Phase Equilibrium and Physical Transformation of Pure and Mixtures_1

1

เอกสารประกอบการสอนรายวชิ า

242311: เคมเี ชิงฟิ สิกส์ 1 (Physical Chemistry I)

สมดุลวัฏภาคและการเปล่ียนแปลงทางกายภาพของสารบริสุทธ์ิและสารผสม

(Phase Equilibrium and Physical Transformation of Pure and Mixtures)

อ.ดร. จกั รสี ทิ ธ์ิ จนิ ดาวงศ์
ภาควชิ าเคมี คณะวทิ ยาศาสตร์ มหาวทิ ยาลยั พะเยา

E-mail: [email protected]

2

วัตถุประสงค์การเรียนรู้ (Learning objectives):

1. การเปล่ียนแปลงทางกายภาพของสารบริสุทธ์ิ (Physical transformations of pure substances)
1. แผนผังวัฏภาคของสารบริสุทธ์ิ (Phase diagrams of pure substances)
2. การทานายการเปล่ียนวัฏภาคทางเทอร์โมไดนามกิ ส์ (Thermodynamics aspects of phase transitions)

2. การเปล่ียนแปลงทางกายภาพของสารผสม (Physical transformations of simple mixtures)
1. การทานายการเปล่ียนวัฏภาคทางเทอร์โมไดนามิกส์ (Thermodynamics aspects of phase transitions)
2. สมบตั ขิ องสารละลาย (The properties of solutions)
3. แผนผังวฏั ภาคของระบบสององค์ประกอบ: ของเหลว (Phase diagrams of binary systems: liquids)
4. แผนผังวฏั ภาคของระบบสององค์ประกอบ: ของแขง็ (Phase diagrams of binary systems: solids)
5. แผนผังวฏั ภาคของระบบสามองค์ประกอบ (Phase diagrams of ternary systems)
6. แอกตวิ ติ ี ้(Activities)

วฏั ภาค (Phase) 3

A phase is a form of matter that is uniform
throughout in chemical composition and physical state.
Examples of P=1:
• Pure gas
• Gas mixture
• Sodium chloride solution
• Ice
Examples of P=2:
• Ice & water mixture
• Oil & water
Examples of P=3:

CaCO3(s) ⎯⎯→ CaO(s) + CO2 (g)

Peter Atkins et.al. Physical Chemistry, 11th ed. 2018.

การเปลย่ี นวฏั ภาค (Phase transition) 4

A phase transition
is the spontaneous conversion of one phase into
another.
A transition temperature, Ttrans
is the temperature at which the two phases are
in equilibrium for a given P.

Peter Atkins et.al. Physical Chemistry, 11th ed. 2018.

5

ศักย์เคมี (Chemical potentials)

For a pure substance:  = Gm = G
n

For a multicomponent system:  = i
i

Chemical potential ( ) is the potential for moving
material.

• from one phase to another (phase change)
• from one substance to another (reaction)

Criterion of phase equilibrium:
At equilibrium the chemical potential of a substance
is the same in and throughout every phase present
in the system.

Peter Atkins et.al. Physical Chemistry, 8th ed. 2006.

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รอยต่อระหว่างวฏั ภาค (Phase boundaries)

A phase diagram is a diagram showing the regions
of pressure and temperature at which its various
phases are thermodynamically stable.
A phase boundary is a line separating the regions

in a phase diagram showing the values of p and T at

which two phases coexist in equilibrium.

A phase transition is the spontaneous conversion
of one phase into another. Phase transition of a
stable phase occurs when P,T cross the phase
boundaries.

a = b

Peter Atkins et.al. Physical Chemistry, 11th ed. 2018.

แผนผงั วัฏภาค (Phase diagrams) 7

melting A phase transition is the spontaneous conversion
of one phase into another.
freezing
Fussion : solid = liquid
vaporization Vaporization : gas = liquid
Sublimation : gas = solid
condensation
The triple point (T3) is a point on a phase diagram at
sublimation which the three phase boundaries meet and all three
phases are in mutual equilibrium.
deposition T3
  = =gas liquid solid

Peter Atkins et.al. Physical Chemistry, 8th ed. 2006.

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รอยต่อระหว่างวัฏภาค (Phase boundaries)

pc A supercritical fluid is a dense fluid phase above
the critical temperature.
melting
The critical pressure (pc) is the vapour pressure
vaporization
at the critical temperature.
freezingcondensation
The critical temperature (Tc) is the temperature at
sublimation which a liquid surface disappears and above which a
liquid does not exist whatever the pressure.
The critical pressure is the vapour pressure at the critical
temperature.

deposition Tc

Peter Atkins et.al. Physical Chemistry, 8th ed. 2006.

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รอยต่อระหว่างวัฏภาค (Phase boundaries)

vapour pressure, p The vapour pressure is the pressure of a vapour
Liquid or solid in equilibrium with the condensed phase. Vapor
pressure increases with temperature.

Boiling is the condition of free vaporization
throughout the liquid.

The boiling temperature is the temperature at
which the vapour pressure of a liquid is equal to the
external pressure.

Peter Atkins et.al. Physical Chemistry, 8th ed. 2006.

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รอยต่อระหว่างวัฏภาค (Phase boundaries)

The boiling temperature:
the pressure is 1 atm is called the normal boiling point (Tb)
the pressure is 1 bar is called the standard boiling point
The normal boiling point is also called the normal boiling point
(1.00 bar = 0.987 atm)

The freezing temperature:
the pressure is 1 atm is called the normal freezing point (Tf)
the pressure is 1 bar is called the standard freezing point
The normal freezing point is also called the normal melting point
(1.00 bar = 0.987 atm)

Peter Atkins et.al. Physical Chemistry, 8th ed. 2006.

กฎของวฏั ภาค (Phase rule) 11

Phase rule

F =C−P+2

F = degrees of freedom
(number of intensive variables, such as T, P, x, that can
be changed without disturbing the number of phase in
equilibrium)
P = the number of phases
C = the number of components in the system
(where components are chemically independent
constituents of a system).

Peter Atkins et.al. Physical Chemistry, 8th ed. 2006.

กฎของวัฏภาค (Phase rule) 12

One-component system (C=1):
F=C–P+2=1–P+2=3–P

Two-component system (C=2):
F=C–P+2=2–P+2=4–P

Three-component system (C=3):
F=C–P+2=3–P+2=5–P

Number of components:
Ethanol water mixture: two constituents & two components
Aqueous NaCl solution: three constituents (Na+, Cl-, H2O);
only 2 components because Na+ and Cl- can’t vary independently
For a pure substance (C=1):
If P=1, then F = 1 - 1+ 2 = 2. Thus, 2 variables, typically T and P, may
be changed independently without changing the number of phases.
If P=2, then F = 1, and only 1 variable can be changed independently
If P=3, then F = 0, and no variable can be changed independently

Peter Atkins et.al. Physical Chemistry, 8th ed. 2006.

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กฎของวัฏภาค (Phase rule)

Ex.1 Which row has the correct number of phases present at each point?

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กฎของวฏั ภาค (Phase rule)

Ex.2 A pure one-component system may exist as a solid, a liquid, or a
gas depending on the conditions; determine the number of degrees
of freedom (F) when the system comprises:

a) A pure solid phase
b) A pure liquid phase
c) A pure gas phase
d) An equilibrium mixture of any two phases
e) An equilibrium mixture of the solid, liquid, and gas phases

a) F = 2, b) F = 2, c) F = 2, d) F = 1, e) F = 0,

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Five representative phase diagrams:

1. Carbon dioxide (CO2)
2. Water (H2O)
3. Ammonia (NH3)
4. Carbon (C)
5. Helium (He)

16

Phase diagrams of carbon dioxide (CO2)

Van T. Lieu. J. Chem. Educ. 1996, 73(9), 837.

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Phase diagrams of carbon dioxide (CO2)

Peter Atkins et.al. Physical Chemistry, 11th ed. 2018.

18

Phase diagrams of water (H2O)

Van T. Lieu. J. Chem. Educ. 1996, 73(9), 837.

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Phase diagrams of water (H2O)

Peter Atkins et.al. Physical Chemistry, 11th ed. 2018.

Phase diagrams of water (H2O) 20

O atom Peter Atkins et.al. Physical Chemistry, 8th ed. 2006.
H-bond
H atom
A fragment of the structure of ice (ice-I).
Each O atom is linked by two covalent bonds to H
atoms and by two hydrogen bonds to a neighbouring
O atom, in a tetrahedral array.

21

Phase diagrams of ammonia (NH3)

Leslie Glasser. J. Chem. Educ. 2009, 86(12), 1457.

22

Phase diagrams of Carbon (C) (1 atm = 0.0001 Gpa)

(1 atm = 105 pa)

Leslie Glasser. J. Chem. Educ. 2009, 86(12), 1457.

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Phase diagrams of Helium (4He)

• Gas and solid never coexist
▪ 2 isotopes: 4He=boson, 3He=fermion
▪ 2 allotropes: solid(bcc), solid(hcp)
▪ 2 liquids: He-I, He-II
▪ -line: liquid-liquid phase boundary
▪ He-II is a superfluid (flows without any viscosity)
▪ Triple point: He-I, He-II, He(gas)

Peter Atkins et.al. Physical Chemistry, 11th ed. 2018.

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Conclusion

Checklist of concepts

❑ A phase is a form of matter that is uniform throughout in chemical composition and physical
state.

❑ A phase transition is the spontaneous conversion of one phase into another.
❑ The thermodynamic analysis of phases is based on the fact that at equilibrium, the chemical

potential of a substance is the same throughout a sample.
❑ A phase diagram is a diagram showing the regions of pressure and temperature at which its

various phases are thermodynamically stable.
❑ The phase rule relates the number of variables that may be changed while the phases of a

system remain in mutual equilibrium.

Peter Atkins et.al. Physical Chemistry, 11th ed. 2018.

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Conclusion Equation Comment
Checklist of equations
 = Gm = G For a single substance
Property n
F = the variance
Chemical potential F =C−P+2 C = the number of components
P = the number of phases
Phase rule

Peter Atkins et.al. Physical Chemistry, 11th ed. 2018.

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Phase stability and phase transitions

For a one-component system, the chemical potential is the same as the molar Gibb energy:

 = Gm = G
n

The Gibb energy and molar Gibb energy varies with temperature and pressure:

dG = −SdT  dGm = −SmdT  d  = −SmdT
dG = Vdp  dGm = Vmdp  d  = Vmdp

By using the notation of partial derivatives:

   = −Sm and    = Vm
  p  
T  p T

Peter Atkins et.al. Physical Chemistry, 11th ed. 2018.

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Phase stability and phase transitions

At equilibrium, the chemical potential of a substance is the
same throughout a sample, regardless of how many phases
are present.

For any system in equilibrium: dG = 0

dG = ( 2 − 1 ) dn

If 2  1 : dG = +  non-sponteneous
If 2 = 1 : dG = 0  equilibrium
If 2  1 : dG = −  spontaneous

the transition temperature (Ttrs) is the temperature at which
the chemical potentials of two phases are equal.

Peter Atkins et.al. Physical Chemistry, 11th ed. 2018.

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The temperature dependence of chemical potentials

❑ General equation:

dG = −SdT +Vdp

❑ Temperature dependence of chemical potential is
determined by the entropy.

 Gm  =    = −Sm
 T  p   p
T

❑ The phase with larger entropy shows steeper change of
chemical potential vs. temperature:

Sm (g)  Sm (l)  Sm (s)

❑ At the temperature where the relative value of chemical
potential changes, a phase transition occurs.

Peter Atkins et.al. Physical Chemistry, 8th ed. 2006.

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The pressure dependence of chemical potentials

❑ General equation:

dG = −SdT +Vdp

❑ Temperature dependence of chemical potential is
determined by the entropy.

 Gm  =    = Vm
 p   
 T  p T

❑ The phase with larger molar volume shows steeper change
of chemical potential vs. pressure:

Vm ( g )  Vm (l )  Vm (s)

❑ In most cases (l) > ( ) , so an increase in P will
increase melting Temp. Water is different!

Peter Atkins et.al. Physical Chemistry, 8th ed. 2006.

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The response of melting to applied pressure

solid liquid solid
liquid

(a) In this case the molar volume of the solid is (b) Here the molar volume is greater for
the solid than the liquid (as for water) (Vm(s) > Vm(l)),
smaller than that of the liquid (Vm(s) < Vm(l)) (s) increases more strongly than (l),
and (s) increases less than (l). and the freezing temperature is lowered.

As a result, the freezing temperature rises. Peter Atkins et.al. Physical Chemistry, 8th ed. 2006.

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The effect of applied pressure on vapour pressure

❑ The quantitative relation between the vapour pressure (p)
when a pressure ΔP is applied and the vapour pressure (p*)

of the liquid in the absence of an additional pressure is

(g) Vm (l )P
 (l)
p = p * e RT

Where p is the observed vapor pressure
∆P is the applied pressure
Vm is the molar volume
p* is the vapour pressure in absence of applied press.

Peter Atkins et.al. Physical Chemistry, 8th ed. 2006.

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ทมี่ าของสมการ (Deriving)

❑ At equilibrium the chemical potentials of the liquid and its vapor are equal:

 ( )p*+P

Vm

p*
 (l) =  (g) l dP = RT p dp
p* p
d (l) = d (g)
  ( )p dpp*+ P p*+ P

( )p* p Vm dP
Vm (l ) dP = Vm ( g ) dp = 1 l dP = Vm l
RT p* RT p*

Vm (l ) dP = RT dp ln p − ln p* = Vm (l ) (( p * +P) − p *)
p
RT
❑ Pressure of liquid (P) and partial pressure (p),
at initial P=p=p*, final P=p*+P: ln p = Vm (l ) P

p * RT

Vm (l )P Effect of applied pressure P
on partial vapour pressure p
 p = p * e RT

Peter Atkins et.al. Physical Chemistry, 8th ed. 2006.

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Slope of phase boundary (Clapeyron equation)

❑ At equilibrium the chemical potentials of phase  = :

 ( p,T ) =  ( p,T )

d  ( p,T ) = d  ( p,T )

−Sm, dT + Vm, dp = −Sm, dT + Vm, dp

( ) ( )Vm, −Vm, dp = Sm, − Sm, dT

( )dp = Sm, − Sm,
( )dT Vm, −Vm,

Clapeyron equation  dp = trsS
dT trsV

Peter Atkins et.al. Physical Chemistry, 8th ed. 2006.

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The solid–liquid boundary

❑ For a solid-liquid phase transition, i.e., melting:

Clapeyron equation

dp = trs S =  fus S =  fus H
dT trsV  fusV Tfus fusV

❑ Because ∆fusV is typically very small and usually positive
(Vm(l) > Vm(s)), the slope of the P-T phase boundary is expected to
be large.

 fus H( )( ) ( )dp =  fusH =

dT Tfus fusV Tfus Vm l −Vm s

❑ When ∆fusV is negative (e.g., water) (Vm(l) < Vm(s)),
the slope of the P-T phase boundary is large, but negative.

Peter Atkins et.al. Physical Chemistry, 8th ed. 2006.

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The solid–liquid boundary

The integration:

dp =  fus H ln(1+ x) = x − 1 x2 + 1 x3 − ....  x
dT Tfus fusV 23

 p dp =  fus H T dT ln T = ln  + T −T*  = T −T*
T* 1 T*  T*
p*  fusV TT* fus  

( )p − p* =  fus H ln T − ln T *  p = p*+  fus H T −T* 
 fusV  fusV  T* 
 

 p = p*+  fus H ln T ❑ This expression is the equation of a steep
 fusV T*
straight line when p is plotted against T

Peter Atkins et.al. Physical Chemistry, 8th ed. 2006.

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The liquid–vapour boundary

❑ For a liquid-vapour phase transition, i.e., vaporization:

Clapeyron equation

dp = trs S = vap S = vap H
dT trsV vapV TvapvapV

❑ Because ∆vapV = Vm(g) - Vm(l) ; Vm(g) >> Vm(l)

( ( ) ( )) ( )dp = vapH = vap H = vap H
= vap H
dT TvapvapV Tvap Vm g −Vm l TvapVm g Tvap  RT 
 p 

 dp = pvap H
dT RT 2

Peter Atkins et.al. Physical Chemistry, 8th ed. 2006.

37

The liquid–vapour boundary

dp = pvap H  ln p ln p = vap H T dT
dT RTv2ap d R T* T2

dx = d ln x ln p*
x
dp = vap H ln p − ln p* = − vap H  1 − 1 
pdT RT 2 R  T T* 
 d ln p = vap H
dT RT 2 ln p = − vap H 1 − 1 
p* R  T T* 
Clausius-Clapeyron equation
p *e− vap H  1 − 1  vap H  1 1 
R  T T*  R  T T* 
p = = p *e− ;  = −

❑ Equation is plotted as the liquid–vapour boundary in Fig. 4.14.
The line does not extend beyond the critical temperature (Tc),
because above this temperature the liquid does not exist.

Peter Atkins et.al. Physical Chemistry, 8th ed. 2006.

38

The solid–vapour boundary

❑ Vapor pressure of solids is called sublimation pressure
e.g., I2 gas above solid I2 ; CO2 solid sublimes directly to
gas at atmospheric pressure

❑ Transition from solid to vapor is called sublimation.

❑ Transition is governed by
❑ Hsub = Hvap + Hfus
❑ because enthalpy is a state function.
❑ Only difference in calculations:
❑ replace Hvap by Hsub.
❑ Because Hsub > Hvap The slope of the P-T boundary

for solid-vapor equilibria is greater than that for
liquid-vapor equilibria.

Peter Atkins et.al. Physical Chemistry, 8th ed. 2006.

39

Conclusion

Checklist of concepts

❑ The chemical potential () of a substance decreases with increasing temperature (T)
in proportion to its molar entropy (Sm).

❑ The chemical potential () of a substance increases with increasing pressure (p) in
proportion to its molar volume (Vm).

❑ The vapour pressure (p) of a condensed phase increases when pressure is applied.

❑ The Clapeyron equation is an exact expression for the slope of a phase boundary.
❑ The Clausius-Clapeyron equation is an approximate expression for the boundary

between a condensed phase and its vapour.

Peter Atkins et.al. Physical Chemistry, 11th ed. 2018.

40

Conclusion

Checklist of equations

Property Equation Comment

Variation of  with temperature    = −Sm  = Gm = G
Variation of  with pressure  T  p n

Vapour pressure in the presence of applied pressure    = Vm
Clapeyron equation  
 p T
Clausius-Clapeyron equation
Vm (l )P P = P − p *

p = p * e RT

dp = trs S Assumes Vm(g) >> Vm(l) or Vm(s)
dT trsV And vapour is a perfect gas
d ln p = vap H
dT RT 2

Peter Atkins et.al. Physical Chemistry, 11th ed. 2018.

41

First-Order Transitions

❑ Many different types of phase transitions e.g. solid-liquid, conducting-superconducting, solid-solid

❑ Can use chemical potential to classify phase transitions into different types using the so-called
EhrenfestClassification

❑ Phase changes are often accompanied by changes of H and V. These changes cause abrupt

changes in the slope of  vs. T and μ vs. P plots.

   −    = Vm, − Vm, = trsV
   
 p T  p T

   −    = −Sm, + Sm, = − trs H
    p T
 T p T

❑ In a first-order transition, ∂μ/∂T is discontinuous. Because H changes by a finite amount for
an infinitesimal change in T, Cp(= dH/dT) at the transition temperature is infinite.

42

First-Order vs Second-order Transitions

First-Order ❑ A transition for which d/dT is discontinuous is
classified as a first-order transition.

Second-Order ❑ Characteristics of first-order transition:
• at the transition temperature, H, S, and V change
discontinuously.
• G is continuous, but its slope is not.
• Cp at transition temperature is infinite.
❑ All the phase transitions considered so far have

been first-order, e.g.
❑ melting/freezing,
❑ vaporization/condensation,
❑ sublimation

Peter Atkins et.al. Physical Chemistry, 6th ed. 2006.

43

First-Order vs Second-order Transitions

First-Order ❑ A transition for which d2/d2T is discontinuous is
classified as a second-order transition.

Second-Order ❑ Characteristics of second-order transitions:
• Finite discontinuity in Cp
• S , V, and H are continuous, but not their 1st
derivatives
• 1st deriv of G is continuous; 2nd deriv. is
discontinuous

Peter Atkins et.al. Physical Chemistry, 6th ed. 2006.

44

First-Order vs Second-order Transitions

❑ First order transitions involve relocation of atoms, molecules and changes energies of interactions.
❑ Second order involve changes in symmetry and order.

Peter Atkins et.al. Physical Chemistry, 11th ed. 2018.

45

Ex. 1 Calculate the pressure exerted by
8.0 g of oxygen in a 500 cm3 container
at 25 0C.

Workbook in Chemistry: Physical Chemistry. 2017.

46

EX.2

1) Calculate the pressure exerted by 6.5 g of nitrogen in a 0.05 dm3 container at 37 0C.
(Ans 12 x 106 Pa)
2) Calculate the volume occupied by 2.00 mole of an ideal gas at standard temperature (0 0C) and
pressure (1 atm) (Ans 0.0448 m3)

Workbook in Chemistry: Physical Chemistry. 2017.

Ex.3 For water at 0 oC, the standard 47
volume of transition of ice to liquid
is -1.6 cm3 mol-1, and the corresponding Peter Atkins et.al. Physical Chemistry, 6th ed. 2006.
standard entropy of transition
is +22 J K-1 mol-1. The slope of the solid-
liquid phase boundary at that
temperature is therefore

Ex.4 48

Peter Atkins et.al. Physical Chemistry, 6th ed. 2006.

dT 49

Ex.5 Estimate Peter Atkins et.al. Physical Chemistry, 6th ed. 2006.

dp

for water at its normal boiling point using the
information in Table 3.2 and

Vm ( g ) = RT
p

50

Solution dp = vap H 109.12 J K-1 mol-1 J K-1
dT T vapV 3.1 x 10-2 m3 mol-1 m3
= = 3.5 x 103

 dp = 3.5 x 103 Pa K-1 or 0.035 atm K-1
dT

and dT = 28 K atm-1
dp

Therefore, a change of pressure of +0.1 atm can be expected to change a boiling temperature
by about +2.8 K.

Peter Atkins et.al. Physical Chemistry, 6th ed. 2006.