Episode 04. Fugacity Estimation and Vapor-Liquid Equilibrium
Summary
When dealing with the equilibrium of a specific system, we often utilize fugacity. In this episode, we discuss methods for estimating the fugacity of components and explore the vapor-liquid equilibrium in mixtures using the methods.
1. Introduction
As we mentioned in the previous episode, when dealing with the equilibrium of a specific system, we often utilize “fugacity”. Hence, to discuss equilibrium in the future, we will require the fugacity values. So, how can we estimate the fugacity of a component?
First, to deduce fugacity, we distinguish whether the component is pure or part of a mixture. Then, we choose the appropriate method based on the phase (typically gas, liquid, or solid) in which it is present. Below, we cover in detail how to estimate fugacity in various situations and then applying this, discuss vapor–liquid equilibrium in mixtures.
2. Fugacity of species
Let’s begin by understanding how we can estimate the fugacity of a pure substance. When estimating the fugacity of species, we often utilize (Eq 1.21) and (Eq 2.10).
\( {\small\displaystyle\begin{align} dG \quad&=\quad VdP\,-\,SdT \tag{Eq 1.21} \\[15pt] G \quad&=\quad g(T)\,+\,RTlnf_i \tag{Eq 2.10} \end{align} } \)
Assuming a constant temperature for the system, we can represent the Gibbs free energy using (Eq 1.21) as shown below. Now, let’s discuss the fugacity of pure species for each phase using the relevant equations.
\( {\small\displaystyle\begin{align} dG \,=\, VdP \quad\longrightarrow\quad G \,=\, \int_0^P V\,dP \tag*{} \end{align} } \)
Fugacity of pure gases
We can estimate the fugacity of pure gases by comparing the Gibbs free energy of real gases with that of ideal gases. The difference in Gibbs free energy for each state, known as residual Gibbs free energy, can be formulated as follows.
\( {\small\displaystyle\begin{align} G^R \quad=\quad \int_0^P\left(V\,-\,V^{ig}\right)\,dP \tag*{} \end{align} } \)
Here, by using the compressibility factor Z to represent the volume of a real gas, we can arrange the residual Gibbs free energy as follows. Due to the constant temperature conditions, we can shift the gas constant and temperature term to the left-hand side as (Eq 4.1).
\( {\small\displaystyle\begin{align} G^R \quad&=\quad \int_0^P\left(\frac{ZRT}{P}\,-\,\frac{RT}{P}\right)\,dP,\qquad PV \,=\, ZRT \\[15pt] \frac{G^R}{RT} \quad&=\quad \int_0^P\left(Z\,-\,1\right)\,\frac{dP}{P} \tag{Eq 4.1} \end{align} } \)
As mentioned in Episode 02, we can represent the residual Gibbs energy of pure components using the fugacity term. Based on (Eq 2.11), we can express \({\small\displaystyle G^R/RT}\) as follows.
\( {\small\displaystyle\begin{align} \frac{G^R}{RT} \quad=\quad ln\frac{f_i^v}{P} \quad=\quad ln\phi_i^v \tag*{} \end{align} } \)
Utilizing the fact that this equation is equivalent to (Eq 4.1), we can derive (Eq 4.2). By substituting the T and P terms on the right-hand side of (Eq 4.2) with Tr and Pr, we can obtain (Eq 4.3).
\( {\small\displaystyle\begin{align} \frac{f_i^v}{P} \quad&=\quad \exp\left[\int_0^P(Z\,-\,1)\,\frac{dP}{P}\right] \tag{Eq 4.2} \\[15pt] T_r\,&=\,\frac{T}{T_c}, \qquad P_r\,=\,\frac{P}{P_c} \quad\longrightarrow\quad dP\,=\,P_cdP_r \\[15pt] \frac{f_i^v}{P} \quad&=\quad \exp\left[\int_0^{P_r}\{Z(T_r,P_r)\,-\,1\}\frac{dP_r}{P_r}\right] \tag{Eq 4.3} \end{align} } \)
Fugacity of pure liquids
We can estimate the fugacity of pure liquids by comparing it to the fugacity of pure species in a saturated state. Just like when estimating the fugacity of pure gases, we use two ways to represent the difference in Gibbs free energy.
| \( {\small\displaystyle\begin{align} &\mathrm{i)}\quad G \quad=\quad \int_0^PV\,dP \\[15pt] &\quad \int_0^P V^l\,dP\,-\,\int_0^{P_i^{sat}} V^l\,dP \quad=\quad \int_{P_i^{sat}}^{P} V^l\,dP \end{align} } \) | \( {\small\displaystyle\begin{align} &\mathrm{ii)}\quad G \quad=\quad g(T)\,+\,RTlnf_i \\[15pt] &\quad G_i^l\,-\,G_i^{sat} \quad=\quad RTln\frac{f_i^l}{f_i^{sat}} \end{align} } \) |
\( {\small\displaystyle\begin{align} \text{By } \mathrm{i})\,\text{and}\,\mathrm{ii})\qquad& \\[15pt] ln\frac{f_i^l}{f_i^{sat}} \quad&=\quad \frac{1}{RT} \int_{P_i^{sat}}^P V^l\,dP \\[15pt] f_i^l \quad&=\quad f_i^{sat} \exp\left[\frac{1}{RT}\int_{P_i^{sat}}^P V^l\,dP\right] \tag{Eq 4.4} \end{align} } \)
The volume of liquids is not significantly affected by pressure compared to gases. Using this characteristic and representing \({\small\displaystyle f_i^{sat}}\) in terms of fugacity coefficient, \({\small\displaystyle \phi_i}\), and vapor pressure of pure species, we rearrange (Eq 4.4) into (Eq 4.5).
\( {\small\displaystyle\begin{align} f_i^l \quad=\quad \phi_i^{sat}P_i^{sat}\exp\left[\frac{V^l(P\,-\,P_i^{sat})}{RT}\right] \tag{Eq 4.5} \end{align} } \)
Fugacity of pure solids
The method for estimating the fugacity of pure solids is quite similar to that for liquids. However, \({\small\displaystyle P_i^{sat}}\) in (Eq 4.6) represents the solid-vapor saturation pressure (or sublimation pressure) value.
\( {\small\displaystyle\begin{align} f_i^s \quad=\quad \phi_i^{sat}P_i^{sat}\exp\left[\frac{V^s(P\,-\,P_i^{sat})}{RT}\right] \tag{Eq 4.6} \end{align} } \)
Fugacity of a component in mixtures
We can estimate the fugacity of a component in mixtures using the activity coefficients, \({\small\displaystyle\gamma_i}\) defined in Episode 02.
\( {\small\displaystyle\begin{align} \gamma_i \,\equiv\, \frac{\hat{f_i}}{x_if_i} \quad\longrightarrow\quad \hat{f_i} \,=\, x_i\gamma_if_i \tag{Eq 4.7} \end{align} } \)
3. Vapor-liquid equilibrium in mixtures
As illustrated in (Figure 4.1), let’s assume that a mixture coexists in the gas and liquid phases at a phase equilibrium state. We know that when a mixture reaches phase equilibrium in the closed system at the given temperature, the fugacity of each component within the mixture is identical in both phases. Therefore, we can establish the equation as (Eq 4.8). Let’s define the mole fractions of component i in the gas phase as yi and in the liquid phase as xi. By substituting the \({\small\displaystyle \hat{f_i}^v}\) terms in (Eq 4.8) with the expressions from (Eq 4.7), we can obtain (Eq 4.9).
\( {\small\displaystyle\begin{align} \hat{f_i}^v \quad&=\quad \hat{f_i}^l \tag{Eq 4.8} \\[15pt] y_i\gamma_i^vf_i^v \quad&=\quad x_i\gamma_i^lf_i^l \tag{Eq 4.9} \end{align} } \)
(Figure 4.1)
Here, instead of the term \({\small\displaystyle y_i\gamma_i^vf_i^v}\) corresponding to the gas phase, we can express it using the system’s pressure and fugacity coefficients of a component in the mixture.
\( {\small\displaystyle\begin{align} y_i\gamma_i^vf_i^v \quad&=\quad y_i\cdot\frac{\hat{f_i}^v}{y_if_i^v}\cdot P\cdot\frac{f_i^v}{P} \\[15pt] &=\quad y_i\cdot P\cdot \frac{f_i^v}{y_iP} \\[15pt] &=\quad y_iP\hat{\phi}^v_i \tag*{} \end{align} } \)
Then, by replacing \({\small\displaystyle f_i^l}\) on the right-hand side of (Eq 4.9) with the expression from (Eq 4.5), we can arrange (Eq 4.9) as follows. Additionally, we can simplify (Eq 4.10) to a more concise form using the newly defined \({\small\displaystyle\Phi_i}\).
\( {\small\displaystyle\begin{align} y_i\hat{\phi}^v_iP \quad&=\quad x_i\gamma_i^l\phi_i^{sat}P_i^{sat}\exp\left[\frac{V^L(P\,-\,P_i^{sat})}{RT}\right] \tag{Eq 4.10} \\[15pt] \Phi_i \quad&\equiv\quad \frac{\hat{\phi_i}^v}{\phi_i^{sat}}\exp\left[\frac{V^l(P_i^{sat}\,-\,P)}{RT}\right] \\[15pt] y_i\Phi_iP \quad&=\quad x_i\gamma_i^lP_i^{sat} \tag{Eq 4.11} \end{align} } \)
Let’s consider that the mixtures present in both the vapor and liquid phases are ideal. In the case of the ideal mixtures, the \({\small\displaystyle \Phi_i}\) and \({\small\displaystyle\gamma_i^l}\) terms in (Eq 4.11) become 1. And we commonly refer to (Eq 4.12) as Raoult’s law.
\( {\small\displaystyle\begin{align} y_iP \quad=\quad x_iP_i^{sat} \tag{Eq 4.12} \end{align} } \)
Nomenclature
\({\small\displaystyle f_i}\) : fugacity of pure species
\({\small\displaystyle \hat{f_i}}\) : fugacity of a component in mixtures
\({\small\displaystyle G}\) : Gibbs free energy
\({\small\displaystyle P}\) : pressure
\({\small\displaystyle T}\) : temperature
\({\small\displaystyle V}\) : volume
\({\small\displaystyle x_i,\,y_i}\) : mole fraction
\({\small\displaystyle\gamma_i}\) : activity coefficient
\({\small\displaystyle\phi_i}\) : fugacity coefficient of pure species
\({\small\displaystyle\hat{\phi_i}}\) : fugacity coefficient of a component in mixtures