Episode 02. Residual Properties, Excess Properties and Fugacity
Summary
There are differences between the actual properties and the ones in the ideal state. In this episode, we discuss the concepts of residual properties and excess properties that refer to the differences. Finally, we cover a new property called fugacity by comparing the property in the real state with the ideal state.
1. Introduction
Assuming a substance is in the ideal state allows us to simplify equations and understand various phenomena easily. However, the properties in the ideal state are not practical when explaining certain phenomena or in situations where high accuracy is required. In this episode, we will introduce several definitions to discuss the properties in the real state and compare them with the ideal state. When comparing properties between real states and ideal states, we deal with properties under various conditions. Typical examples include mixture properties, properties of pure components, and properties of individual components in mixtures. Table 2.1 summarizes these properties. M is the molar value of extensive properties.
| Condition | Mixture property | Pure species property | Partial molar property |
|---|---|---|---|
| Real state | \( \displaystyle M \) | \( \displaystyle M_i \) | \( \displaystyle \left[\frac{\partial (nM)}{\partial n_i}\right]_{P,T,n_j} \quad=\quad \bar{M}_i \) |
| Ideal-gas state | \( \displaystyle M^{ig} \) | \( \displaystyle M^{ig}_i \) | \( \displaystyle \left[\frac{\partial (nM^{ig}_i)}{\partial n_i}\right]_{P,T,n_j} \quad=\quad \bar{M}^{ig}_i \) |
| Ideal-solution state | \( \displaystyle M^{id} \) | \( \displaystyle M^{id}_i \) | \( \displaystyle \left[\frac{\partial (nM^{id}_i)}{\partial n_i}\right]_{P,T,n_j} \quad=\quad \bar{M}^{id}_i \) |
The mixture properties in (Table 2.1) are equivalent to the ssum of the individial component’s partial molar properties.
\( {\small\displaystyle\begin{align} nM \quad=\quad \sum_i n_i\bar{M}_i \quad\text{or}\quad M \quad=\quad \sum_i x_i\bar{M}_i \tag{Eq 2.1} \end{align} } \)
Furthermore, at the same temperature and partial pressure of the component, the partial molar properties (except for volume) in the ideal gas mixture are equivalent to the properties of the pure species. However, for ideal gases, mixture molar volume, pure species volume, and partial molar volume are identical under the same temperature and pressure conditions.
\( {\small\displaystyle\begin{align} M^{ig}_i(T,p_i) \quad=\quad \bar{M}^{ig}_i(T,P) \tag{Eq 2.2} \end{align} } \)
\( {\small\displaystyle\begin{align} ^*\, V^{ig}_i \quad=\quad \bar{V}^{ig}_i \quad=\quad V^{ig} \quad=\quad \frac{RT}{P} \tag*{} \end{align} } \)
2. Residual properties & Excess properties
Let’s compare the properties in the real state and those in the ideal gas states under the same temperature and pressure. We define the difference between these two properties as the residual property. The residual property is expressed as shown in (Eq 2.3).
\( {\small\displaystyle\begin{align} M^R \quad&\equiv\quad M\,-\,M^{ig} \tag{Eq 2.3} \end{align} } \)
\( {\small\displaystyle\begin{pmatrix} V^R &=& V\,-V^{ig}\\ H^R &=& H\,-\,H^{ig}\\ G^R &=& G\,-\,G^{ig}\\ &\vdots& \end{pmatrix} \tag*{} } \)
When (Eq 2.3) is applied to the pure species, it can be expressed as follows.
\( {\small\displaystyle\begin{align} M^R_i \quad&=\quad M_i\,-\,M^{ig}_i \tag{Eq 2.4} \end{align} } \)
What about the residual property of a mixture? Let’s consider a mixture with n moles. Based on (Eq 1.6), we can multiply both sides of (Eq 2.3) by the mole number n. Then, by taking the partial derivative of (Eq 2.5) with respect to the mole number of each component. We can express it in terms of partial molar property as (Eq 2.6).
\( {\small\displaystyle\begin{align} nM^R \quad&=\quad nM\,-\,nM^{ig} \tag{Eq 2.5} \\[15pt] \left[\frac{\partial (nM^R)}{\partial n_i}\right]_{P,T,n_j} \quad&=\quad \left[\frac{\partial (nM)}{\partial n_i}\right]_{P,T,n_j}\,-\,\left[\frac{\partial (nM^{ig})}{\partial n_i}\right]_{P,T,n_j} \\[15pt] \bar{M}^R_i \quad&=\quad \bar{M}_i\,-\,\bar{M}^{ig}_i \tag{Eq 2.6} \end{align} } \)
Excess properties refer to the difference in properties between real solutions and ideal solutions. Excess properties can also be represented in terms of partial molar properties. The deriving process is identical to that of obtaining partial molar residual properties.
\( {\small\displaystyle\begin{align} M^E \quad&\equiv\quad M\,-\,M^{id} \tag{Eq 2.7} \\[15pt] \bar{M}^E_i \quad&=\quad \bar{M}_i\,-\,\bar{M}^{id}_i \tag{Eq 2.8} \end{align} } \)
3. Comparison of Gibbs energy in the real and ideal state
Gibbs energy often becomes an important indicator when discussing thermodynamics. To utilize it in various scenarios, we need to define the Gibbs energy in each state.
Pure species in the ideal gas state & Pure species in the real state
Assuming a constant temperature and no change in composition, we can simplify (Eq 1.24) as follows.
\( {\small\displaystyle\begin{align} d(nG) \quad=\quad (nV)dP\,-\,(nS)dT\,+\,\sum_i\mu_i\,dn_i \tag{Eq 1.24} \end{align} } \)
\( {\small\displaystyle\begin{align} d(nG) \quad=\quad (nV)dP \tag*{} \end{align} } \)
Let’s apply the equation to the pure species in the ideal gas state. Through the following steps, we can derive the equation for the Gibbs energy under these specific conditions.
\( {\small\displaystyle\begin{align} dG^{ig}_i \quad=\quad V^{ig}dP \quad=\quad \frac{RT}{P}dP \quad\longrightarrow\quad \int dG^{ig}_i \quad=\quad \int \frac{RT}{P}dP \tag*{} \end{align} } \)
\( {\small\displaystyle\begin{align} G^{ig}_i(T,P) \quad&=\quad g(T)\,+\,RTlnP \tag{Eq 2.9} \end{align} } \)
What about the Gibbs energy of pure substances in the real state? In such cases, the ideal gas equation is not applicable. Therefore, we introduce a property known as fugacity(f) instead of the pressure(P) in (Eq 2.9). Fugacity(f) substitutes for pressure, so it has the same unit as pressure.
\( {\small\displaystyle\begin{align} G_i(T,P) \quad&=\quad g(T)\,+\,RTlnf_i \tag{Eq 2.10} \end{align} } \)
Based on (Eq 2.9) and (Eq 2.10), we can calculate the residual Gibbs free energy of pure species. \(\phi_i\) in (Eq 2.11) refers to the fugacity coefficient of pure species.
\( {\small\displaystyle\begin{align} G^{R}_i \quad&=\quad G_i\,-\,G^{ig}_i \tag*{} \end{align} } \)
\( {\small\displaystyle\begin{align} G^R_i \quad&=\quad RTln\left(\frac{f_i}{P}\right)\quad=\quad RTln\phi_i,\quad \phi_i\quad\equiv\quad\frac{f_i}{P} \tag{Eq 2.11} \end{align} } \)
Mixtures in the ideal gas state & Mixtures in the real state
Let’s extend this concept to the Gibbs free energy of a component in a mixture. As mentioned in the introduction, we can set the following relationship based on (Eq 2.2).
\( {\small\displaystyle\begin{align} G^{ig}_i(T,p_i) \quad=\quad \bar{G}^{ig}_i(T,P) \tag*{} \end{align} } \)
By (Eq 2.9) and the above relationship, we can express the partial molar Gibbs free energy in the ideal gas state as shown in (Eq 2.12). When a mixture is in the ideal gas state, we can represent the partial pressure as the product of the mole fraction and the total pressure.
\( {\small\displaystyle\begin{align} \bar{G}^{ig}_i(T,P) \quad&=\quad g(T)\,+\,RTlnp_i \tag{Eq 2.12} \\[15pt] &=\quad g(T)\,+\,RTln(y_iP) \end{align} } \)
By the definition of partial molar property, the partial molar Gibbs free energy is equivalent to the chemical potential. By rearranging (Eq 2.12), we derive the relationship between the chemical potential and Gibbs free energy of a pure component in the ideal gas state.
\( {\small\displaystyle\begin{align} \mu_i \quad=\quad \left[\frac{\partial(nG)}{\partial n_i}\right]_{P,T,n_j} \quad\equiv\quad \bar{G}_i \tag*{} \end{align} } \)
\( {\small\displaystyle\begin{align} \longrightarrow\quad \mu^{ig}_i \quad&=\quad g(T)\,+\,RTln(y_iP) \\[15pt] &=\quad g(T)\,+\,RTlnP\,+\,RTlny_i \\[15pt] \mu^{ig}_i \quad&=\quad G^{ig}_i(T,P)\,+\,RTlny_i \tag{Eq 2.13} \end{align} } \)
For a mixture in the real state, the partial molar Gibbs energy (or chemical potential of a pure component) is defined as (Eq 2.14). where \(\hat{f}_i\) denotes the fugacity of component i in the mixture.
\( {\small\displaystyle\begin{align} \bar{G}_i(\text{or }\mu_i) \quad&=\quad g(T)\,+\,RTln\hat{f}_i \tag{Eq 2.14} \end{align} } \)
Similarly, by substituting (Eq 2.12) and (Eq 2.14) into (Eq 2.3), we can obtain the equation for the individual’s residual Gibbs energy of the mixture. We define the ratio \(\small\hat{f}_i/y_iP\) as \(\hat{\phi}_i\), the fugacity coefficient of component i in the mixture.
\( {\small\displaystyle\begin{align} \bar{G}^{R}_i(\text{or }\mu^R_i) \quad=\quad \bar{G}_i\,-\,\bar{G}^{ig}_i \tag*{} \end{align} } \)
\( {\small\displaystyle\begin{align} \bar{G}^R_i(\text{or }\mu^R_i) \quad=\quad RTln\left(\frac{\hat{f}_i}{y_iP}\right) \quad=\quad RTln\hat{\phi}_i ,\quad \hat{\phi}_i\equiv\frac{\hat{f}_i}{y_iP} \tag{Eq 2.15} \end{align} } \)
Mixtures in the ideal solution state & Mixtures in the real state
An ideal gas mixture is an ideal solution consisting of ideal gas components. If we extend (Eq 2.13) to the ideal-solution model, partial molar Gibbs energy in the ideal solution state is defined as (Eq 2.16). Instead of using gas mole fractions(yi), we substitute them with mole fractions(xi) and replace the Gibbs energy of pure species in the ideal-gas state with that in the real state. By substituting (Eq 2.10) into the Gibbs energy term of pure species, we can derive (Eq 2.17).
\( {\small\displaystyle\begin{align} \bar{G}^{id}_i(\text{or }\mu^{id}_i) \quad&=\quad G_i(T,P)\,+\,RTlnx_i \tag{Eq 2.16} \\[15pt] \bar{G}^{id}_i(\text{or }\mu^{id}_i) \quad&=\quad g(T)\,+\,RTln(x_if_i) \tag{Eq 2.17} \end{align} } \)
Now, let’s dertermine the excess partial molar Gibbs energy in an ideal solution by utilizing (Eq 2.14) and (Eq 2.17). We define the ratio \(\small\hat{f}_i/x_if_i\) as \(\gamma_i\), the activity coefficient of component i in the mixture.
\( {\small\displaystyle\begin{align} \bar{G}^E_i(\text{or }\mu^E_i) \quad=\quad \bar{G}_i\,-\,\bar{G}^{id}_i \tag*{} \end{align} } \)
\( {\small\displaystyle\begin{align} \bar{G}^E_i(\text{or }\mu^E_i) \quad=\quad RTln\left(\frac{\hat{f}_i}{x_if_i}\right) \quad=\quad RTln\gamma_i, \quad \gamma_i \,\equiv\, \frac{\hat{f}_i}{x_if_i} \tag{Eq 2.18} \end{align} } \)
Understanding the newly defined properties in this episode will be valuable going forward. Particularly, we utilize these properties in predicting equilibrium. Detail discussion about their application will be covered in another episode.
Nomenclature
\({\small\displaystyle f}\) : fugacity
\({\small\displaystyle G}\) : Gibbs free energy
\({\small\displaystyle H}\) : enthalpy
\({\small\displaystyle M}\) : molar extensive property in the real state
\({\small\displaystyle n}\) : mole numbers
\({\small\displaystyle P}\) : pressure
\({\small\displaystyle p_i}\) : partial pressure of component i
\({\small\displaystyle R}\) : ideal gas constant
\({\small\displaystyle T}\) : temperature
\({\small\displaystyle V}\) : molar volume
\({\small\displaystyle x,\,y}\) : mole fraction
\({\small\displaystyle \gamma_i}\) : activity coefficient of component i in the mixture
\({\small\displaystyle \mu}\) : chemical potential
\(\phi\) : fugacity coefficient