Ch. 5 Simple Mixtures
The thermodynamic description of mixtures
The properties of solutions
Phase diagrams of binary systems
Phase diagrams of ternary systems
Activities
The activities of ions
Prof. Yo-Sep Min
Physical Chemistry I, Spring 2017
• For a two-component system, F = 4  P.
• If T is constant, the remaining variance is F’ = 3  P, which has
a maximum value of 2.  pressure and composition
Therefore, one form of the phase diagram is a map of pressure
and composition
• If p is constant, the phase diagram can be depicted in terms of
temperature and composition.
Prof. Yo-Sep M in
Physical Chemistry I, Spring 2017
Lecture 20, Ch. 5-2
• According to Raoult’s law, the partial
vapor pressures of the components of an
ideal solution of two volatile liquids (A and
B) are related to the composition of the
mixture.
and
• The total vapor pressure (p) of the
mixture is expressed as:
y-intercept
Slope
• The total vapor pressure (at a fixed T) increases linearly from
to
as xA changes from 0 to 1.
Prof. Yo-Sep M in
Physical Chemistry I, Spring 2017
Lecture 20, Ch. 5-3
• The compositions of the liquid (xJ) and vapor (yJ) in mutual
equilibrium are not necessarily the same.
• In the vapor phase, the more volatile component should be
richer than the less volatile.
• From Dalton’s law for a mixture of gases A and B, the mole
fraction in the gas phase (yA and yB) are given by:
and
• If the mixture of A and B is an ideal solution, yJ may be
expressed in terms of xJ.
and
Prof. Yo-Sep M in
Physical Chemistry I, Spring 2017
Lecture 20, Ch. 5-4
• y-x plot for various values of
volatile than B).
• In all cases, yA > xA when
• yA = xA when
(A is more
.
.
• If B is non-volatile (
), B does not
contribute to the vapor (yB = 0 and yA = 1).
Prof. Yo-Sep M in
Physical Chemistry I, Spring 2017
Lecture 20, Ch. 5-5
Total vapor pressure
Prof. Yo-Sep M in
Physical Chemistry I, Spring 2017
Lecture 20, Ch. 5-6
• p-y plot for various values of
volatile than B).
(A is more
• When
(equal volatility), the total vapor
pressure is independent of yA.
• When
, the total vapor pressure
increases with yA.
Prof. Yo-Sep M in
Physical Chemistry I, Spring 2017
Lecture 20, Ch. 5-7
• The pressure-composition diagram of a two-component
mixture can be drawn by the combination as below:
Prof. Yo-Sep M in
Physical Chemistry I, Spring 2017
Lecture 20, Ch. 5-8
• For a mixture of two volatile liquids (C = 2),
when two phases (P = 2) are in equilibrium,
• However, at a given T, the remaining variance
is one (F’ = 1).
• Therefore if the composition (xA or yA) is specified, the
pressure at which the two phases are in equilibrium is fixed.
• And also, if the pressure of the coexisting two phases is
specified, the composition (xA and yA) is fixed.
Prof. Yo-Sep M in
Physical Chemistry I, Spring 2017
Lecture 20, Ch. 5-9
L+V
Because the applied pressure is higher than the vapor
pressure of the system, only liquid phase exists.
Because the applied pressure is lower than the vapor
pressure of the system, only vapor phase exists.
• Note that there is a similar story for p-yA
diagram.
Prof. Yo-Sep M in
Physical Chemistry I, Spring 2017
Lecture 20, Ch. 5-10
• The horizontal axis shows the overall mole
fraction (zA) of A in the entire system.
L+ V
• Above the upper line, only liquid is present. 
zA = xA
• Below the lower line, only vapor is present.
 zA = yA
• In the region between the upper and the lower lines, two
phases (L + V) are present.
• Therefore, in this region if p is also given, at constant T, the
remained variance is zero (F = C – P + 2 = 2; F’ = 2 – 1 – 1 = 0).
 The liquid and vapor phases coexisting in this region have
fixed compositions.
Prof. Yo-Sep M in
Physical Chemistry I, Spring 2017
Lecture 20, Ch. 5-11
• Consider lowering the applied pressure on
a liquid mixture of overall composition zA = a.
• The vertical line through “a” is called an
isopleth (from Greek words for ‘equal
abundance’) in which the overall composition
is constant.
• p > p1: A single liquid phase.
• p = p1 (point a1): A tiny amount of vapor coexists with liquid.
The line from a1 to a1’ is called a tie line.
yA is given by point a1’
xA ~ zA
Prof. Yo-Sep M in
Physical Chemistry I, Spring 2017
Lecture 20, Ch. 5-12
• p3 < p < p1: for example, consider p2.
• Because the remaining variance is zero
for a given p2 at the point a2’’, xA and yA are
fixed.
xA is given by a2
yA is given by a2’.
• p = p3 (point a3’):
A tiny amount of liquid coexists with vapor.
xA is given by point a3.
yA ~ zA
• p < p3 (point a4): only vapor phase present.
Prof. Yo-Sep M in
Physical Chemistry I, Spring 2017
Lecture 20, Ch. 5-13
• A point in the two-phase region indicates
not only qualitatively that both liquid and
vapor are present, but represents
quantitatively the relative amounts of each
phase.
• To find the relative amounts of two phases  and  in
equilibrium, we measure the distance l and l along the tie line.
Lever Rule:
n: total amount of A and B in phase .
n: total amount of A and B in phase .
Prof. Yo-Sep M in
Physical Chemistry I, Spring 2017
m
m
l
l
Lecture 20, Ch. 5-14
• The lever rule is readily proved as below:
n: total amount of A and B in phase .
n: total amount of A and B in phase .
The overall amount of A in both  and  phases (nzA) is
the sum of its amounts in the two phases.
Multiplying
Prof. Yo-Sep M in
by zA,
Physical Chemistry I, Spring 2017
Lecture 20, Ch. 5-15
• From the lever rule, the relative
amounts of each phase can be given by
• Following a similar procedure,
Relative amount of α phase
Prof. Yo-Sep M in
Relative amount of β phase
Physical Chemistry I, Spring 2017
Lecture 20, Ch. 5-16
• Another choice for phase diagram of a two-component mixture
is the temperature-composition diagram (at a fixed pressure).
• This diagram is used to discuss distillation of the mixture.
V
V+L
L
• Consider an ideal mixture of A and B (A is
more volatile than B).
• Note that the liquid phase now lies in the
lower part of the diagram.
• The region between the lines is a twophase (V + L) region where F’ = 1 (fixed p).
• Therefore, at a given T (F’=0), the compositions of the phases
in equilibrium are fixed.
• The vapor phase lies in the upper part of the diagram.
Prof. Yo-Sep M in
Physical Chemistry I, Spring 2017
Lecture 20, Ch. 5-17
Tb,B
V
V+L
L
• Consider heating the ideal mixture of A
and B (A is more volatile than B).
• By heating a liquid (a1 ), when T reaches
T2, the liquid mixture boils.
xA ~ a1 = a2 (Most of A and B are liquid.)
yA = a2’ (A trace of A and B is vapor.)
If p = 1 atm, this T is Tb,A.
Dew point line
Bubble point line
Prof. Yo-Sep M in
Physical Chemistry I, Spring 2017
Lecture 20, Ch. 5-18
• In a simple distillation, all the hot vapors
produced are immediately channeled into
a condenser which cools and condenses
the vapors.
• Therefore, the distillate will not be pure - its composition will
be identical to the composition of the vapors at the given
temperature and pressure, and can be calculated from Raoult’s
law.
• As a result, simple distillation is usually used only to separate
liquids whose boiling points differ greatly (rule of thumb is
25 °C), or to separate liquids from involatile solids or oils.
Prof. Yo-Sep M in
Physical Chemistry I, Spring 2017
Lecture 20, Ch. 5-19
• In fractional distillation, the boiling and
condensation cycle is repeated successively.
• This technique is used to separate volatile liquids.
Prof. Yo-Sep M in
Physical Chemistry I, Spring 2017
Lecture 20, Ch. 5-20
• In a fractional distillation of A from A/B
mixture, when a first condensate of
composition a3 is reheated, this mixture
boils at T3 and yields a vapor of
composition a3’.
• Then the vapor is drawn off, and the first
drop condenses to a liquid of composition
a4.
• The cycle can then be repeated until almost pure A is obtained.
Prof. Yo-Sep M in
Physical Chemistry I, Spring 2017
Lecture 20, Ch. 5-21
• The efficiency of a fractionating column is expressed in terms of
the number of theoretical plates (the number of effective
vaporization and condensation steps).
• To achieve a condensate of a specified composition from a
given distillate, the fractionating column must have plates of
which the number corresponds to the number of theoretical
plates.
3 theoretical plates
5 theoretical plates
• More similar partial pressures, more plates.
Prof. Yo-Sep M in
Physical Chemistry I, Spring 2017
Lecture 20, Ch. 5-22
• T-z phase diagram of ideal solution: A is more volatile than
B.
Prof. Yo-Sep M in
Physical Chemistry I, Spring 2017
Lecture 20, Ch. 5-23
• When the favorable A-B interactions reduce the vapor pressure
of the mixture below the ideal value, a maximum in the T-z
phase diagram (minimum in the P-z diagram) may occur.
• The A-B interactions stabilize the liquid.
• GE = HE < 0 (more favorable to mixing than ideal)
Chloroform/tetrahydrohuran
Prof. Yo-Sep M in
Physical Chemistry I, Spring 2017
Lecture 20, Ch. 5-24
• When the unfavorable A-B interactions increase the vapor
pressure of the mixture above the ideal value, a minimum in the
T-z phase diagram (maximum in the P-z diagram) may occur.
• The A-B interactions destabilize the liquid.
• GE = HE > 0 (less favorable to mixing than ideal)
Ethanol/toluene
Prof. Yo-Sep M in
Physical Chemistry I, Spring 2017
Lecture 20, Ch. 5-25
• Consider a liquid mixture of compositions (a2
~ a4) on the right of the maximum.
• After removal (condensation elsewhere) of
the vapor (a2’), VLE moves to a composition
(a3) that is richer in B.
• After repeating the processes, VLE reaches the composition b
in which the vapor and liquid have the same composition b.
• In the red point, evaporation occurs without change of
composition.  the mixture is said to form an azeotrope. (From
the Greek words for ‘boiling without changing’)
• When the azeotropic composition has been reached, distillation
cannot separate the two liquids.
Prof. Yo-Sep M in
Physical Chemistry I, Spring 2017
Lecture 20, Ch. 5-26
• A high-boiling azeotrope: When the liquid of
composition a is distilled, the composition of
remaining liquid changes towards b but no
further.
Ex) A mixture of HCl/H2O is azeotropic at 80
w% of water, and boils unchanged at 108 oC.
• A low-boiling azeotrope: When the mixture at a
is fractionally distilled, the vapor in equilibrium in
the fractionating column moves towards b and
then remains unchanged.
Ex) A mixture of ethanol/H2O is azeotropic at 4
w% of water, and boils unchanged at 78 oC.
Prof. Yo-Sep M in
Physical Chemistry I, Spring 2017
Lecture 20, Ch. 5-27
• Consider the distillation of two immiscible liquids,
such as octane and water.
• Both liquids are saturated with a tiny amount of
the other component, so the total vapor pressure of
the mixture is close to:
Boil at
different T
A-rich phase
B-rich phase
• The distillation of two immiscible liquids can be regarded the
joint distillation of the separated components.
• For the saturated solutions, the mixture boils at a lower
temperature than either component would alone.
 because boiling begins when the total vapor pressure reaches
the atmospheric pressure, not when either vapor pressure does.
Prof. Yo-Sep M in
Physical Chemistry I, Spring 2017
Lecture 20, Ch. 5-28
• The steam distillation enables some heat-sensitive , waterinsoluble organic compounds to be distilled at a lower
temperature than their Tb.
• The only disadvantage is that the composition of the
condensate is proportional to the vapor pressures of the
components.  Low volatility, low abundance.
Prof. Yo-Sep M in
Physical Chemistry I, Spring 2017
Lecture 20, Ch. 5-29
• Consider a system consisting of pairs of partially miscible
liquids (partially immiscible, P = 2) which are liquids that do not
mix in all proportions at all temperatures.
Ex) hexane and nitrobenzene
• When P = 2,
• The selection of a temperature implies
that the compositions of the immiscible
liquid phases (P = 2) are fixed.
Hexane/nitrobenzene
at 1 atm
Prof. Yo-Sep M in
• When P = 1 (fully mixed phase), F = 3
and F’ = 2 at a constant p.  Composition
may be adjusted.
Physical Chemistry I, Spring 2017
Lecture 20, Ch. 5-30
• The region below the curve corresponds
to the compositions and temperatures at
which the liquids are partially miscible.
• The upper critical solution temperature
(Tuc) is the temperature above which the
two liquids are miscible in all proportions.
• Consider adding B into A.
Completely
miscible
(P = 1)
Prof. Yo-Sep M in
Partially miscible (P = 2)
The major (A saturated with B): a’
The minor (B saturated with A): a’’
Relative abundance: Lever rule
Physical Chemistry I, Spring 2017
Lecture 20, Ch. 5-31
• Consider raising T which increases the
miscibility.
Completely miscible (P = 1)
Partially miscible (P = 2)
Tiny amount of the B-rich phase
The other is the A-rich phase
Partially miscible (P = 2)
The major (A saturated with B)
The minor (B saturated with A)
(See Ex. 5C.2)
Prof. Yo-Sep M in
Physical Chemistry I, Spring 2017
Lecture 20, Ch. 5-32
• The upper critical solution
temperature (Tuc) is the highest T at which
phase separation occurs.
• Above Tuc, phase separation does not
occur whatever the composition.  also
called the upper consolute temperature.
• This temperature exists because the greater thermal motion
overcomes the tendency of like molecules to stick together and
therefore to form two phases.
Prof. Yo-Sep M in
Physical Chemistry I, Spring 2017
Lecture 20, Ch. 5-33
• In the formation of an ideal solution by mixing of liquids,
• The driving force for mixing is the increasing
entropy of the system, and the enthalpy of
mixing is zero.
• The average energy of A-B interactions in the
ideal solution is the same as the average
energy of A-A and B-B interactions in the pure
liquids.
• mixG < 0 for all compositions and T, so the
mixing is spontaneous in all proportions.
Prof. Yo-Sep M in
Physical Chemistry I, Spring 2017
Lecture 20, Ch. 5-34
• In the formation of a regular solution (HE  0 but SE = 0) by
mixing liquids,
• If  < 0, mixing is exothermic (A
-B interactions more favorable).
• If  > 0, mixing is endothermic
(A-B interactions less favorable).
• For  > 2, two minima separated by a maximum.  The
system will separate spontaneously into two phases with
compositions corresponding to the minima. Partially miscible
Prof. Yo-Sep M in
Physical Chemistry I, Spring 2017
Lecture 20, Ch. 5-35
• For  > 2, phase separation occurs in the
compositions corresponding to the minima.
• The compositions for the minima are obtained
by
Prof. Yo-Sep Min
Physical Chemistry I, Spring 2017
Lecture 20, Ch. 5-36
• As  decreases, two minima
move together and merge
when =2.
Prof. Yo-Sep M in
Physical Chemistry I, Spring 2017
Lecture 20, Ch. 5-37
• Some systems show a lower critical solution
temperature (Tlc, also called the lower
consolute temperature).
• Below Tlc, liquids mix in all proportions.
Ex) water/trimethylamine mixture
• At low T, the two components are more miscible because they
form a weak complex.
• At high T, the complexes break up and the two components are
less miscible.
Prof. Yo-Sep M in
Physical Chemistry I, Spring 2017
Lecture 20, Ch. 5-38
• Some systems have both upper and lower
critical solution temperatures.
• T < Tlc: the two components form a weak
complex.
• Tlc < T < Tuc: the weak complexes have
been disrupted.
• T > Tuc: the thermal motion homogenizes
the mixture again.
Ex) water/nicotine system
Prof. Yo-Sep M in
Physical Chemistry I, Spring 2017
Lecture 20, Ch. 5-39
• Consider a pair of liquids that are partially miscible and form a
low-boiling azeotrope.
• This combination is quite common because both properties
reflect the tendency of unlilke molecules to avoid each other.
• A-B interaction is less favorable than A-A and B-B interactions.
Prof. Yo-Sep M in
Physical Chemistry I, Spring 2017
Lecture 20, Ch. 5-40
• There are two possibilities:
1. The liquids become fully miscible before their boiling.
2. Boiling occurs before mixing is complete.
Prof. Yo-Sep M in
Physical Chemistry I, Spring 2017
Lecture 20, Ch. 5-41
• The phase diagram for two components that
become fully miscible before they boil.
• In the simple distillation of a mixture of
composition a1, the 1st drop of distillate may
have single- or two-phase depending on the
cooling T.
Prof. Yo-Sep M in
Physical Chemistry I, Spring 2017
Lecture 20, Ch. 5-42
• The phase diagram for a binary system in
which boiling occurs before the two liquids
are fully miscible.
• There is no Tuc.
• The distillate from composition a1 has
composition b3 and is a two-phase mixture.
• A system at e1 forms two phases, which persist up to e2 (but
the composition of each phase changes).
• Above e2, the vapor of this azeotropic mixture has the same
overall composition as the liquid.
Prof. Yo-Sep M in
Physical Chemistry I, Spring 2017
Lecture 20, Ch. 5-43
• The phase diagram for two almost
immiscible solids and their completely
miscible liquids.
• a1a2: Pure solid B is deposited.
• a2a3a4: More and more B is deposited.
• At a4,: The remaining liquid has the eutectic composition (e).
• Now this liquid freezes to give a two-phase system of almost
pure A (a5’) and almost pure B (a5’’).
Prof. Yo-Sep M in
Physical Chemistry I, Spring 2017
Lecture 20, Ch. 5-44
• The isopleth at e corresponds to the
eutectic composition.
• The name comes from the Greek words for
‘easily melted’.
• A liquid with the eutectic composition freezes
at a single temperature without previously
depositing solid A or B.
• A solid with the eutectic composition melts at the lowest T,
without change of composition.
• For the eutectic point, F’ = 0 at constant p, when C=2 and P=3.
Ex 1) Solder (Sn 67 w%, Pb 33 w%), melting at 183 oC.
Ex 2) NaCl 23 w%, H2O 77 w%, melting at – 21.1 oC
Prof. Yo-Sep M in
Physical Chemistry I, Spring 2017
Lecture 20, Ch. 5-45
• Consider the rate of cooling down the
isopleth through a1.
• a1a2: Liquid cools readily.
• a2a3a4: The cooling is slower because
the deposition of B is exothermic.
• At a4: When the remaining liquid reaches the
eutectic composition (e), T remains constant
(eutectic halt), until the whole sample has
been solidified.
• If the liquid has the eutectic composition
initially, the liquid cools steadily down to the
freezing temperature of the eutectic.
Prof. Yo-Sep M in
Physical Chemistry I, Spring 2017
Lecture 20, Ch. 5-46
• The eutectic halt is longest for the eutectic
isopleth, which gives the location of the
eutectic composition and its melting
temperature.
• This cooling curves at different overall
compositions are used to construct the phase
diagram.
Prof. Yo-Sep M in
Physical Chemistry I, Spring 2017
Lecture 20, Ch. 5-47
• Many binary mixtures react to produce
compounds. Ex) Ga/As system
• 3 constituent (S = 3), 2 component (C = 2)
• If prepared by mixing an excess of B with A,
the system consists of C and unreacted B.
• The component C is a true compound, not just an equimolar
mixture.
• The solid deposited on cooling along the isopleth a is the
compound C.
• The pure compound C melts congruently (along x=0.5)  the
composition of the melted liquid is the same to the solid compound.
Prof. Yo-Sep M in
Physical Chemistry I, Spring 2017
Lecture 20, Ch. 5-48
• Some reaction compounds (Na2K) are not stable as a liquid.
Incongruent melting
Prof. Yo-Sep M in
Physical Chemistry I, Spring 2017
Lecture 20, Ch. 5-49
• Reading:
5D Phase diagrams of ternary systems (p.216 ~ 219)
• Homework
Exercises: 5C. 5(b), 10(b)
Prof. Yo-Sep M in
Physical Chemistry I, Spring 2017
Lecture 20, Ch. 5-50