5.6 Vapor pressure diagrams

Key points 1.  Raoult’s law is used to calculate the total vapor

pressure of a binary system 2.  The composition of the vapor in equilibrium

with a binary mixture is calculated by using Dalton’s law

3.  The compositions of the vapor and the liquid phase in equilibrium are located at each end of a tie line

4.  The lever rule is used to deduce the relative abundances of each phase in equilibrium

•  The partial pressures of the components of an ideal solution of two volatile liquids are related to the composition of the liquid mixture by Raoult’s law

•  The total vapor pressure (at some fixed T) changes linearly with the composition from pB* to pA* as xA changes from 0 to 1

5.6 Vapor pressure diagrams

pA = xApA∗ pB = xBpB

∗

p = pA + pB = xApA∗ + xBpB

∗ = pB∗ + (pA

∗ − pB∗ )xA

(a) The composition of the vapor •  The vapor should be richer in the more volatile component than

the liquid •  From Dalton’s law that the mole fractions in the gas, yA and yB are

5.6 Vapor pressure diagrams

yA =xApA

∗

pB∗ + pA

∗ − pB∗( ) xA

yA =pAp

yB =pBp

yB =1− yA

p = pA∗ pB

∗

pA∗ + pB

∗ − pA∗( ) yA

(a) The composition of the vapor

5.6 Vapor pressure diagrams

p = pA∗ pB

∗

pA∗ + pB

∗ − pA∗( ) yA

yA =xApA

∗

pB∗ + pA

∗ − pB∗( ) xA

(b) The interpretation of the diagrams

5.6 Vapor pressure diagrams

•  The point a : the vapor pressure of a mixture of composition, xA

•  The point b : the composition of the vapor that is in equilibrium with the liquid at that pressure

•  Let’s interpret the horizontal axis as showing the overall composition, zA

•  All the points down to the solid diagonal line is under such high pressure that it contains only liquid

•  All points below the lower curve correspond to a system that is under such low pressure that it contains only a vapor phase

•  Points that lie between two lines correspond to where two phases exist

(b) The interpretation of the diagrams

5.6 Vapor pressure diagrams

•  Consider the effect of lowering p on a liquid mixture of overall composition a

•  The vertical line with the fixed composition is “isopleth”

•  At a1 : the liquid can exist in equilibrium with its vapor. The composition of the vapor phase is given by point a1’.

•  Tie line : the line that join a1 and a1’ •  At p1: there is virtually no vapor present;

however the tiny amount of vapor that is present has the composition a1’

•  At p2: the overall composition is a2’’. The composition of liquid and vapor phases are a2 and a2’, respectively

•  At p3: there is virtually no liquid •  At a4: only vapor is present

(c) The lever rule

5.6 Vapor pressure diagrams

•  To find the relative amounts of two phases α and β that are in equilibrium, we measure the distance lα and lβ along the horizontal tie line and use the lever rule

nαlα = nβlβ

5.7 Temperature-composition diagrams

Key points 1.  A phase diagram can be used to discuss the

process of fractional distillation 2.  Depending on the relative strengths of the

intermolecular forces, high- or low-boiling azeotropes may be formed

3.  The vapor pressure of a system composed of immiscible liquids is the sum of the vapor pressure of the pure liquids

4.  A phase diagram may be used to discuss the distillation of partially miscible liquids

(a) The distillation of mixtures

5.7 Temperature-composition diagrams

•  The liquid phase lies in the lower part of the diagram

•  Consider what happens when a liquid of composition a1 is heated

•  The liquid boils at T2 ; the liquid has composition a2 and the vapor has composition a2’

•  Simple distillation : The vapor is withdrawn and condensed

•  Fractional distillation : the boiling and condensation cycle is repeated successively.

•  The cycle can be repeated until in due course almost pure A is obtained in the vapor and pure B remains in the liquid ; theoretical plates

(b) Azeotropes

5.7 Temperature-composition diagrams

•  A maximum in the phase diagram may occur when the favorable interactions between A and B molecules reduce the vapor pressure of the mixture

•  Consider a liquid of composition a on the right of the maximum

•  The vapor (at a2’) of the boiling mixture is richer in A. If the vapor is removed, the remaining liquid will move to a composition that is richer in B, a3

•  As evaporation proceeds, the composition of the remaining liquid shifts towards B as A is drawn off

• The liquid reached the composition b • Azeotrope : the mixture for which the evaporation occurs without change of composition ; the distillation cannot separate the two liquids

(b) Azeotropes

5.7 Temperature-composition diagrams

•  A minimum in the phase diagram may occur when the unfavorable interactions between A and B molecules increase the vapor pressure of the mixture

•  Consider a liquid of composition a on the right of the minimum

•  The mixture boils at a2 to give a vapor composition a2’.

•  This vapor condenses to a liquid of a3 with the same composition.

•  That liquid reaches equilibrium with its vapor at a3’

•  The fractionation shifts the vapor towards b, but not beyond

(c) Immiscible liquids

5.7 Temperature-composition diagrams

•  Distillation of two immiscible liquids •  At equilibrium, there is a tiny amount

of A dissolved in B and a tiny amount of B dissolved in A

•  p ≈ pA* + pB* •  At temperature where p is equal to the

atmospheric pressure, dissolved substances are purged from their solution.

•  The presence of the saturated solutions means that the ‘mixture’ boils at a lower T than either componet would alone because boiling begins when the total vapor pressure reaches 1atm, not when either vapor pressure reaches 1 atm.

•  Steam distillation