10.5 Liquid and Solid Standard States
– Phase Diagram for Transformation between Raoultian
Liquid and Solid Solutions
– The activities and Choice of Standard States
Choice of Standard States
Standard States of GM = 0 is chosen at liquid A and solid B.
Standard States of GM = 0 is chosen at liquid A and liquid B.
Standard States of GM = 0 is chosen at solid A and solid B.
Choice of Standard States
e
f
solid A and solid B.
e
f
liquid A and solid B.
e
f
liquid A and liquid B.
The choice of standard states will not alter the positions of the double tangent.
The Activities of B with Composition
The Activities and Standard States
The Activities with Chosen Standard States
aB ( w.r.t. solid B)
e
aB ( w.r.t. liquid B)
Gmo ( B )
RT
 Liquid B
aB (w.r.t solid B)  aB (w.r.t liquid B)
aB( solid )  aB(liquid)
1
 Solid B
1
 Liquid B
 Solid B
e
0

o
Gm
(B)
RT
aB ( liquid )
0
aB ( solid )
e
Gmo ( B )
RT
The Activities of A with Composition
GAo (l )  GAo ( s )  Gmo ( A)  RT ln(
a A w.r.t solid A
)
a A w.r.t liquid A
aA (w.r.t solid B)  aA (w.r.t liquid B)
aA( solid )  aA(liquid)
Solid A 
a A( liquid )
a A( solid )
Solid A 1 
e
Gmo ( A )
RT
Liquid A 1
e

o
Gm
( A)
Liq. A
o
Gm
( A)
e
RT
0
0
0
RT
The Activity Values with Temperature
G
a A( liquid )
G
a A( solid )
o
A( s )
s
GAo (l )
e
Gmo ( A )
G
GBo (l )
RT
GBo ( s )
s
l
l
Tm( A)
T
T
T
X B (l )
(
X B (s )
Tm(B )
)
If temperature is decreased,
Solid A 1 
Liq. A
 Liquid B
1
 Solid B
o
Gm
(B)
o
Gm
( A)
e
e
RT
0
0
RT
T
The Double Tangents with Temperature
G
c
c
c
c
c
c
c
c
c
c
c
o
A( s )
 Gmo ( A)
G
o
A( l )
a
0
f
f
e
e
A
0 GBo (l )
d
b
b
b
b
b
b
b
b
b
b
b
Gmo ( B )
GBo ( s )
f
e
e f
e f
ef
e
f e e e e f
f
f f
XB
B
X B (l )
X B (s )
10.6 Phase Diagrams, Gibbs Free Energy, and
Thermodynamic Activity
– Phase Diagram for Transformation between Raoultian
Liquid Solution and Solid Solution with Miscibility Gap
– Phase Diagram for Transformation between Raoultian
Liquid Solution and Insoluble Solid Solutions
– Dependence of Liquidus Line on Positive Deviation of
Liquid Solution from Raoultian behavior
Miscibility Gap
1
2
3
4
Two solid solutions (I and II) are competing with one liquid solution.(III)
Two solid solutions (I and II)
have different crystal structures.
Two solid solutions (I and II)
have the same crystal structure.
10.6 Phase Diagrams, Gibbs Free Energy, and
Thermodynamic Activity
– Phase Diagram for Transformation between Raoultian
Liquid Solution and Solid Solution with Miscibility Gap
– Phase Diagram for Transformation between Raoultian
Liquid Solution and Insoluble Solid Solutions
– Dependence of Liquidus Line on Positive Deviation of
Liquid Solution from Raoultian behavior
Effect of Temperature on Molar Gibbs
Free Energy of Mixing
Effect of Temperature on Activities
Activities at T1
Activity B for solution III follows the diagonal line, since
solution III is ideal liquid solution and liquid B is chosen as
standard state.
Activity A for solution III does not follow the diagonal line,
not because that solution III is not an ideal solution, but
because solid A is chosen as standard state.
Activity B for solution I positively deviates from the diagonal
line, because that the solution I is an positively deviated nonideal solid solution and should positively deviates from the
diagonal line, even if solid B in phase I is chosen as standard
state. Since liquid B is chosen as standard state, the curve
deviates even more positive.
Activity A for solution I positively deviates from the diagonal
line, because that the solution I is an positively deviated nonideal solution and should positively deviates from the
diagonal line, as solid A in phase I is chosen as standard state.
However, the curve should approach diagonal line as XA
approaches 1 to conform to Raoultian behavior of pure
element.
Activities at T2
Activity B for solution I positively deviates from the diagonal
line, because that the solution I is an positively deviated nonideal solid solution, and should positively deviates from the
diagonal line, even if solid B in phase I is chosen as standard
state. Since solid B in phase II is chosen as standard state, the
curve deviates even more positive.
Activity B for solution III does not follow the diagonal line
even though solution III is an ideal liquid solution, since solid
B in phase II, and not the liquid B, is chosen as standard state.
However, extrapolation of the activity curve passes through
the original point A, suggesting the ideal solution.
Activity B for solution II positively deviates from the
diagonal line, because that the solution II is an positively
deviated non-ideal solid solution and should positively
deviates from the diagonal line, as solid B in phase II is
chosen as standard state. However, the curve should approach
diagonal line as XB approaches 1 to conform to Raoultian
behavior of pure element.
Activities at TE (Eutectic)
Activity B for solution I positively deviates from the diagonal
line, because that the solution I is an positively deviated nonideal solid solution, and should positively deviates from the
diagonal line, even if solid B in phase I is chosen as standard
state. Since solid B in phase II is chosen as standard state, the
curve deviates even more positive.
Activity B for solution III remains as one single point, with its
value equal to that of its conjugate solid solutions of phase I
and that of its conjugate solid solutions of phase II.
Activity B for solution III and that of its conjugate solid
solutions is slightly higher than activity A for solution III and
that of its conjugate solid solutions, as can be deduced from
the difference of the intercepts of the triple tangent line to the
Gibbs free energy of mixing curves with A and B axes.
Activities at T3
Activity B for solution I positively deviates from the diagonal
line, because that the solution I is an positively deviated nonideal solid solution, and should positively deviates from the
diagonal line, even if solid B in phase I is chosen as standard
state. Since solid B in phase II is chosen as standard state, the
curve deviates even more positive.
Activity B for the conjugate solid I and II solutions is slightly
higher than activity A for the conjugate solid I and II solutions,
as can be deduced from the difference of the intercepts of the
double tangent line to the Gibbs free energy of mixing curves
with A and B axes.
10.6 Phase Diagrams, Gibbs Free Energy, and
Thermodynamic Activity
– Phase Diagram for Transformation between Raoultian
Liquid Solution and Solid Solution with Miscibility Gap
– Phase Diagram for Transformation between Raoultian
Liquid Solution and Insoluble Solid Solutions
– Dependence of Liquidus Line on Positive Deviation of
Liquid Solution from Raoultian behavior
Gibbs Free Energy of Mixing Curve for
Insoluble Solutions
Phase Diagram for Insoluble Solid Solutions
(l)
Also note that the equation above can be derived from
the equation below with XA(s) = 1.
The Calculation of Bi Liquidus Line in Cd-Bi System
The Bi Liquidus Line
Bi
Cd
The Calculation of Cd Liquidus Line in Cd-Bi System
The Cd Liquidus Line
Bi
Cd
Deviation from Actual Liquidus Lines
Positive deviation
ideal
10.6 Phase Diagrams, Gibbs Free Energy, and
Thermodynamic Activity
– Phase Diagram for Transformation between Raoultian
Liquid Solution and Solid Solution with Miscibility Gap
– Phase Diagram for Transformation between Raoultian
Liquid Solution and Insoluble Solid Solutions
– Dependence of Liquidus Line on Positive Deviation of
Liquid Solution from Raoultian behavior
Liquidus Lines and 
For  > cr
Imminent Immiscibility at cr
Monotectic Phase Diagram
10.7 The Phase Diagrams of Binary Systems
That Exhibit Regular Solution Behavior
in The Liquid and Solid States
Hypothetical Regular Solution
Hypothetical Regular Solution
Tm,( A)  T  Tm,( B)
T  Tm,( A)  Tm,( B)
Hypothetical Regular Solution
With increasingly negative values of l and
increasingly positive values of s the temperature of
the point of contact of the liquidus curve with the
solidus curve decreases and the critical temperature
in the solid state increases, which eventually
produces a eutectic system.
Hypothetical Regular Solution
Effect of l and s on Phase Diagrams
Assume regular liquid and solid solutions.