6.1 Phase Equilibria and VLE Fundamentals

6.1 Phase Equilibria and VLE Fundamentals

Quick Answer: At equilibrium, each component has equal fugacity in every phase. For ideal VLE use Raoult’s law $y_i P = x_i P_i^{\mathrm{sat}}(T)$; for dilute solutes use Henry’s law. The Gibbs phase rule tells how many intensive variables you may set freely. Know critical point vs triple point as pure-fluid definitions.

Domain B (thermodynamics and phase equilibria) carries about 15% of this guide's planning allocation. Chapter 5 covered laws, equations of state, and property relations. This chapter applies that machinery to phase equilibria and chemical reaction equilibrium—the language of flash drums, distillation feeds, and equilibrium-limited reactors in Qatar’s gas processing, refining, and petrochemical plants.

Gibbs Phase Rule

For a nonreactive multiphase system of (C) independent components and (P) phases, the number of degrees of freedom (F) (intensive variables you may independently specify) is:

[ F = C - P + 2 ]

The “+2” accounts for temperature and pressure as the usual intensive coordinates. If additional constraints exist (azeotrope condition, specified quality, chemical reaction equilibria), (F) decreases.

Components count independent chemical species after accounting for reactions and any fixed stoichiometric relations the problem imposes. Phases are homogeneous regions separated by interfaces (vapor, liquid, solid, second liquid, etc.).

Exam reading of F

• F = 0: invariant—system sits at a fixed intensive state (triple point of a pure substance).
• F = 1: univariant—one free variable (pure-fluid saturation curve).
• F = 2: bivariant—typical single-phase pure fluid or binary two-phase with two free specs.

If a stem says “binary mixture at fixed T and P in VLE,” check whether compositions are free or fixed: for binary L+V, F = 2, so fixing T and P determines both liquid and vapor compositions (bubble/dew loci), not arbitrary x and y.

Vapor–Liquid Equilibrium Criteria

Thermal and mechanical equilibrium require equal T and P in coexisting phases (neglecting interfacial and gravity effects). Chemical (phase) equilibrium for each component (i) requires equal fugacity:

$$f_i^{\mathrm{V}} = f_i^{\mathrm{L}}$$

Fugacity has units of pressure and measures “escaping tendency.” For an ideal gas, fugacity equals partial pressure $y_i P$. Real gases and liquids need corrections.

Ideal VLE: Raoult’s Law

For an ideal liquid solution with ideal vapor (or vapor at low P where fugacity ≈ partial pressure):

$$y_i P = x_i P_i^{\mathrm{sat}}(T)$$

where:

• $x_i$ = liquid mole fraction of (i)
• $y_i$ = vapor mole fraction of (i)
• (P) = system pressure
• $P_i^{\mathrm{sat}}(T)$ = pure-component vapor pressure of (i) at the system temperature

Sum rules: $\sum x_i = 1$ and $\sum y_i = 1$. For a binary ideal system:

$$P = x_1 P_1^{\mathrm{sat}} + x_2 P_2^{\mathrm{sat}} = x_1 P_1^{\mathrm{sat}} + (1-x_1) P_2^{\mathrm{sat}}$$ $$y_1 = \frac{x_1 P_1^{\mathrm{sat}}}{P}$$

Raoult’s law is the default exam model when the stem says “ideal solution,” “ideal mixture,” or gives only pure vapor pressures with no activity coefficients.

Dilute Solutes: Henry’s Law

When a component is dilute in the liquid (often a sparingly soluble gas), the partial pressure is linear in liquid mole fraction with a Henry’s constant $H_i$ (units of pressure; sometimes defined per molality or concentration):

$$p_i = y_i P = H_i x_i \quad (x_i \to 0)$$

Henry’s constant depends on T, solvent, and solute. Do not replace $P_i^{\mathrm{sat}}$ with $H_i$ for a concentrated solvent—use Raoult (or activity-corrected Raoult) for the solvent and Henry for the dilute solute when the problem is framed that way.

Activity, Fugacity, Ideal vs Nonideal

Fugacity coefficient $\hat{\phi}_i$ relates vapor fugacity to partial pressure:

$$f_i^{\mathrm{V}} = y_i \hat{\phi}_i P$$

Activity coefficient $\gamma_i$ relates liquid fugacity to an ideal-solution reference (Lewis–Randall often used for solvents):

$$f_i^{\mathrm{L}} = x_i \gamma_i f_i^{0}$$

At moderate pressure with $f_i^{0} \approx P_i^{\mathrm{sat}}$ and $\hat{\phi}_i \approx 1$:

$$y_i P = x_i \gamma_i P_i^{\mathrm{sat}}(T)$$

Conceptual role on the exam: activity/fugacity explain why real K-values differ from ideal Raoult predictions. You are rarely asked to compute Wilson or NRTL parameters; you are expected to know that $\gamma_i > 1$ means positive deviation (easier to vaporize than ideal) and that azeotropes require nonideality.

K-value (distribution coefficient):

$$K_i = \frac{y_i}{x_i}$$

For ideal Raoult: $K_i = P_i^{\mathrm{sat}}/P$. Light components have large $K_i$; heavies have small $K_i$.

Critical Point vs Triple Point

These pure-component landmarks appear as definition items and as context for EOS/critical property use from Chapter 5.

• Above $T_c$ you cannot liquefy a pure gas by pressure alone; the fluid is supercritical if also above $P_c$ in the usual jargon for the supercritical region.
• Triple point of water is the practical fixed point near 0.01 °C and about 611 Pa—not the same as the normal melting point at 1 atm.
• Do not confuse critical point with azeotrope: an azeotrope is a mixture VLE feature where y = x at a given P (or T); critical point is a pure-fluid (or mixture critical locus) end of phase distinction.

Worked Conceptual/Numerical: Ideal Binary VLE

Data at T = 80 °C:

• Component A: $P_A^{\mathrm{sat}} = 120\,\mathrm{kPa}$
• Component B: $P_B^{\mathrm{sat}} = 60\,\mathrm{kPa}$
• Liquid: $x_A = 0.40$ (ideal solution, ideal vapor)

Bubble pressure:

$$P = x_A P_A^{\mathrm{sat}} + x_B P_B^{\mathrm{sat}} = 0.40(120) + 0.60(60) = 48 + 36 = 84\,\mathrm{kPa}$$

Vapor composition:

$$y_A = \frac{x_A P_A^{\mathrm{sat}}}{P} = \frac{0.40 \times 120}{84} = 0.571$$ $$y_B = 1 - y_A = 0.429$$

Check: $y_B P = 0.429 \times 84 \approx 36 = x_B P_B^{\mathrm{sat}}$ ✓

K-values: $K_A = 120/84 = 1.43$, $K_B = 60/84 = 0.71$. A is the more volatile component.

Worked Henry’s Law Sketch

Natural gas sweetening and sour-water problems often treat H₂S or CO₂ as dilute solutes. Suppose at process T, Henry’s constant for solute S in water is $H_S = 50\,\mathrm{MPa}$ (mole-fraction basis), liquid $x_S = 0.002$, and total P = 2.0 MPa. Partial pressure of S in the equilibrium vapor is:

$$p_S = H_S x_S = 50 \times 0.002 = 0.10\,\mathrm{MPa}$$ $$y_S = p_S / P = 0.10 / 2.0 = 0.05$$

If the stem gave Raoult’s law with a huge pure vapor pressure instead, you would over-predict solubility behavior for a sparingly soluble gas—match the model to the regime.

UPDA Exam Checklist for Section 6.1

1. Count C and P, then apply (F = C - P + 2) (adjust if reactions are independent constraints).
2. Ideal VLE ⇒ Raoult; dilute gas ⇒ Henry; “activity coefficient given” ⇒ modified Raoult.
3. Bubble/dew calculations (next section) rest on these same equations.
4. Critical point = end of pure L–V curve; triple point = three-phase pure-fluid invariant.
5. K-values and relative volatility (Section 6.2 / distillation chapter) flow directly from $K_i = y_i/x_i$.

Master fugacity equality and the two ideal limiting laws first. Section 6.2 turns them into bubble, dew, and flash problem language used on process thermo MCQs.