Lecture 18 Multicomponent Phase Equilibrium 1 Thermodynamics of Solutions onsider forming a solution. Start with X A moles of pure A and X B moles of pure B at a , and T and mix them to form a single phase mixture of A and B at that same P and T. nge in the G in forming the mixture is the G of the solution minus G for each of the mponents: G' M G' soln X A G' A 0 X B G' B 0 Since G' soln X A G A X B G B Then, G' M X A G A G' A 0 X B G B G ' B 0 G' M X A G A X B G B sing the definition or the partial molar G of mixing i i 0 kT ln a i N Av i N Av i 0 G i G i 0 G i RT ln a i G' M X A RT ln a A X B RT ln a B G' M X i G i i G' M RT X i ln a i i N Av i N Av i 0 RT ln a i G’ X B 0 1 G' A 0 G A G B G' B 0
Lecture 18Multicomponent Phase Equilibrium1 Thermodynamics of Solutions Let’s consider forming a solution. Start with X A moles of pure A and X B moles.
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Lecture 18 Multicomponent Phase Equilibrium 1
Thermodynamics of Solutions
Let’s consider forming a solution. Start with XA moles of pure A and XB moles of pure B at a given P, and T and mix them to form a single phase mixture of A and B at that same P and T.
The change in the G in forming the mixture is the G of the solution minus G for each of the pure components:
G' M G' so ln XAG' A0 XBG' B
0
Since G' so ln XAGA XBGB
Then, G' M XA GA G' A0 XB GB G' B
0
G' M XAG A XBGBUsing the definitionfor the partial molar G of mixing
i i0 kT ln ai
NAvi NAvi0 Gi Gi 0 Gi RT ln ai
G' M XART ln aA XBRT ln aB
G' M XiGii
G' M RT Xi ln aii
NAvi NAvi0 RT ln ai
G’
XB0 1
G' A0
G A
G B
G' B0
Lecture 18 Multicomponent Phase Equilibrium 2
Extensive Properties Versus Composition Diagrams
G’
0 1
G' A0
G A
G B
G' B0
XB
V’
0 1
V' A0
V A
V B
V' B0
XB
G’
0 1
G' A0
G A
G B
G' B0
XB
G’
0 1
G' A0
G A
G B
G' B0
XB
What happens if we change T?
Lecture 18 Multicomponent Phase Equilibrium 3
Extensive Properties Versus Composition Diagrams
G’
0 1
G' A0
G A
G B
G' B0
XB
G’
0 1
G' A0
G A
G B
G' B0
XB
G’
0 1
G' A0
G A
G B
G' B0
XB
G’
0 1
G' A0
G A
G B
G' B0
XB
G’
0 1
G' A0
G A
G B
G' B0
XB
G’
0 1
G' A0
G A
G B
G' B0
XB
Lecture 18 Multicomponent Phase Equilibrium 4
Construction of Phase Diagrams
GM
0
G' A0
GA
GB
G' B0
XB 1
L
GM
0
G' A0
GA
GB
G' B0
XB 1
L
GM
0
G' A0
GA
GB
G' B0
XB 1
L
E
Lecture 18 Multicomponent Phase Equilibrium 5
Phase Diagram Construction
G’
0 1XB
L
0 1XB
L
0 1XB
L
0 1XB
L
0 1XB
T
0 1XB
T
L
L+
Lecture 18 Multicomponent Phase Equilibrium 6
Construction of a Eutectic Phase Diagram
GM
0
G' A0
GA
GB
G' B0
XB
T
1
0 XB 1
+
+L
L
+L
L
GM
0
G' A0
GA
GB
G' B0
XB 1
L
GM
0
G' A0
GAGB
G' B0
XB 1
L
GM
0
G' A0
GA
G' B0
XB
1
L
GM
0
G' A0
G' B0
XB 1
at TM(A)at TM(B)
Above TE and belowmelting points
at TEBelow TE
Lecture 18 Multicomponent Phase Equilibrium 7
Example: Construction of a Phase Diagram
GM
0 XB
T
1
0 XB1
+
L
+L
Low T
0 XB 1
L
XB
XB
L
XB
XB
L
L
L
L
+
+L+ L
High T
Lecture 18 Multicomponent Phase Equilibrium 8
Phase Diagrams
T
0 XB1
+
L
+
L +
+L+ L
T
0 XB 1
+
L
+L
T
0 XB1
+
L
+
+L+ L
L +
+
T
0 XB 1
+
L
+L
+
+LT
0 XB 1
+
L
+L
+
+L
+
T
0 XB 1
L
+
+L
+ L
+ L +
Lecture 18 Multicomponent Phase Equilibrium 9
Flawed Phase Diagrams
T
0 XB1
+
L
+
L +
+L+ L
T
0 XB 1
+
L
+L
T
0 XB1
+
L
+
+L+ L
L +
+
T
0 XB 1
+
L
+L
+
+LT
0 XB 1
+
L
+L
+
+L
+
T
0 XB 1
L
+
+L
+ L
+ L +
+
+L
Lecture 18 Multicomponent Phase Equilibrium 10
Lever Rule
T
0 XB 1
+
L
+L
F XB
XBXB
XB
F XB XB
XB XB
The fraction of each phase present in equilibrium isdetermined by doing a mass balance of a component. The resulting expressions are called the lever rule:
Lecture 18 Multicomponent Phase Equilibrium 11
Example: Lever Rule Example
T
0XB
1
+
L
+L
F XB
XBXB
XB
F XB XB
XB XB
The fraction of each phase present in equilibrium isdetermined by doing a mass balance of a component. The resulting expressions are called the lever rule:
T
0 XB1
+
L
+
+L+ L
What are the fractions of each phase present at pointsshown on the two phase diagrams to the right?
+L
Lecture 18 Multicomponent Phase Equilibrium 12
Temperature Processing
T
0 XB 1
+
L
+L
Heating from (a) first results in less and less until you have only the phase. Furtherheating results in melting over a range ofT. During this process the additional liquidthat is formed becomes richer and richer inA until all of phase is melted.
a b
Heating from (b) first results in making boththe and phases less pure, although thefraction of each phase remains relatively constant.At the eutectic T heating results in melting and above the eutectic T there is equilibriumbetween the phase and the liquid phase.Further heating results in melting over a range ofT. During this process the additional liquidthat is formed becomes richer and richer inA until all of phase is melted.
c
Heating from (c) first results in making boththe and phases less pure, and thefraction of each phase remains relatively constant.At the eutectic T heating results in melting both and . Above the eutectic T there is only the liquid phase.
Lecture 18 Multicomponent Phase Equilibrium 13
Two-Phase Fields in Binary Phase Equilibria
GM
0
G' A0
GA
GB
G' B0
XB 1
L
GM
0
G' A0
GA
GB
G' B0
XB 1
L
GM
0
G' A0
GA
GB
G' B0
XB 1
L
T
0 XB 1
L
GA
GB
Lecture 18 Multicomponent Phase Equilibrium 14
Miscibility Gap in Binary Systems
GM
0
GA
GB
XB 1
For systems with negative enthalpies of mixing, both the entropy of mixing and enthalpy of mixing will lead to decreases in G of mixing at intermediate compositions. This will lead to complete miscibilityof A and B.
SM
0
S ASB
XB 1
HM
0
H AH B
XB 1
GA H A TS A
GB HB TSB
GM XAGB XBGB
Increasing T
GM HM TSM
SM XASB XBSB
HM XAHB XBH B
Lecture 18 Multicomponent Phase Equilibrium 15
Miscibility Gap in Binary Systems
GM
0
GA
GB
XB 1
L
T
0 XB 1
1+ 2
21
For systems with positive enthalpies of mixing, the entropy of mixing will become less importantas T decreases. This will lead to phase separation into two phases rich in A and B at low T. As thetemperature is lowered further, the positive enthalpy of mixing dominates and less and less mixingoccurs, thus leading to a larger miscibility gap at lower temperatures.
Lecture 18 Multicomponent Phase Equilibrium 16
Spinodal Decomposition
GM
0
GA GB
XB 1
T
0 XB 1
1+ 2
21
GM
XB
GM
0
GA GB
XB 1
The spinodal compositions are defined by the inflection points in the G vs X diagram.
For a G vs X diagram as shown, compositions within the spinodal range lead to uphilldiffusion, but in the spinodal region two phase equilibrium will be kinetically hindered since uphill diffusion is not favorable in this range.
Note that phase separation initially leads to increases for G outside the spinodal, and decreases in G inside the spinodal.
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