Chapt. 8 Phase diagrams

1
Chapt. 8 Phase diagrams

• In this chapter we describe a systematic
  way of discussing the physical changes mixtures
  undergo when they are heated or cooled and when
  their compositions are changed. In particular, we
  see how to use phase diagrams to judge whether
  two substances are mutually miscible, whether an
  equilibrium can exist over a range of conditions,
  or whether the system must be brought to a
  definite pressure, temperature, and composition
  before equilibrium is established. Phase diagrams
  are of considerable commercial and industrial
  significance, particularly for semiconductors,
  ceramics, steels, and alloys. They are also the
  basis of separation procedures in the petroleum
  industry and of the formulation of foods and
  cosmetic preparations.

2
? Phases, components, and degrees of freedom

• 8.1 Definitions
• Phase

(described in chpt. 6)
The number of phases in a system is denoted P.
A gas, or a gaseous mixture, is a single
phase, a crystal is a single phase, and two
totally miscible liquids form a single phase.
A solution of sodium chloride in water is
a single phase. Ice (Single phase,
P1) Alloy (Single phase, P1)
Ice is equilibrium with water (Double phase,
P2)
3
Constituent The number of chemical species (an
ion or a molecule) presented in a system.
Labeled as S.

• Component A chemically independent constituent
  of a system.
• The number of components, C, in a system
  is the minimum number of independent species
  necessary to define the composition of all the
  phases present in the system.

4
Discussion

• The relation between S and C
• (1) When no reaction takes place, the number of
  components is equal to the number of
  constituents. (Alcohol and water)
• (2) If there are reaction equilibriums exist, the
  number of components is equal to the difference
  between number of constituents and the number of
  equilibriums, R. That is
• CS-R
  (8.1s)

Example HAcHAc- in water If we above
equilibrium does not consider, then S2,
R0, C2-02 If the equilibrium is considered,
then S4, R1, C4-13, which is different from
above result. In fact, in above system, the
concentration of H and Ac- is equal to each
other, then a concentration condition should be
considered, therefore, C4-1-12
(3) If the concentrations of species are equal to
each other, then the number of equal relations is
labeled as R, then CS-R-R
(8.2s)
5
Degrees of freedom

• The variance, F, of a system is the number
  of intensive variables that can be changed
  independently without disturbing the number of
  phases in equilibrium.
• Example (a) Taking account of a system as
  liquid water at 25oC and 1 bar.
• Here, S1, C1, the number of phase, P1. It
  is clear the pressure and temperature may be
  changed independently without changing the number
  of phases, so F 2. We say that such a system is
  bivariant, or that it has two degrees of freedom.
  
• (b) water at 100oC and 1 bar.
• Also, S1, C1, but P2(liquid and
  gas). It is clear, if T is given, p is also given
  out, it can not change freely. Therefore, F1.

6
8.2 The phase rule

• First we consider a one-component system(C1).
• (1) One phase, P1, F2, T, p
• (2) Two phase, P2, mJ(a,p,T) mJ(b,p,T),
  which means a function between p and T exists.
  F1, T or p.
• (3) Three phase, P3, mJ(a,p,T) mJ(b,p,T) and
  mJ(a,p,T) mJ(g,p,T), F0.
• It is clear that the maximum number of
  phase for a one-component system is three.
• Second for a two-component system(C2).
• Except for T, p, x1, or x2 is another
  variable.

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For an arbitrary components system, the
number of variable occurred by composition is
C-1. If there is P phase, then the total number
of variable is P(C-1)2

• However, they are dependent to each other among P
  phases.
• For example, P2 then
  Relation
• m1(a) m1(b), m2(a) m2(b), mC(a) mC(b),
  i.e., C?1
• P3
• m1(a) m1(b), m1(a) m1(g), mC(a) mC(g),
  i.e., C?2
• General speaking, there is C(P-1) relation, then
  we have
• F P(C-1)2- C(P-1) C-P2
  (8.1)
• Above equation is phase rule.

8
Discussion

• (1) To deduce phase rule, we assume the specie
  exists in all phase. In fact this assumption is
  unnecessary.
• (2) From phase rule, one can predict the max
  number of Phase and min number of Freedom Degree
  of a system.
• Because F P(C-1)2- C(P-1) C-P2
• We understand that the minimum of F is Fmin0,
  then the maximum number of P is PmaxC2
• If C1, then Pmax3 which is same that described
  previously.
• On the other hand, Pmin1, then FmaxC1.
• If C1, then Fmax2.
• (3) In eqn. 1, the number 2 is variable of T, p.
  If T or p is given, then we have F C-P1 or
  F C-P. Both F and F are named as
  conditional freedom degree.
• (4) General formula for phase rule is F C-Pn,
  where n is variables except for compositions.

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(a) One-component systems
FC-P2

• Here C1,
• (1) P1,
• Then F2
• T, p variables.
• (2) P2
• Then F1
• T or p variable.
• (3) Pmax3
• Then Fmin0
• No variable.

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(b) Experimental procedures

• To get the phase diagram, two techniques
  of thermal analysis, which takes advantage of the
  effect of the enthalpy change during a
  first-order transition, and differential scanning
  calorimetry are sused.
• In thermal analysis, a sample is allowed
  to cool and its temperature is monitored.

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High pressure

• Modern work on phase transitions often
  deals with systems at very high pressures, and
  more sophisticated detection procedures must be
  adopted. Some of the highest pressures currently
  attainable are produced in a diamond-anvil cell
  like that illustrated in Fig. 8.5. The sample is
  placed in a minute cavity between two gemquality
  diamonds, and then pressure is exerted simply by
  turning the screw. The advance in design this
  represents is quite remarkable for, with a turn
  of the screw, pressures of up to about I Mbar can
  be reached which a few years ago could not be
  reached with equipment weighing tons.

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Two-component systems

• C2, then F4-P.
• If T is given, then
• F3-P
• The Maximum value of F is 2.
• Here, the degree of freedom of T is
  discarded. Another degrees of freedoms are p and
  x1 or x2. Therefore, the phase diagram is a map
  of pressures and compositions at which each phase
  is stable. On the other hand, if p is given,
  then a map of T versus x will be obtained.

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8.3 Vapour pressure diagrams

• Ideal solution
• (a) The composition of the vapour
• pApAxA, pBpBxB (8.2)
• The total pressure of the vapour is
• ppApB pAxA pBxB
• pB (pA - pB) xA (8.3)
• Then
• yApA/p, yBpB/p (8.4)
• Inserting eqn. 2 and 3 into 4

From eqn.3, xA(p-pB)/(pA-pB)
Inserting above and eqn. 3 into eqn.5, then
The plot of eqn.3 and 5 listed at 199.
14
Discussion

• According to eqn.5,

If pAgtpB, i.e., pB/pAlt1, then
for pAgtpB
for pAltpB
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(b) The interpretation of the diagrams
Overall composition a, a2, a4 Phase
composition a1,a1,a2,a2, a3,a3

• In distillation, the compositions of liquid and
  vapour are equal interest.

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Notes

1. Points that lie between the two lines correspond
   to a system in which there are two phases
   present, one a liquid and the other a vapour.
2. (2) Among the area between black and blue line
   and at two lines, F1, therefore, if p is given,
   phase composition is given out, too.

(3) If the pressure of the system is decreased,
the state of the system will move down the
vertical line because the overall composition
doesn't change. The vertical line is called
isopleth. (4) A line joining two points
representing phases in equilibrium is called a
tie line. (5) The exist of a phase no matter the
amount of the phase itself.
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(c) The lever rule
nnanb (8.1s)
where n is overall amount of the system
included A and B. For A, nzA na xA nb yA
(8.2s) Inserting eqn.1s into 2s, we have
(nanb )zA na xA nb yA (8.3s)
Rearrange above formula, we have na(zA
- xA) nb(yA- zA) (8.4s) Then,
na la nb lb
(8.7) Eqn.7 is the lever rule.

• As shown right, the state of the system
  moves from a to O by pressure descending. There
  are two phase, a and b, at this state. The amount
  of a and b are na and nb, respectively. It is
  clear that

18
Discussion
na la nb lb (8.7)

• (1) Eqn. 7 can be used when a equilibrium between
  two phase exists.
• (2) At O1, la0, from eqn. 7, nb should be zero,
  however, the b phase does exist, perhaps tiny.
  Similarly, at O2, a phase is tiny.

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8.4 Temperature-com-position diagrams

• In previously description, we described
  the pz diagram at given temperature. On the
  other hand, one can get the diagram of Tz. From
  the figure of pz at different T, Tz figure can
  be obtained as shown right. In shadow, gas is in
  equilibrium with liquid.

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(a) The distillation of mixtures

• T?, from a1 to a2.
• At a2, gas occurred. L-G coexisted.
• The vapour is richer in the more volatile
  component A (the component with the lower boiling
  point).

Simple distillation the vapour is withdrawn and
condensed. This technique is used to
separate a volatile liquid from a non-volatile
solute or solid. Fractional distillation Here,
boiling and condensation cycle is repeated
successively. This technique is used to
separate volatile liquids.
21
Theoretical plates

• Theoretical plates The number of effective
  vaporization and condensation steps that are
  required to achieve a condensate of given
  composition from a given distillate.
• It is clear that if the composition of gas is
  similar that of the liquid, the theoretical
  plates is more to achieve the same degree of
  separation.

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Real solution
(1) Miscible liquids
(a) Slightly (mostly positive or negative)
deviation
(b) Large positive deviation

• We discussed the diagrams of the ideal solution.
  For real solution, there are two cases. The first
  is two components are solved to each other during
  any compositions, and the second they are solved
  each other during some ranges of the
  compositions. Now we first discuss the first
  case, that is the two components are solved in
  whole concentration.

There is maximum vapour at the top of the curve.
Corresponding, the boiling temperature has the
lowest value. From the theoretical analyses, at
this point, the composition of vapour is equal to
that of liquid. Therefore, one cannot separates
the material by simple distillation. We call the
solution as azeotrope.
As shown in figure, because the lowest boiling
temperature for azeotrope, we call it as
low-boiling azeotrope. Because the boiling
temperature of azeotrope is determined, therefore
the composition of azeotrope also has the only
value at a given pressure. Example H2O-C2H5OH,
351.28K, 95.57 at po.
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(c) Large negative deviation

• Here, a highest boiling temperature is observed.
• Similarly, the composition is constant at given
  pressure. The conditional degree of freedom is
  ZERO.
• ExampleH2O-HCl, 381.65K, 20.24 at po.
• Note (1) At right side, pure A and azeotrope can
  be obtained by distillation. However, at left
  side, only pure B and azeotrope can be obtained.
• (2) Azeotrope is considered as standard for
  titration.
• Please read the text at 325 in Physical
  chemistry written by Nanjing Univ.
• (3) Most of systems, if A is negative, B
  also is negative.

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(c) Immiscible liquids

• For immiscible liquids, at equilibrium, there is
  a tiny amount of A dissolved in B, and similarly
  a tiny amount of B dissolved in A both liquids
  are saturated with the other component (see right
  Figure).
• The total pressure, p is
• ppApBgtpA or pB

Therefore, the boiling temperature of mixture
is lower than that of both A or B. This
distinction is the basis of steam distillation,
which enables some heat-sensitive,
water-insoluble organic compounds to be distilled
at a lower temperature than their normal boiling
point.
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8.5 Liquid-liquid phase diagrams

• (a) Phase separation As shown as right, two
  liquid is miscible at higher temperature.
  However, when the temperature decreases, for
  example, lower than Tuc, two phases are found.
  This system is named partially miscible.
• (b) Critical solution temperatures Point O is
  called critical point. The temperature at
  critical point is called the critical solution
  temperature. As shown as right figure, we call it
  as upper critical solution temperature, Tuc. If
  TgtTuc, in any ratio of A and B, two kinds of
  liquid solved to each other. Only single phase is
  observed.

When TltTuc, if the overall composition
locates out side the cap, only single phase
obtained. However, in the cap, there are two
phases. Phase1 is the a saturated solution of A
in B, and phase 2 the saturated solution of B in
A. At a given temperature and pressure, the
compositions of both saturated solution are
determined. F2-211(T or z). If T is given,
then F0( error at 204). The amount of phase1
or phase2 can be calculated by lever rule.
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Lower critical solution temperature

• As shown as right, some system has a lower
  critical solution temperature, which means that
  when TgtTlc, the mixture is composed of two phases
  among some ratio of A and B. However when TltTlc,
  the two kinds of liquid are miscible.
• The next figure is an example of solid solution.

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With both Tuc and Tlc system

• The diagram is shown as right.

Theoretical interpretation is described from
the lelft figure.
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(c) The distillation of partially miscible
liquids

• Mole fraction of B,xB

Mole fraction of B,xB
29
8.6 Liquid-solid phase diagrams

• Above figure is a typical liquid-solid phase
  diagram of two component. The mixture at c is
  called eutectic mixture. E is named eutectic
  point.

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Box 8.1 Liquid crystals

• No required for you.

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Box 8.2 Ultrapurity and controlled impurity

• To get materials of extreme purity for
  special uses, one usually uses zone refining
  illustrated as right.

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(a) Eutectics

• (1) Point e is eutectic point. The temperature at
  e has the lowest value comparing both of melting
  point of both pure A and B.
• (2) The composition at e is called eutectic
  composition.

(3) For any mixture, when it is cooled a platform
is observed, which corresponding to the freeze of
the eutectic mixture. (4) Solutions of
composition to the right of e deposit B as they
cool, and solutions to the left deposit A only
the eutectic mixture (apart from pure A or pure
B) solidifies at a single definite temperature
(F' 0 when C 2 and P 3) with-out gradually
unloading one or other of the components from the
liquid.
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(b) Reacting systems

• With a stable compound
• A typical diagram with a stable compound
  produced by the reaction of two compo-nents is
  shown as right.
• It looks like a combine of two simple
  diagrams.

With an unstable compound.
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The diagram of H2SO4

• Following contents please reference the text
  book written by Nanjing university at 342-

35
Salt-H2O diagram

• This diagram is usually used for the purified of
  salt.

36
Combine of gas-liquid and liquid-solid diagrams
for purifying the mixture
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Other types of the diagrams

• Form solid solution through out any
  composition

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Partially miscible
40
8.7 The diagrams of tri-component

• Here C3,
• Then FC-P2
• The lowest value of P is 1,therefore
• Fmax3-124
• It is impossible to express the value by figure
  in four dimensions. For convenient one let both
  of T and p at a given value, then Fmax3-102
• The two variables are two of x1, x2 or x3. Now
  one can displays the diagram in two variables as
  before. However it is not convenient. One prefer
  to use triangle coordinate as shown next page.

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(a) Triangle coordinate

• The composition at black point are
• 30A, 40B, 30C

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(b) The properties of the triangle coordinate

• (1) At the side the concentration of the
  substance located against the side is zero.
• (2) The concentration of the substance at a top
  of the triangle is same when the system locates
  the line parallels with the side against the
  substance.

The contents of B at 1, 2, and 3 are 50
43
(3) As shown the next figure, if we connect the
point 1 with the vertex A, then ratio of B and C
is same at this connect line.

• The ratio of B and C is same for 1, 2 and 3, but
  the A increases from 1 to 3.

44
(4) There are two tricomponents systems as shown
in next figure, D and E. The mixture of these two
systems locate at the

• connection of D and E. The position of the
  new system can be calculated by lever ruler.
• WDODWEOE
• WWDWE
• O closed to E if WEgtWD

45
Justifications

• For C, it is clear that
• WO,CWD,CWE,C
• WO,CWoad
• WD,CWDab
• WE,CWEaf
• Then
• Woad WDabWEaf
• WOWDWE
• (WDWE)ad WDabWEaf
• WD(ad-ab)WE(af-ad)
• i.e. WDbdWedf
• then WDDOWeOE

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Deduce
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(5) Application

• In a special case, if one system is
  mixture, S, another is PURE substance, for
  example, A. Then when A is added into S
  gradually, the new system will run along with the
  connection of SA and close to A which as shown
  left figure. In contrast, If A is take our, for
  example, evaporate, occur crystal, and so on,
  then the residue will locate the extend of the
  connection of AS and run far from the A, as shown
  as right.

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The diagram for tri-component

• 1. Partially miscible of pair of liquids
• In the figure, out the cap is single phase,
  F3-12
• in the cap, two phase coexist, for example, a1,
  b1, a2, b2, , where a1, b1, or a2, b2 are
  conjugating solution,respectively. Normally, the
  connection line between two conjugating solution
  is not parallel with the bottom line of triangle.
  The cap line is called bi-nodal curve. And O is
  called isothermal consolute point or plait point.

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2. Partially miscible of more than one pair of
liquids
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Extraction
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3 The water-two salts system
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Complicated diagrams
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