Vapour-liquid equilibrium: bubble and dew point calculations

Raoult's and modified Raoult's law, and the BUBL P, DEW P, BUBL T and DEW T calculations with Antoine vapour pressures and relative-volatility iteration.

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Why it matters

Every distillation column, flash drum and condenser is specified by the temperatures at which a mixture starts to boil (bubble point) and starts to condense (dew point). Tray temperatures, condenser and reboiler conditions, and the column pressure all come from these four standard calculations, so they are the bread and butter of separation design and a staple of GATE.

Key ideas

The equilibrium relation. At low to moderate pressure, equality of fugacities between vapour and liquid becomes yᵢP = xᵢγᵢPᵢ^sat (modified Raoult's law). For mixtures of similar molecules (benzene-toluene, hexane-heptane) γᵢ ≈ 1 and it reduces to Raoult's law, yᵢP = xᵢPᵢ^sat. Assumptions of Raoult's law: ideal-gas vapour, ideal liquid solution, and negligible Poynting effect, so it is a low-pressure, similar-components law.

Vapour pressures. Pᵢ^sat depends on temperature only and is usually given by the Antoine equation, ln Pᵢ^sat = A − B/(T + C). The constants are substance-specific and come from a data book, with their own units for P and T; always check which ones a table uses.

The bubble point. A liquid of known composition xᵢ is heated (or depressurised) until the first bubble appears. The bubble has a different composition yᵢ, richer in the more volatile component. Summing yᵢ = 1 gives the bubble-point equation P = ΣxᵢPᵢ^sat (Raoult).

The dew point. A vapour of known composition yᵢ is cooled (or compressed) until the first drop forms. Summing xᵢ = 1 gives 1/P = Σyᵢ/Pᵢ^sat.

The four calculations.

  • BUBL P: given T and x, find P and y. Direct, no iteration.
  • DEW P: given T and y, find P and x. Direct for Raoult's law; iterative with γ because γ depends on the unknown x.
  • BUBL T: given P and x, find T and y. Iterative, because Pᵢ^sat(T) is nonlinear.
  • DEW T: given P and y, find T and x. Iterative. A good iteration uses the relative volatility α₁₂ = P₁^sat/P₂^sat, which changes slowly with T: calculate one P^sat from the sum, get T from its Antoine equation, update α and repeat.

Diagrams. At fixed T, the P-x-y diagram has a straight bubble line (for Raoult) above a curved dew line; at fixed P, the T-x-y diagram has the bubble curve below the dew curve. Between them lies the two-phase region where the lever rule gives the phase split. For a pure component the bubble and dew points coincide; for a mixture they differ, which is the temperature glide that distillation exploits.

Raising the pressure raises both bubble and dew temperatures (the vapour pressures must rise to match) and usually reduces the relative volatility, making separation harder.

Formulas

yᵢ·P = xᵢ·Pᵢ^sat (Raoult's law) yᵢ·P = xᵢ·γᵢ·Pᵢ^sat (modified Raoult's law) P = Σ xᵢ·Pᵢ^sat (BUBL P, Raoult) 1/P = Σ yᵢ/Pᵢ^sat (DEW P, Raoult) P = Σ xᵢ·γᵢ·Pᵢ^sat, 1/P = Σ yᵢ/(γᵢ·Pᵢ^sat) (modified Raoult) Kᵢ = yᵢ/xᵢ = Pᵢ^sat/P (Raoult K-value) α₁₂ = K₁/K₂ = P₁^sat/P₂^sat P₂^sat = P / (x₁·α₁₂ + x₂) (BUBL T iteration, binary) ln Pᵢ^sat = A − B/(T + C) (Antoine)

Symbols: xᵢ liquid and yᵢ vapour mole fractions; P total pressure (kPa); Pᵢ^sat vapour pressure of pure i at T (kPa); γᵢ liquid activity coefficient; Kᵢ vapour-liquid distribution ratio; α₁₂ relative volatility; A, B, C Antoine constants (from a data book, in the units the table states).

Worked examples

For benzene (1) and toluene (2) take Antoine constants (P in kPa, T in °C): benzene A = 13.7819, B = 2726.81, C = 217.572; toluene A = 13.9320, B = 3056.96, C = 217.625. Assume Raoult's law.

Example 1 (standard): BUBL P and DEW P at 90 °C. Given: T = 90 °C; (a) liquid with x₁ = 0.4, (b) vapour with y₁ = 0.4.

  1. P₁^sat = exp(13.7819 − 2726.81/307.572) = 136.50 kPa; P₂^sat = exp(13.9320 − 3056.96/307.625) = 54.31 kPa.
  2. (a) P = x₁P₁^sat + x₂P₂^sat = 0.4 × 136.50 + 0.6 × 54.31 = 54.60 + 32.59 = 87.18 kPa.
  3. y₁ = x₁P₁^sat/P = 54.60/87.18 = 0.626.
  4. (b) 1/P = y₁/P₁^sat + y₂/P₂^sat = 0.4/136.50 + 0.6/54.31 = 0.002930 + 0.011048 = 0.013978, so P = 71.54 kPa.
  5. x₁ = y₁P/P₁^sat = 0.4 × 71.54/136.50 = 0.210. Answer: (a) bubble P = 87.2 kPa, y₁ = 0.626; (b) dew P = 71.5 kPa, x₁ = 0.210

Example 2 (GATE level): BUBL T at 101.33 kPa. Given: liquid with x₁ = 0.4 at P = 101.33 kPa. Find the bubble temperature and the first-bubble composition.

  1. Start from the mole-fraction-weighted normal boiling points: T₀ = 0.4 × 80.1 + 0.6 × 110.6 = 98.4 °C.
  2. At 98.4 °C, α₁₂ = P₁^sat/P₂^sat = 2.443. Then P₂^sat = P/(x₁α₁₂ + x₂) = 101.33/(0.977 + 0.6) = 64.25 kPa.
  3. Toluene Antoine solved for T: T = B/(A − ln P₂^sat) − C = 3056.96/(13.9320 − 4.1628) − 217.625 = 95.29 °C.
  4. Repeat: at 95.29 °C, α₁₂ = 2.468, P₂^sat = 63.84 kPa, T = 95.09 °C. Once more: α₁₂ = 2.470, P₂^sat = 63.81 kPa, T = 95.07 °C (converged).
  5. P₁^sat at 95.07 °C = 157.61 kPa; y₁ = x₁P₁^sat/P = 0.4 × 157.61/101.33 = 0.622.
  6. Check: 0.4 × 157.61 + 0.6 × 63.81 = 63.04 + 38.29 = 101.33 kPa. Answer: bubble T = 95.1 °C, y₁ = 0.622 (The same mixture as a vapour, y₁ = 0.4, has its dew point at about 101.4 °C: the 6 °C glide between bubble and dew points.)

Common mistakes

  • Using the bubble-point sum for a dew-point problem (or vice versa): bubble sums xᵢPᵢ^sat, dew sums yᵢ/Pᵢ^sat.
  • Mixing Antoine constants for log₁₀ with ln, or °C with K, or mmHg with kPa.
  • Taking the bubble-point vapour composition equal to the liquid composition.
  • Applying Raoult's law to non-ideal mixtures (water-ethanol, acetone-chloroform) or at high pressure.
  • Forgetting that DEW P with activity coefficients needs iteration because γ depends on the unknown liquid composition.

For GATE CH

  • NAT problems on BUBL P, DEW P, y from x (or x from y) with vapour pressures given, and relative volatility.
  • BUBL T or DEW T where Antoine constants are given and a short iteration (or a check of given options) is expected.
  • Conceptual MCQs on the shape of P-x-y and T-x-y diagrams, assumptions of Raoult's law and the effect of pressure.
  • Practise one full BUBL T iteration by hand with the relative-volatility method.

Quick check

  1. Write the bubble-pressure equation for an ideal binary.
  2. For P₁^sat = 120 kPa, P₂^sat = 40 kPa and y₁ = 0.5, what is the dew pressure?
  3. Which of the four calculations need iteration when Raoult's law holds?
  4. What is α₁₂ in terms of vapour pressures for an ideal system?
  5. At fixed P, which is higher for a mixture: the bubble temperature or the dew temperature?

Answers: 1. P = x₁P₁^sat + x₂P₂^sat. 2. 60 kPa. 3. BUBL T and DEW T. 4. P₁^sat/P₂^sat. 5. The dew temperature.

Interview questions

All Chemical Engineering Thermodynamics interview questions

Try answering each one aloud before you open it.

  1. 1.What is vapour-liquid equilibrium (VLE) in the context of chemical engineering thermodynamics?Concept

    Vapour-liquid equilibrium (VLE) refers to the state where a liquid and its vapour are in equilibrium with each other at a given temperature and pressure. At this point, the rate of evaporation equals the rate of condensation, and the composition of the liquid and vapour phases remains constant over time.

  2. 2.Explain the significance of bubble point and dew point in VLE calculations.Concept

    The bubble point is the temperature at which the first bubble of vapour forms from a liquid mixture at a given pressure. Conversely, the dew point is the temperature at which the first drop of liquid forms from a vapour mixture at a given pressure. These points are crucial for designing separation processes like distillation, as they help determine the conditions under which phase changes occur.

  3. 3.How do Raoult's Law and Dalton's Law apply to VLE calculations?Concept

    Raoult's Law is used to calculate the partial pressures of components in a liquid mixture, assuming ideal behaviour. It states that the partial pressure of a component is equal to the product of its mole fraction in the liquid phase and its pure component vapour pressure. Dalton's Law is used for the vapour phase, stating that the total pressure is the sum of the partial pressures of all components. These laws are fundamental in calculating bubble and dew points for ideal mixtures.

  4. 4.Why is the concept of activity coefficients important in non-ideal VLE calculations?Application

    Activity coefficients account for deviations from ideal behaviour in liquid mixtures. They modify Raoult's Law to better predict the behaviour of real mixtures by considering interactions between different molecules. This is important for accurate VLE calculations in non-ideal systems, where interactions significantly affect phase equilibria.

  5. 5.What happens if the pressure is increased in a system at its bubble point?Application

    At constant temperature, raising the pressure above the bubble pressure makes the mixture a subcooled (compressed) liquid, so the vapour disappears. To bring it back to its bubble point at the higher pressure, the temperature must rise until ΣxᵢPᵢ^sat(T) equals the new pressure. So bubble (and dew) temperatures increase with pressure, and the relative volatility usually falls.

  6. 6.Why is it important to know the bubble and dew points in the design of distillation columns?Application

    Knowing the bubble and dew points is crucial in distillation column design because they define the temperature and pressure conditions under which phase changes occur. This information helps in determining the number of stages required, the reflux ratio, and the energy requirements for the separation process, ensuring efficient and cost-effective operation.

  7. 7.What is the effect of azeotropes on VLE calculations?Application

    Azeotropes are mixtures that exhibit the same composition in both liquid and vapour phases at a certain temperature and pressure, making them challenging for separation by distillation. In VLE calculations, azeotropes represent a point where the usual methods of separation fail, requiring alternative techniques like pressure-swing distillation or the use of entrainers.

  8. 8.Calculate the bubble point temperature of a binary mixture with components A and B, given: x_A = 0.4, P = 101.3 kPa, P_A^0 = 80 kPa, P_B^0 = 120 kPa (vapour pressures at a trial temperature).Numerical

    With Raoult's law, the bubble pressure at the trial temperature is ΣxᵢPᵢ^sat = 0.4 × 80 + 0.6 × 120 = 32 + 72 = 104 kPa. This exceeds the system pressure of 101.3 kPa, so the trial temperature is above the bubble temperature. Lower T, recompute the vapour pressures from Antoine equations, and repeat until the sum equals 101.3 kPa; the bubble temperature cannot be found from one set of vapour pressures alone.

  9. 9.Determine the dew point temperature for a binary mixture with y_A = 0.5, P = 101.3 kPa, P_A^0 = 90 kPa, P_B^0 = 110 kPa (vapour pressures at a trial temperature).Numerical

    With Raoult's law, the dew pressure at the trial temperature is 1/(0.5/90 + 0.5/110) = 99.0 kPa. The system pressure (101.3 kPa) is higher, so at this temperature the vapour would already be partly condensed: the dew temperature at 101.3 kPa is higher than the trial value. Raise T, update the vapour pressures (Antoine) and iterate until Σyᵢ/Pᵢ^sat = 1/P.

  10. 10.Explain how temperature-composition (T-x-y) diagrams are used in VLE analysis.Concept

    Temperature-composition (T-x-y) diagrams graphically represent the relationship between temperature and the compositions of liquid and vapour phases at equilibrium. They help visualize how the composition of each phase changes with temperature, providing insights into the separation process and aiding in the design of distillation columns by showing the bubble and dew point lines.

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