Mixtures and Phases — Molecular Foundations to Engineering Models (Nonreactive, Reactive, and Electrolytes)

Scope: first-principles derivations for mixtures. Unifies macroscopic thermodynamics, statistical mechanics, and chemical-engineering practice. Covers partial molar properties, Gibbs–Duhem, entropy of mixing, VLE/LLE/SLE criteria, γ–φ and φ–φ methods, EOS mixing rules, reactive equilibria and reactive flash, and electrolyte solution theory (Debye–Hückel and Pitzer). Includes algorithms and worked templates.

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1. Preliminaries and Notation

• Components $i = 1…N$, moles $n_i$, total $n = \sum_i n_i$, mole fractions $y_i$ (vapor), $x_i$ (liquid), $z_i$ (feed).
• Chemical potential: $\mu_i(T,P,\{n_j\})$. For pure species at $(T,P)$: $\mu_i = \bar{g}_i = \text{molar Gibbs free energy}$.
• Differential of $G$ for a closed, multicomponent, simple-compressible system: $dG = -S\,dT + V\,dP + \sum_i \mu_i\,dn_i.$ Thus $\mu_i = (\partial G/\partial n_i)_{T,P,n_{j\ne i}}.$
• Partial molar property $\overline{M}_i = (\partial (nM)/\partial n_i)_{T,P,n_{j\ne i}}.$

Gibbs–Duhem: at constant T,P, $\sum_i n_i d\mu_i = 0$ or in mole-fraction form $\sum_i x_i d\mu_i = 0.$

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2. Statistical Basis of Mixture Entropy and $\mu_i$

2.1 Entropy of Mixing from Microstates

Microcanonical counting for two ideal gases A,B initially separated then mixed at identical T,P: $\Delta S_{mix} = -R \sum_i n_i \ln x_i.$ Origin: indistinguishability and combinatorics ($\ln N!$ via Stirling’s approximation). Removes Gibbs paradox.

2.2 Canonical Derivation of $\mu_i$

For mixture partition function $Z = (\prod_i Z_i^{N_i}/N_i!) Z_{conf}$. Helmholtz energy $A=-k_B T\ln Z$. Then at fixed T,V,N_j: $\mu_i = \left(\frac{\partial A}{\partial N_i}\right)_{T,V,N_{j\ne i}}.$ For ideal gas mixture with no configurational interactions: $\mu_i^{ig}(T,P,y_i) = \mu_i^{\circ}(T) + RT\ln(y_i P/P^\circ).$

2.3 Excess Properties

Define excess relative to ideal mixture at same T,P,x: $M^E = M - M^{id}.$ Then $G^E$ encodes nonideality; activity coefficients follow: $\ln \gamma_i = (\partial (G^E/RT)/\partial n_i)_{T,P,n_{j\ne i}}.$

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3. Phase Equilibrium Criteria and Working Forms

3.1 Fundamental Conditions

At fixed T,P: equality of chemical potentials of each component across phases $\alpha,\beta$: $\mu_i^{(\alpha)}(T,P,\{x_j^{(\alpha)}\}) = \mu_i^{(\beta)}(T,P,\{x_j^{(\beta)}\}).$

3.2 Fugacity and Activity

• Fugacity of component i in phase $\phi$: $f_i^{(\phi)}$. For a pure fluid: $\mu = \mu^{ig}(T,P) + RT\ln(f/P).$
• In mixtures: $\mu_i = \mu_i^{ig}(T,P,y_i) + RT\ln \phi_i$ with fugacity coefficient $\phi_i$ from an EOS; or $\mu_i = \mu_i^{\circ}(T,P) + RT\ln a_i$ with activity $a_i = \gamma_i x_i$ in liquids.

3.3 $\gamma$–$\phi$ Formulation for VLE

At equilibrium between liquid L and vapor V for each i: $f_i^{(L)} = f_i^{(V)} \Rightarrow \gamma_i x_i f_i^{\circ}(T,P) = y_i \phi_i P.$ Choose standard state so that $f_i^{\circ}$ is convenient:

• Lewis–Randall (ideal solution): $f_i^{\circ} = f_i^{pure}(T,P)\Rightarrow a_i=x_i.$
• Raoult (solvent at low P): $f_i^{\circ} = P_i^{sat}(T)\Rightarrow y_i P = x_i \gamma_i P_i^{sat}\phi_i.$
• Henry (dilute solute): $f_i^{\circ} = H_i(T)\Rightarrow y_i P = x_i \gamma_i H_i.$

3.4 $\phi$–$\phi$ with a Single EOS

Use a cubic EOS for both phases. Compute $\phi_i^{(L)}, \phi_i^{(V)}$ via mixing rules. Solve $y_i \phi_i^{(V)} P = x_i \phi_i^{(L)} P.$

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4. Activity–Coefficient Models ($G^E$ Models)

• Margules/Van Laar/Wilson: closed forms for $G^E/RT$ → $\ln \gamma_i$.
• NRTL: captures nonrandomness with parameter $\alpha$; widely used for nonelectrolytes and some associating systems.
• UNIQUAC: separates combinatorial and residual contributions using surface-area and volume parameters.
• UNIFAC: group-contribution variant of UNIQUAC.

Relation to excess Gibbs: $G^E = RT \sum_i x_i \ln \gamma_i$. Consistency with Gibbs–Duhem must hold.

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5. EOS Mixing and Combining Rules

For cubic EOS with mixture parameters $a(T), b$: $a = \sum_i\sum_j x_i x_j a_{ij},\quad a_{ij} = (1-k_{ij})\sqrt{a_i a_j},\quad b = \sum_i x_i b_i.$ Two-parameter (vdW-1f) rules above; alternatives include Huron–Vidal and Wong–Sandler, which match excess-Gibbs behavior at infinite pressure to couple EOS with $G^E$ models.

Fugacity coefficients from EOS: $\ln \phi_i = \int_0^P \left(\frac{\bar v_i}{RT} - \frac{1}{P'}\right) dP' + \text{EOS-specific terms}.$ Closed forms exist for Peng–Robinson and SRK.

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6. Flash and Phase-Split Calculations

6.1 Isothermal–Isobaric VLE Flash (Two Phases)

Unknowns: $\beta$ (vapor fraction), $x_i, y_i$. Define $K_i = y_i/x_i$. Overall balance and Rachford–Rice: $\sum_i \frac{z_i (K_i-1)}{1 + \beta (K_i-1)} = 0.$ Given T,P: compute K_i from EOS (φ–φ) or from $\gamma$–$\phi$ with $K_i = \frac{\gamma_i f_i^{\circ}}{\phi_i P}.$

6.2 Multiphase and LLE

Extend with stability analysis (tangent plane distance). Solve for multiple $\beta^{(\alpha)}$ and compositions that minimize G at fixed T,P,n.

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7. Reactive Mixtures

7.1 Equilibrium Conditions

For reactions $r = 1…R$ with stoichiometric matrix $\nu_{ir}$ (positive for products): chemical equilibrium at T,P minimizes G subject to element balances, or equivalently sets affinity to zero: $\mathcal A_r = -\sum_i \nu_{ir} \mu_i = 0.$

7.2 Equilibrium Constants and Activities

Standard state definition gives $\Delta_r G^{\circ}(T) = -RT \ln K_r(T).$ For ideal gases: $K_p = \prod_i (y_i P/P^{\circ})^{\nu_{ir}}.$ For real systems, replace with activities or fugacities: $K = \prod_i a_i^{\nu_{ir}} = \prod_i (\phi_i y_i P/P^{\circ})^{\nu_{ir}} \text{ (gas)},\quad a_i = \gamma_i x_i \text{ (liquid)}.$

7.3 Extent of Reaction Formulation

Let $\xi_r$ be extents. Species moles $n_i = n_i^0 + \sum_r \nu_{ir} \xi_r.$ Minimize $G(T,P,\{n_i\})$ with respect to $\xi_r$ under bounds; or solve equilibrium conditions $\mathcal A_r=0$ with activities. Newton–Raphson on $\xi$ is standard.

7.4 Reactive Flash

Combine Rachford–Rice with reaction stoichiometry. Unknowns: $\beta, x_i, y_i, \{\xi_r\}.$ Solve nested: outer loop on $\beta$ and $\xi$, inner on phase composition with $K_i$ that depend on activities of updated compositions.

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8. Electrolyte and Ionic Solution Theory

8.1 Electrochemical Potential and Electroneutrality

For ion $i$ with charge $z_i$: $\tilde\mu_i = \mu_i + z_i F \Phi,$ where $\Phi$ is the electric potential and F is Faraday’s constant. Only differences in electrochemical potential are measurable. Macroscopic solutions satisfy electroneutrality: $\sum_i z_i m_i = 0$ for molalities $m_i$.

8.2 Activities and Mean Ionic Activity Coefficient

Define mean ionic quantity for electrolyte $A_{\nu_+} B_{\nu_-}$: $a_{\pm} = (a_+^{\nu_+} a_-^{\nu_-})^{1/(\nu_++\nu_-)}, \quad \gamma_{\pm} = (\gamma_+^{\nu_+} \gamma_-^{\nu_-})^{1/(\nu_++\nu_-)}.$ Chemical equilibria use activities $a_i = \gamma_i m_i$ (molality standard) or $\gamma_i x_i$ (mole fraction standard).

8.3 Debye–Hückel Theory (Dilute Limit)

Ionic atmosphere yields $\log_{10} \gamma_i = -\frac{A z_i^2 \sqrt{I}}{1 + B a_i \sqrt{I}}, \quad I = \tfrac{1}{2} \sum_j z_j^2 m_j.$ $A$ and $B$ depend on T and solvent dielectric/ density; $a_i$ is an ion-size parameter. Limiting law for very dilute: $\log_{10} \gamma_i = -A z_i^2 \sqrt{I}.$

8.4 Extended Debye–Hückel and Davies

For moderate $I$, use extended forms or Davies equation: $\log_{10} \gamma_i = -A z_i^2 \left[ \frac{\sqrt{I}}{1+\sqrt{I}} - 0.3 I \right].$

8.5 Pitzer Equations (High Ionic Strength)

Excess Gibbs energy expressed via virial-like terms in molality with binary and ternary interaction parameters ($\beta^{(0)}, \beta^{(1)}, C^\phi$). Provides $\phi$ (osmotic coefficient) and $\gamma_\pm$: $G^E/(RT) = \sum_{i,j} m_i m_j f_{ij}(I) + \sum_{i,j,k} m_i m_j m_k g_{ijk}(I).$ Differentiation gives activity coefficients and osmotic coefficients consistent with Gibbs–Duhem and electroneutrality. Calibrate to osmotic and EMF data.

8.6 Speciation, Acid–Base, and Complexation

Mass-action with activities: $K(T) = \prod_i a_i^{\nu_{ir}}.$ Solve coupled equilibrium and electroneutrality for species including water autoionization, acid–base pairs, and complexes. Ionic strength appears in $\gamma_i$. Use Newton iterations on logarithms of primary variables (e.g., pH, log $m_i$).

8.7 Osmotic Pressure and Water Activity

From $\phi$ and solvent chemical potential: $\mu_w = \mu_w^\circ + RT\ln a_w$. Osmotic pressure $\Pi$ relates via: $\Pi = - (RT/\bar v_w) \ln a_w.$

8.8 Transport Note

Electrical conductivity and diffusion follow from Nernst–Planck; thermodynamic factors come from activity-coefficient matrix $\Gamma_{ij} = \partial \ln a_i / \partial \ln m_j$.

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9. Azeotropy and Phase Behavior

Azeotrope at T,P where $y_i = x_i\,\forall i$. For $\gamma$–$\phi$ with Raoult standard: $K_i = \frac{\gamma_i P_i^{sat}}{\phi_i P}.$ Criterion for binary minimum-boiling azeotrope at given P: $d(\ln K_1)/d x_1 = d(\ln K_2)/d x_1$ at $x_1=x_1^{az}$; equivalent tangent-plane degeneracy in $G^E$. For LLE, use equality of tangent planes to $G^E$ surface.

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10. Algorithms and Numerical Templates

10.1 $\gamma$–$\phi$ Isothermal Flash (Two Phases)

1. Guess $\beta$. 2) Compute $K_i$ using current $x$ via model: $\gamma(x,T)$, $\phi(T,P,y)$. 3) Solve Rachford–Rice for $\beta$. 4) Update $x,y$. 5) Iterate to convergence; enforce nonnegativity and $\sum x_i=\sum y_i=1$.

10.2 $\phi$–$\phi$ with Cubic EOS

1. Stability test via tangent-plane distance. 2) If split, initialize compositions from Wilson $K_i$. 3) Solve equality of component fugacities with EOS fugacity coefficients and mixing rules. 4) Update phase fractions.

10.3 Reactive Flash

Minimize $G(T,P)$ with variables $(\beta, \text{compositions}, \xi_r)$. Alternative: nested loop. Use element conservation matrix $A$ (elements × species) to reduce variables. Apply line search on $\xi$ to keep $n_i \ge 0$.

10.4 Electrolyte Speciation + VLE/LLE

Loop: given T,P, total analytical concentrations, solve speciation with activity model (DH/Pitzer). Then compute phase split using effective activities/fugacities. Iterate until consistency between speciation and phase equilibrium.

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11. Worked Examples (Symbolic Templates)

1. Binary VLE with NRTL at fixed T,P: derive $\ln \gamma_i$. Compute $K_i$ and solve RR for $\beta$. Plot y–x and T–x if sweeping T.
2. EOS $\phi$–$\phi$ flash for PR: compute $a,b$ via mixing rules, solve cubic for compressibility factors $Z^{(L)}, Z^{(V)}$, then $\ln \phi_i^{(\phi)}$, iterate on compositions.
3. Reactive equilibrium $A + B \rightleftharpoons C$ at T,P with nonideality: write Gibbs, $K(T)$, express activities using $\gamma(x)$ or $\phi(y)$. Solve for $\xi$.
4. Electrolyte: aqueous NaCl at 25 °C. Compute $I, \gamma_\pm$ via extended Debye–Hückel. Compare to Pitzer at molalities up to 6 m. Derive osmotic coefficient and water activity.

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12. Summary Equations

• Gibbs–Duhem: $\sum x_i d\mu_i = 0$.
• $\mu$ and activities: $\mu_i = \mu_i^{\circ} + RT\ln a_i$.
• VLE $\gamma$–$\phi$: $\gamma_i x_i f_i^{\circ} = y_i \phi_i P$.
• Rachford–Rice: $\sum z_i (K_i-1)/(1+\beta(K_i-1)) = 0$.
• Reaction: $\Delta_r G^{\circ} = -RT\ln K$; $\mathcal A = -\sum \nu_i \mu_i = 0$ at equilibrium.
• Debye–Hückel: $\log_{10} \gamma_i = -A z_i^2 \sqrt{I}/(1+B a_i \sqrt{I})$.
• Pitzer: osmotic and activity via binary/ternary interaction terms in molality.

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13. Cross-Links

• See pure-substances.md for EoS fundamentals and residual functions.
• See second-law.md for stability and Legendre transforms.
• Upcoming chemical-thermodynamics.md will expand electrolyte models and transport.