# **Step-by-Step Solution: Analytical Expressions for Fugacity and Fugacity Coefficient** Below, I present the general analytical expressions for the **fugacity** (\( f_i \)) and **fugacity coefficient** (\( \phi_i \)) for a species \( i \) in a mixture, for both the **Ideal Gas Equation of State (IGEoS)** and a **Generic Cubic Equation of State (GCEoS)**. I provide the fundamental equations, and explain how you would implement or adapt them in **MS Excel** or **MATLAB**. --- ## **1. Ideal Gas Equation of State (IGEoS)** For an ideal gas, the fugacity is simply the partial pressure, and the fugacity coefficient is always 1. \[ \boxed{ f_i^{\text{IG}} = y_i P } \] where \( y_i \) is the mole fraction of \( i \), and \( P \) is the total pressure. \[ \boxed{ \phi_i^{\text{IG}} = 1 } \] **Excel/Matlab Implementation:** No advanced coding needed; just use \( f_i = y_i \cdot P \) and \( \phi_i = 1 \). --- ## **2. Generic Cubic Equations of State (GCEoS)** For a **generic cubic EOS**, such as vdW, RK, SRK, or PR, the fugacity coefficient is given by: \[ \ln \phi_i = \int_{}^{P} \left( \frac{\bar{V}_i - RT/P}{RT} \right) dP \] But for cubic EOS, a closed form is derived from the EOS parameters: ### **General GCEoS Formulation** The GCEoS is: \[ P = \frac{RT}{V_m - b} - \frac{a}{V_m^2 + u b V_m + w b^2} \] (Parameters \( a, b, u, w \) depend on the EOS type, as in your tables.) #### **Fugacity Coefficient Expression** For species \( i \) in a mixture: \[ \boxed{ \ln \phi_i = \frac{b_i}{b} (Z - 1) - \ln(Z - B) - \frac{A}{2\sqrt{2}B} \left[ \frac{2 \sum_j y_j a_{ij}}{a} - \frac{b_i}{b} \right] \ln \left( \frac{Z + (1 + \sqrt{2})B}{Z + (1 - \sqrt{2})B} \right) } \] Where: - \( Z \) is the compressibility factor: \( Z = PV_m / RT \) - \( A = \frac{aP}{R^2T^2} \) - \( B = \frac{bP}{RT} \) - \( a = \sum_i \sum_j y_i y_j a_{ij} \) - \( b = \sum_i y_i b_i \) - \( a_{ij} = \sqrt{a_i a_j}(1-k_{ij}) \) (binary interaction, \( k_{ij} \) often zero) - \( a_i, b_i \) are species parameters (see your tables/formulas) - \( y_i \) = mole fraction of \( i \) #### **Fugacity** \[ f_i = \phi_i y_i P \] --- ### **How to Apply or Implement in Excel/Matlab** 1. **Input Parameters:** \( T, P, y_i, a_i, b_i \) for all components. 2. **Calculate mixture parameters:** \( a, b \). 3. **Solve for \( Z \):** Use the cubic EOS to solve for \( Z \) (root-finding in Excel or Matlab). 4. **Calculate \( A, B \):** Using mixture \( a \), \( b \). 5. **Calculate \( \ln \phi_i \)** using the big boxed formula above. 6. **Find \( \phi_i \) and \( f_i \):** \[ \phi_i = \exp(\ln \phi_i) \] \[ f_i = \phi_i y_i P \] 7. **Excel:** Use spreadsheet formulas and Solver for cubic roots. 8. **Matlab:** Use built-in root solvers (e.g., `roots()`, `fzero()`). --- ### **Editable MathType Equations** You can copy the following equations into MathType or a Word Equation editor: **Fugacity Coefficient:** ``` ln φ_i = (b_i/b)(Z-1) - ln(Z-B) - (A/(2√2 B)) * [2 Σ_j y_j a_ij / a - b_i/b] * ln((Z+(1+√2)B)/(Z+(1-√2)B)) ``` **Fugacity:** ``` f_i = φ_i y_i P ``` --- ## **Summary Table** | EOS | \( \phi_i \) Formula | |----------|----------------------| | IGEoS | \( \phi_i = 1 \) | | GCEoS (vdW, RK, SRK, PR) | \( \ln \phi_i = \text{see boxed formula above} \) | --- ## **References** - Smith, J.M., Van Ness, H.C., Abbott, M.M. **Introduction to Chemical Engineering Thermodynamics** - Sandler, S.I. **Chemical and Engineering Thermodynamics** --- If you want to **implement this in Excel or Matlab**, program each step above as indicated. The key is to solve for \( Z \) first, then substitute all values into the fugacity coefficient expression. **Let me know if you need a sample Matlab or Excel code for the cubic EOS root and fugacity computation!**