Thermodynamics of Solutions

1
Thermodynamics of Solutions
Lets 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
Since
Then,
0
1
XB
Using the definition for the partial molar G of
mixing
2
Extensive Properties Versus Composition Diagrams
G
V
0
1
XB
0
1
XB
G
G
What happens if we change T?
0
1
XB
0
1
XB
3
Extensive Properties Versus Composition Diagrams
G
G
G
0
1
XB
0
0
1
XB
1
XB
G
G
G
0
0
1
XB
1
XB
0
1
XB
4
Construction of Phase Diagrams
?GM
?GM
?GM
L
L
L
?
?
E
?
?
?
XB
0
XB
0
XB
0
1
1
1
5
Phase Diagram Construction
G
L
L
?
?
?
L
?
L
0
0
1
1
0
XB
XB
XB
0
1
1
XB
T
T
L
L?
?
0
XB
1
0
XB
1
6
Construction of a Eutectic Phase Diagram
Above TE and below melting points
at TE
Below TE
?GM
?GM
?GM
L
?
?
?
?
?
?
L
XB
0
XB
0
XB
1
0
1
1
at TM(A)
at TM(B)
?GM
?GM
T
L
?L
?L
?
?
?
?
?
??
?
L
L
0
1
0
XB
XB
0
1
1
XB
7
Example Construction of a Phase Diagram
Low T
?GM
L
L
?
?
?
?
?
?
?
?
L
L
?
?
?
?
XB
XB
0
0
XB
XB
XB
XB
1
1
High T
T
L
? L
? L
?
?L
? ?
?
?
? ?
?
?
L
?
?
L
0
1
XB
8
Phase Diagrams
T
T
L
T
L
L
? L
? L
? L
? L
?L
?
?
?
?
L ?
L ?
?
? ?
? ?
?
? ?
?
?
? ?
??
? ?
0
1
XB
0
1
XB
0
XB
1
T
T
L
T
L
L
? L
? L
? L
? L
?
? L
?
?
? L
?
?
L ?
? ?
?
? ?
? ?
? ?
? ?
? ?
?
? ?
XB
0
XB
0
XB
1
0
1
?
1
9
Flawed Phase Diagrams
T
T
L
T
L
L
? L
? L
?L
? L
? L
?L
?
?
?
?
L ?
L ?
?
? ?
? ?
?
? ?
?
?
? ?
??
? ?
0
1
XB
0
1
XB
0
XB
1
T
T
L
T
L
L
? L
? L
? L
? L
?
? L
?
?
? L
?
?
?
L ?
? ?
?
? ?
? ?
? ?
?
?
? ?
? ?
?
? ?
? ?
XB
0
XB
0
XB
1
0
1
1
?
10
Lever Rule
The fraction of each phase present in equilibrium
is determined by doing a mass balance of a
component. The resulting expressions are called
the lever rule
T
L
?L
?
?
??
XB
0
1
11
Example Lever Rule Example
The fraction of each phase present in equilibrium
is determined by doing a mass balance of a
component. The resulting expressions are called
the lever rule
T
L
?L
?
?
??
0
1
XB
What are the fractions of each phase present at
pointsshown on the two phase diagrams to the
right?
T
L
? L
? L
?
?L
? ?
?
?
? ?
0
1
XB
12
Temperature Processing
Heating from (a) first results in less and less ?
until you have only the ? phase. Furtherheating
results in melting ? over a range of T. During
this process the additional liquid that is formed
becomes richer and richer inA until all of phase
? is melted.
T
L
?L
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 equilibrium between the ?
phase and the liquid phase. Further heating
results in melting ? over a range of T. During
this process the additional liquid that is formed
becomes richer and richer inA until all of phase
? is melted.
?
?
??
c
a
b
XB
0
1
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.
13
Two-Phase Fields in Binary Phase Equilibria
?GM
?GM
?GM
L
?
?
?
?
L
?
L
XB
0
XB
XB
1
0
0
1
1
T
L
?
XB
0
1
14
Miscibility Gap in Binary Systems
?HM
?GM
?SM
Increasing T
XB
XB
XB
0
0
0
1
1
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 miscibility of A and B.
15
Miscibility Gap in Binary Systems
?
?GM
T
?1
?2
?1 ?2
?
L
0
XB
XB
0
1
1
For systems with positive enthalpies of mixing,
the entropy of mixing will become less
important as T decreases. This will lead to phase
separation into two phases rich in A and B at low
T. As the temperature is lowered further, the
positive enthalpy of mixing dominates and less
and less mixing occurs, thus leading to a larger
miscibility gap at lower temperatures.
16
Spinodal Decomposition
?GM
?GM
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.
XB
0
XB
0
1
1
?
T
?GM
?1
?2
?1 ?2
XB
1
XB
0