Expressions for Fugacity Calculation

It is clear that, if we want to take advantage of the fugacity criteria to perform equilibrium calculations, we need to have a means of calculating it. Let us develop a general expression for fugacity calculations. Let us begin with the definition of fugacity in terms of chemical potential for a pure component shown in (16.21a):

d μ = R T d In f @ const T

(16.26)

The Maxwell’s Relationships presented in equation (15.27c) is written for a pure component system as:

{(\frac{\partial μ}{\partial P})}_{r} = \bar{V} = \tilde{v}

(16.27)

Consequently,

d μ = \tilde{v} d P @ const T

(16.28)

Substituting (16.28) into (16.26),

R T d In f = \tilde{v} d P @ const T

(16.29)

Introducing the concept of fugacity coefficient given in equation (16.23a),

ϕ = \frac{f}{P}

(16.23a)

ln ϕ = \ln f- ln P

(16.30)

We end up with:

R T d ln ϕ = \tilde{v} d P - R T d ln P

(16.31a)

or equivalently,

R T d ln ϕ = \tilde{v} d P - R T \frac{d P}{P}

(16.31b)

Integrating expression (16.31b),

\int_{\ln ϕ^{m}}^{\ln ϕ} d ln ϕ = \int_{P^{m}}^{P} {\frac{\tilde{v}}{R T} - \frac{1}{P}} d P

(16.32)

It is convenient to define the lower limit of integration as the ideal state, for which the values of fugacity coefficient, volume, and compressibility factor are known.

At the ideal state, in the limit $P - > 0$ ,

ϕ^{*} - > 1 ∴ \ln ϕ^{*} - > 0

(16.33)

Substituting into (16.32),

\ln ϕ = \int_{0}^{P} {\frac{\tilde{v}}{R T} - \frac{1}{P}} d P

(16.34)

Equation (16.34) is the expression of fugacity coefficient as a function of pressure, temperature, and volume. Notice that this expression can be readily rewritten in terms of compressibility factor:

\ln ϕ = \int_{0}^{P} (\frac{\frac{P \tilde{v}}{R T} - 1}{P}) d P = \int_{0}^{P} {\frac{Z - 1}{P}} d P

(16.35)

Let us also derive the expression for the fugacity coefficient for a component in a multicomponent mixture. Following a pattern similar to that which we have presented, beginning with the definition of fugacity for a component in terms of chemical potential:

d μ_{i} = R T d ln f_{i} @ c o n s t T

(16.36)

This time, it is more convenient to use the Maxwell’s Relationships presented in equation (15.27d):

{(\frac{\partial μ_{i}}{\partial V})}_{T, n} = - {(\frac{\partial P}{\partial n_{i}})}_{T, V, n_{i \neq 1}}

(16.37)

After you introduce the definitions of fugacity coefficient and compressibility factor:

ϕ_{i} = \frac{f_{i}}{y_{i} P}

(16.38a)

P = \frac{Z n R T}{V}

(16.38b)

and recalling that our lower limit of integration is the ideal state, for which, at the limit $P - > 0$ :

V^{*} - > \infty

(16.39c)

ϕ i^{*} - > 1 and hence ln ϕ i^{*} - > 0

(16.39a)

z^{*} - > 1 and hence \ln Z^{*} - > 0

(16.39b)

it can be proven that:

\ln ϕ_{i} = \frac{1}{R T} \int_{\infty}^{v} {\frac{R T}{V} - {(\frac{\partial P}{\partial n_{i}})}_{T, V, n_{i \neq 1}}} d V - \ln Z

(16.40)

The multi-component mixture counterpart of equation (16.35) becomes:

\ln ϕ_{i} = \int_{0}^{P} {{\bar{Z}}_{i} - 1} \frac{d P}{P}

(16.41a)

where:

{\bar{Z}}_{i} = {(\frac{\partial Z}{\partial n_{i}})}_{P, T, n_{i \neq 1}} = \frac{P}{R T} {(\frac{\partial V}{\partial n_{i}})}_{P, T, n_{i \neq 1}} = \frac{P \bar{V_{i}}}{R T}

(16.41b)

Equations (16.34), (16.35), (16.40), and (16.41) are very important for us. Basically, they show that fugacity, or the fugacity coefficient, is a function of pressure, temperature and volume:

f = f (P, V, T)

This tells us that if we are able to come up with a PVT relationship for the volumetric behavior of a substance, we can calculate its fugacity by solving such expressions. It is becoming clear why we have studied equations of state — they are just what we need right now: PVT relationships for various substances. Once we have chosen the equation of state that we want to work with, we can calculate the fugacity of each component in the mixture by applying the above expression. Now that we know how to calculate fugacity, we are ready to apply the criteria for equilibrium that we just studied! That is the goal of the next module.

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