to fluids. 

Fugacity Formula 1. If along an isotherm T, we have a complete Pvx equation of state for our mixture, then we can compute from the definition of the fugacity coefficient (4.3.18). Here we write that definition in the form... 

In Chapter 7 we introduced fugacity and the fugacity coefficient as auxiliary variables related to the Gibbs energy of pure component. Here we extend these definitions to mixtures. The fugacity of component in mixture,/, is defined as... 

This definition implies that the fugacity coefficient for an ideal gas is always 1. 

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