Two-Component Vapour–Liquid Diagrams

Introduction

A binary liquid mixture does not generally boil at one fixed temperature over its whole composition range. Its liquid and vapour usually have different compositions, so both phase amounts and compositions change during evaporation. Pressure–composition and temperature–composition diagrams record those paired equilibrium states. Reading the axes, the bubble and dew boundaries, and the horizontal tie line is essential before doing material-balance or distillation calculations.

Core explanation

Let component A be the more volatile of two miscible liquids A and B at a chosen temperature. Write x A for the mole fraction of A in liquid and y A for its mole fraction in vapour. They refer to different phases and need not be equal. Under ideal-solution and ideal-gas approximations, Raoult's law gives p A=x A p A , p B=(1−x A)p B , and total pressure P=x A p A +(1−x A)p B . Dalton's law then gives y A=p A/P=x A p A /P. If p A >p B and both components are present, y A>x A: the vapour is enriched in the more volatile A.

On a fixed-temperature P–composition diagram, the bubble-pressure curve gives the pressure at which a liquid of composition x first produces a vapour bubble. The dew-pressure curve gives the pressure at which a vapour of composition y first produces a liquid drop. A horizontal tie line through the two-phase region connects equilibrium liquid and vapour endpoints at the same temperature and pressure. At high enough pressure the system is liquid; lowering pressure crosses the bubble boundary into the two-phase region and eventually crosses the dew boundary into vapour. Exact orientation should always be read from labelled axes and curves.

On a fixed-pressure T–composition diagram, the bubble-temperature curve gives the temperature at which a liquid of composition x starts to boil. The dew-temperature curve gives the temperature at which a vapour of composition y starts to condense. Between them, liquid and vapour coexist. For a common nonazeotropic mixture, heating a liquid at fixed pressure first reaches the lower bubble curve, then moves through a two-phase interval, and becomes all vapour on the upper dew curve. A horizontal tie line at a chosen temperature identifies the paired x and y values. The overall composition z lies between them, and the lever rule determines phase proportions.

At a two-phase equilibrium of C=2 components, the ordinary phase rule gives F=2. Specifying temperature and pressure fixes the equilibrium compositions of both phases for a simple system. In a T–x–y diagram pressure has already been fixed, leaving one degree of freedom, often selected as temperature. Overall z does not alter the tie-line endpoints at given T and pressure in the ideal simple model; it controls how much liquid and vapour occur. This is the conceptual split between equilibrium and material balance.

Real mixtures can deviate from Raoult's law. Unlike-interaction strengths change activities, curve shapes and possible azeotropes. The ideal equations therefore serve as an illustrative calculation, not a universal prediction. Labels matter: a bubble curve is indexed by liquid composition x, a dew curve by vapour composition y, and a diagram may print both on the same horizontal mole-fraction scale.

Step-by-step reasoning

Identify the fixed variable: T for a P–x–y diagram or pressure for a T–x–y diagram. Locate the overall composition z, then determine whether the point lies in a one-phase or two-phase region. If two phases occur, draw a horizontal tie line and read its liquid x and vapour y endpoints. Use material balance separately for phase amounts.

Visual explanation

On a T–composition plot at fixed pressure, imagine a lower bubble curve and upper dew curve enclosing a lens-shaped region. A horizontal line through that lens touches liquid x A on one side and vapour y A on the other. A vertical line at overall z A need not pass through either equilibrium phase composition.

Real-world analogy

An overall fruit-juice blend may separate into two cups with different flavour proportions; the label on the original bottle is neither cup's exact recipe. Likewise, overall z is not automatically x or y. The analogy captures composition accounting, though vapour–liquid separation is set by chemical-potential equilibrium rather than deliberate mixing.

Real-world example

In an ethanol–water still, vapour above a boiling liquid can be richer in the more volatile ethanol over much of the composition range. Condensing that vapour changes the collected composition. A real ethanol–water system is nonideal and has an azeotrope, so an ideal Raoult-law sketch cannot predict its entire separation path.

Why?

Why do x A and y A differ? A molecules escape and persist in the vapour relative to B according to their liquid activities and vapour-phase fugacities. For an ideal mixture with p A >p B , the ratio y A/x A=p A /P exceeds one within the binary two-phase range, enriching vapour in A.

Common misconception

The overall composition z, liquid composition x and vapour composition y are three different quantities. Setting all three equal destroys the material-balance meaning of a tie line. Another error is to assume that any binary mixture boils at one temperature regardless of its evolving composition; that is true only in special contexts, such as an azeotropic composition at fixed pressure.

Worked example

At a fixed temperature, ideal A and B have pure-component vapour pressures p A =80 kPa and p B =40 kPa. A liquid has x A=0.25. Its bubble pressure is P=0.25(80)+0.75(40)=50 kPa. The equilibrium vapour has y A=x A p A /P=0.25(80)/50=0.40. Thus vapour is richer in A than liquid. These numbers are paired endpoints of a tie line at this temperature and pressure; 0.25 is not the overall z unless specified.

Quick check

1. On a T–x–y diagram at fixed pressure, what does a horizontal tie line connect? Answer: It connects the compositions x of coexisting liquid and y of coexisting vapour at the same equilibrium temperature and pressure.

Exam focus

Label every mole fraction with x, y or z. State whether T or pressure is held fixed. For ideal calculations, write Raoult's and Dalton's laws before inserting numbers. Locate bubble and dew curves from their definitions instead of memorising an unlabelled diagram's visual orientation.

Advanced insight

For nonideal liquid mixtures replace mole fractions in Raoult's form by activities, often x i γ i, and account for vapour nonideality through fugacity when needed. The equilibrium condition is equality of each component's chemical potential across phases. Bubble and dew curves are graphical projections of those coupled equations, not independent empirical rules.

Summary

Binary VLE diagrams show how paired liquid x and vapour y compositions change with temperature or pressure. Bubble and dew curves bound the coexistence region, and a tie line connects its equilibrium endpoints. The overall composition z governs amounts, while equilibrium determines endpoints. Ideal Raoult–Dalton equations illustrate vapour enrichment but real mixtures can deviate.

Practice questions

1. For an ideal mixture with p A >p B , is vapour generally richer or poorer in A than liquid? Answer: Richer, because y A=x A p A /P and P lies below p A for a true binary mixture. 2. At fixed T, x A=0.5, p A =100 kPa and p B =60 kPa. Find bubble pressure. Answer: P=0.5(100)+0.5(60)=80 kPa under ideal Raoult-law assumptions. 3. What does a point between bubble and dew boundaries represent? Answer: Two coexisting phases whose separate compositions are the horizontal tie-line endpoints at that T and pressure. 4. Does an overall composition z A=0.30 force both x A and y A to equal 0.30? Answer: No. z A is a material-balance average of the distinct equilibrium phase compositions.