1. 3. Equlibria

2. 3. Equilibra 3.1 Equlibrium Constant 3.1.1 Equilibrium Constant
3.1.2 Effect of Temperature on Equilibrium Constant
3.1.3 Effect of Pressure on Equilibrium Constant
3.1.4 Le chatelier’s Principle
3.1.5 Alternative Standard State
3.1.6 Interaction Coefficients
3.1.7 Ellingham Diagram
3.2 Phase Equilibria
3.2.1 Phase Rule
3.2.2 Phase Transformations
3.2.3 Phase Equilibria and Free Energies

3. Equilibrium Constant
Consider a general chemical reaction occurring at constant temperature and pressure;
aA + bB + … = mM + nN + …
Where a,b, …,m,n, … are stoichiometric coefficients indicating the number of moles of respective species A,B, …,M,N, …
Let ΔGr denote the free energy change associated with the above reaction.
For any species on solution
The free energy change of reaction when all reactants and products are at their respective standard states – Standard free energy change of reaction.

4. 3.1.1. Equilibrium Constant Now we define the activity quotient, Q as
Thus we have
At equilibrium, ΔG=0
Where Qeq=Q at eqilibrium
K is called equilibrium constant.
Note that the value of ai’s in this expression for K are the values at equilibrium, and hence ΔGr becomes zero when these values are plugged in the equation for ΔGr

5. Equilibrium Constant
Note that since ai’s are dimensionless, both K and Q are also dimensionless
Note that K can be determined from the knowledge of either ΔGr0 or ai’s at equilibrium, whereas Q has nothing to do with ΔGr0
We may obtain the following equation by proper combination of equations for ΔGr0,Q and K
Note that
if Q/K < 1 → the reaction is spontaneous from left to right as written, if Q/K = 1 → the reaction is at equilibrium, and if Q/K > 1 → the reaction is spontaneous from right to left as written.

6. 3.1.2. Effect of Temperature on Equilibrium Constant
At equilibrium
This equation is known as the van’t Hoff equation.

7. 3.1.2. Effect of Temperature on Equilibrium Constant
This equation expresses the temperature dependence of the equilibrium constant in
terms of the heat of reaction. This equation tells us that
If the reaction is endothermic, i.e., ∆Hr0 > 0, K increases with increasing T,
If the reaction is exothermic, i.e., ∆Hr0 < 0, K decreases with increasing T.

8. 3.1.2. Effect of Temperature on Equilibrium Constant
If ∆Hr0 is independent* of T, the integration of van’t Hoff equation yields
* This is generally the case when the range of temperatures involved is not appreciable, and in the absence of any phase changes in the participating species.
Temperature dependence of an equilibrium constant may be examined by plotting lnK versus 1/T
From van’t Hoff equation

9. · · · · · · 3.1.2. Effect of Temperature on Equilibrium Constant
The temperature dependence of an equilibrium constant can be determined by plotting lnK versus 1/T :
From the relationship
lnK
the slope of the line is
1/T
As the slope of the this line is positive as shown, the reaction is exothermic.
In other words, ∆Hr0 < 0.

10. 3.1.3. Effect of Pressure on Equilibrium Constant
The equilibrium constant depends on the value of the standard free energy of a reaction:
Recall that in the equation ΔGr0 is the free energy change when all the reaction and products are at their respective standard states in which Ptotal = 1atm
Therefore K is independent of pressure.
This conclusion does not necessarily mean that the equilibrium composition is independent of pressure. Consider the reaction
K is independent of pressure . This does not mean that the individual partial pressures are independent of the total pressure, but the ratio
Is independent of the total pressure

11. 3.1.3. Effect of Pressure on Equilibrium Constant
Recall that Pi=NiPtotal Substitution of this equation in to the equation for K yields
Values of the individual mole fractions change in such a way that the ratio cancle
If the total pressure Ptotal changes
Independent of pressure
dependent on pressure
For the general reaction
Note that Kc is independent of pressure, only if m+n-a-b=0 i.e., no net change in the number of moles by the reaction.
=Kc , is the equilibrium constant expressed in concentration.

12. 3.1.4. Le Chartelier’s Principle
Consider the general reaction which is at equilibrium:
Le Chartelier’s Principle provides a concenient way of predicting the direction in which the reaction proceeds toward a new equilibrium state.
Le Chartelier’s Principle
Perturbation of a system at equilibrium will cause the equilibrium position to change in such a way as to tend to remove the perturbation.
Example If a reaction is exothermic, the reaction will be promoted by lowering the temperature If a reaction results in a change in volume, then increase in pressure will cause the reaction to proceed in the direction which results in decrease in volume.
Le Chartelier’s Principle provides a good guide to the effects of pressure, temperature and concentration. For a quantitative analysis, however, more rigorous treatments are required as seen in the previous three sections (sections 3.1.1~3)

13. 3.1.5. Alternative Standard States
Recapitulation
The choice of a standard state is arbitrary, and the activity is always unity at the standard state chosen.
The activity of a component in a solution is essentially a relative quantity. From the definition of activity it follows that the numerical value of the activity of a particular component is dependent on the choice of the standard state. There is no fundamental reason for preferring one standard state over another. Convenience dictates the choice of the standard state.

14. Raultian standard state
Alternative Standard States
The particular choice is known as the Raultian standard state.
Raultian standard state
But there are some inconvinience and limitations associated with this standard state:
Raultian activity scale
If the pure component exist in a physical state which is different from that of the solution at the temperature of interest, how can the pressure term in the definition of activity be determined?
With the Raultian standard state, it is found not in frequently that the activity of a solute in a dilute solution is very small
Activities expressed according to this scale are based on the Raultian standard state.
To resolve these problems, we now define a new standard state called the Henrian standard state which originates from Henry’s law

15. Henrian standard state
Alternative Standard States
we now define a new standard state called the Henrian standard state which originates from Henry’s law.
Raultian standard state
Raultian activity scale
Henrian standard state
Henrian activity scale
Henry’s law

16. 3.1.5. Alternative Standard States
The Henrian standard state is a hypothetical, non-physical state for component j. suppose that we are interest in the composition marked x in the following figure:
Raultian activity scale
Henrian activity scale
Two important points
If j obeys Henry’s law, the value of the activity on the Henrian scale is numerically equal to the mole fraction of j.
The numerical value of the activity of j at the Henrian standard state is 1 on the Henrian activity scale, but γi0 on the Raultian activity scale.
Henry’s law
Activity of j on the Raultian scale
Activity of j on the Henrian scale
Note that the numerical values are different for the same composition.

17. 3.1.5. Alternative Standard States
The activity of j on the Henrian activity scale is X
j obeys Henry’s law at the mole fraction X

18. 3.1.5. Alternative Standard States
Recall that
Raultian scale
Henrian scale
This equation relates the activity on the Henrian scale to the activity on the Raultian scale.
The Henrian standard state is sometimes called the infinitely dilute solution standard state because it is mostly used for dilute solutions.

19. 3.1.5. Alternative Standard States
From a practical point of view, therefore, it is more convinient to use the concentration scale expressed in weight percent (wt%j) rather than in mole fraction (Nj). We thus define a new standard state called 1wt% standard state.
Raultian standard state
Raultian activity scale
Henrian standard state 2 1
1 wt% activity scale 1
100
Wt%j
1 wt% Standard state
The position of 1wt% is a bit exaggerated to help visualization

20. Relationship between Nj and wt%j (i-j binary solution)
Alternative Standard States
Note also that in dilute solutions in which j obeys henry’s law the value of aj(wt%) is numerically the same as (wt%j).
Relationship between Nj and wt%j (i-j binary solution)
Base : 100g of solution
Mi and Mj : Molecular weight of i and j, respectively ni and nj : Number of moles of I and j, respectively.
Combination yields
(1-wt%j)

21. Summary of the relationships
Alternative Standard States
From the relationship between Henrian and the 1wt% activity scales in the range of dilute solutions,
Combination of the last two equations yields
Note that the 1 wt% standard state is real if the solution obeys Henry’s law upto 1 wt%, otherwise it is hypothetical.
Summary of the relationships

22. 3.1.5. Alternative Standard States
Recall that the partial molar free energy or chemical potential of a component in a solution is independent of the standard state chosen. In other words, it is an absolute thermodynamic property of the component in the solution.
Free energy axis
Thus

23. 3.1.5. Alternative Standard States
In thermodynamic considerations of A-B binary solution, if the standard state of B is changed from the Raultian standard state to the Henrian standard state, the standard molar free energy changes accordingly.
but
and
Combination yields
In a similar way we may find the free energy change associated with the change in the standard state from the Raultian to the 1 wt% standard state.

24. 3.1.5. Alternative Standard States
Consider the heterogeneous reaction
The standard free energy change at temperature T for the reaction is ΔGr0 when Raultian standard states, i.e., pure liquid A, gaseous B and pure solid M at 1 atm are used.
It is sometimes more convenient to use alternative standard states for the species involved in the reaction. When the standard state of liquid A is changed from Raultian to Henrian standard state, the free energy change of the reaction
Can be obtained as follows:
(1)
(2)
(3)
(1)-(3)

25. 3.1.5. Alternative Standard States
Similarly, if the standard state of A is changed from Raultian to 1 wt% standard state
Where MA and MX are molecular weights of A and solvent X in the A-X binary solution
Recall that
This equation is valid for all standard states. That is
We often use symbol f to denote the activity coefficient for 1 wt% standard state:
As both the Henrian and 1 wt% standard states are based on Henry’s law, for diluter solutions,

26. 3.1.6. Interaction Coefficients
The activity of B in dilute solution with respect to the Henrian standard state is given by
where fB is the activity coefficient of B on the Henrian activity scale.
First, suppose that we have the A-B binary solution, and let fBB denote the activity coefficient of B in the A-B-C ternary solution in which B is the only solute. Then
Next, consider the A-B binary solution. Holding the concentration of B constant, we add a small amount of C. C will then influence the interaction between A and B in the solution, and hence alter the activity coefficient of B.
where fBC is the effect of C on the activity coefficient of B.

27. 3.1.6. Interaction Coefficients
For the A-B-C-D quaternary solution,
This relationship it is assumed that addition of D in the solution does not give any influence on fBC and vice versa. This condition is generally satisfactory in most practical systems.
For a multicomponent system in general,
Taking logarithms,

28. 3.1.6. Interaction Coefficients
Since lnfB is some function of the mole fractions of B, C, D, ..., the Tayor-series expansion yields
Put
is known as interaction coefficient and experimentally determined.
From a practical point of view it is often more convenient to use weight percent for expressing the concentrations in conjunction with the 1 wt% standard state and logarithms to the base ten.
The last two equations offer an important means to calculate the activity coefficient of a dilute solute in multicomponent solutions.

29. 3.1.6. Interaction Coefficients
These equations are valid only for dilute solution because of approximations taken in the Tayor-series expansion :
Approximations
 As the Henrian standard state is chosen, NB → 0, fB = fBo =  For dilute solutions, the second order terms are negligible.
The following relations exist between interaction coefficients :
where A, B : solutes S : solvent

30. Ellingham Diagram
The standard free energy of formation (△Gfo) of a compound varies with temperature. The variation with temperature is usually presented by means of a table or some simple equations like
Ellingham presented the variation of △Gf0 with temperature in a graphical form in that △Gf0 was plotted against temperature.
He found that relationships in general were linear over temperature ranges in which no change in physical state occurred.
The relations could well be represented by means of the simple equation : T

31. Ellingham Diagram
In plotting △Gf0 -T diagrams Ellingham made use of which is the standard free energy of formation of the compound, not per mole of the compound, but per mole of the gaseous element consumed. For instance,
Consider a general reaction of formation :
This figure is known as the Elligham diagram which graphically shows the change in with T.
T, oC

32. 3.1.7. Ellingham Diagram Recall that and But when M and MO2 are pure.
Thus
This equation relates to the oxygen pressure in equilibrium with M(s) and MO2(s) at temperature T.

33. Ellingham Diagram
Each discontinuity point in the diagram indicates the phase change of a species involved in the formation reaction. For instance,
M + O2 = MO2
M(g) + O2(g) = MO2(s)
M(l) + O2(g) = MO2(s)
M(s) + O2(g) = MO2(s)
M(s)
M(l)
M(g)
T, oK

34. 3.1.7. Ellingham Diagram Recall that
Thus the slope of the line in the Ellingham diagram is and the intercept of the line at 0K is

35. Ellingham Diagram
Simplifying the notations in the diagram, we have
M + O2 = MO2
Where M and B stands for the melting and boiling points respectively. B M
T, oC
As we may add as many formation reactions as we want, this method of presentation provides a large amount of thermodynamic data and shows the relative stability of compounds for given conditions.
The Ellingham diagram for some oxides with Richardson nomographic scales is attached.

36. Ellingham Diagram

37. 3.2.1.Phase Rule Suppose that we have one mole of ideal gas A.
Are we free to vary temperature? Yes.
Are we free to vary both temperature and pressure independently? Yes.
Can we than vary temperature, pressure and volume independently? No.
We must specify only two variables, i.e (T,P), (P,V) or (V,T). The third variable is uniquely determined by the equation of state In other words, the equilibrium condition of the system is fully defined by specifying two variables. A
We may say that we have two variables under our control.
We rhen say number of degrees of freedom = 2

38. 3.2.1.Phase Rule
What if the gas phase is the mixture of the ideal gases A and B?
Is the number of degrees of freedom still the same? No.
In other words, cam we fully define the thermodynamic state of the system by specifying two variables, say, T and P?
Specifying T and P will determine V, but the composition of the gas phase is left undetermined until the concentration of either A or B is fixed.
A B
That is, even after fixing T and P we can freely vary the concentration of either A or B (but not both because NA + NB = 1), and hence have one additional degree of freedom.
Thus, the number of degrees of freedom is three in this case.

39. 3.2.1.Phase Rule What if a liquid phase coexists with the gas phase?
What if a solid phase coexists as well?
Would it be thermodynamically feasible to have another solid phase in equilibrium?
Are there system methods available to answer these questions?
A B Gas
Liquid A B
Solid A B
Before deriving a thermodynamic methods for the characterization of equilibrium states of heterogeneous systems involving any number of substances.

40. 3.2.1.Phase Rule
Before deriving a thermodynamic method for the characterization of equilibrium states of heterogeneous systems, we define two important terms :
Phase : A physically distinct, homogeneous and mechanically separable part of a system
Examples
The ice-water-steam system at 0 ℃ has three distinct phases : solid (ice), liquid (water) and gas (steam).
Vapors and gases, either pure or mixed, constitute a single phase because the component gases are miscible.
Solutions (liquid and solid) are single phases.
Immiscible liquids constitute separate phases (e.g., liquid slag and metal in a furnace)

41. 3.2.1.Phase Rule
Component : The number of components of a system is the minimum number of composition variables that must be specified in order to completely define the composition of each phase in the system.
The number of components is not necessarily the same as the number of elements, species or compounds present in the system, but given by
c = s - r
c : Number of components s : Number of chemically distinct constituents r : Number of algebraic relation

42. 3.2.1.Phase Rule Examples In the nitrogen-hydrogen-ammonia system
A nonreactive mixture of N2(g), H2(g) and NH3(g) at a low temperature
s = 3, r = 0 → c = 3
A mixture of N2(g), H2(g) and NH3(g) at a high temperature where the following equilibrium is established :
Thus, s = 3, r = 1 → c =2
* Each independent chemical reaction at equilibrium gives rise to a restrictive condition.

43. 3.2.1.Phase Rule
If the above mixture was initially obtained by heating NH3(g), there exists an additional restrictive condition :
Thus, s = 3, r = 2 → c = 1
* Note that each stoichiometric relation or constraint gives rise to a restrictive condition.
We now consider more on the equality of chemical potentials at equilibrium.
Total free energy G
For a closed system
For a open system
at constant T & P
This equation applies to all phase in a system

44. 3.2.1.Phase Rule Total free energy change : From mass balance : and
Combination yields :
Since dn1α and dn2α in the equation are arbitrary and hence not zero,
At equilibrium, therefore, the chemical potential of each component is constant throughout the entire system.
We now ready to derive an equation known as the phase rule which applies to both homogeneous and heterogeneous systems at equilibrium.

45. Number of restricting equations
3.2.1.Phase Rule
Consider a system composed of c components that are distributed between p phase.
Number of variables
There are ( c – 1 ) component variables in each phase, because knowing the concentrations (e.g., mole fractions) of ( c – 1 ) components the last one can be found from.
N1 + N NC = 1
Since there are p phases, the total number of component variables of the system is p(c – 1)
There are two additional variables, temperature and pressure.
Thus the total number of variables is p(c – 1) + 2
Number of restricting equations
The chemical potential of component i is constant throughout the system.
Hence the number of independent equations for each component is (p – 1).
Since there are c components in the system, the total number of restricting equation is c(p – 1)

46. 3.2.1.Phase Rule
Therefore, the number of undetermined variables is given by,
Number of undetermined variables =
This is called the number of degrees of freedom and denoted f
Thus,
This equation is called the Gibbs phase rule or the phase rule
The phase rule offers a simple means of determining the minimum number of intensive variables that have to be specified in order to be specified in order to unambiguously determine the thermodynamic state of the system.
The application of the phase rule does not require a knowledge of the actual constituents of a phase.
The phase rule applies only to systems which are in equilibrium.

47. 3.2.2.PhaseTransformation
The intensive properties of a system include temperature, pressure and the chemical potentials (or partial molar free energies) of the various species present.
If there is a temperature difference
The transfer of energy occurs as heat.
Temperature : A measure of the tendency of heat to leave the system
If there is a pressure difference
The transfer of energy occurs as PV work.
Pressure : A measure of the tendency toward mechanical work.
If there is a Chemcial Potential difference
The transfer of energy occurs as transfer of matter.
Temperature : A measure of the tendency of the species to leave the phase, to react or to spread throughout the phase.
Through chemical reaction, diffusion, etc.

48. Gas chemical potential has the key to these questions.
3.2.2.PhaseTransformation
A substance with a higher chemical potential has a spontaneous tendency to move to a state with lower chemical potential.
Why do solid substances melt upon heating? Why do liquid substances vaporize rather than solidify upon heating? Why do phase transitions occur?
Solid
Liquid
Gas
Gas chemical potential has the key to these questions.
Consider a one-component system :
This equation shows that, because entropy is always positive, the chemical potential of a pure substance decrease as the temperature is increased.

49. 3.2.2.PhaseTransformation
As S(g)>S(l)>S(s), the slope of the polt of μ versus T is steeper for the vapor than for the liquid, and steeper for the liquid than for the solid.
This characterastic temperature is called the transition temperature
solid
Liquid
Gas Tm Tb
Below this temperature, μ(s)<μ(l) and hence the solid phase is more stable. Above this temperature, on the other hand, the liquid is more stable. At Tm the chemical potential of the solid and the liquid are the same and hence both phases coexist. Tm : Melting temperature
Below this temperature, μ(l)>μ(g) and hence the liquid phase is more stable. Above this temperature, on the other hand, the gas is more stable. At Tb the chemical potential of the liquid and the gas are the same and hence both phases coexist. Tb : Boiling temperature

50. 3.2.2.PhaseTransformation
Next examine the effect of pressure in the chemical potential.
As V is always positive, an increase in pressure increases the chemical potential of any pure substance. For most substances, V(l)>V(s) and hence, from the equation, an in crease in pressure increases the chemical potential of the liquid more than of the solid.
High P
Low P
High P
Low P
solid
Liquid Tm Tm
Low P
High P
Thus an increase in pressure results in an increase in the melting temperature. If V(l)<V(s), however, an increase in pressure effect a decrease in Tm. (e.g.,Vwater<Vice)

51. 3.2.2.PhaseTransformation
If would be useful to combine the effect of temperature and pressure on phase transition of a substance in a same diagram.
Liquid
solid t
Gas
This figure is known as the phase diagram of a substance. It shows the thermodynamically stable phases at different pressures and temperatures. The lines separating phases are called as phase boundaries at which two phases coexist in equilibrium. The point t is the triple point at which three phases coexist in equilibrium.

52. To answer the above question, we make use of the fact that
3.2.2.PhaseTransformation
Now, a question arises :
Is there a way to quantitatively describe the phase boundaries in terms of P and T
The phase rule predicts the existence of the phase boundaries, but does not give any clue on the shape (slope) of the boundaries.
To answer the above question, we make use of the fact that
At equilibrium the chemical potential of a substance is the same in all phases present.

53. 3.2.2.PhaseTransformation
Consider the phases α and β which are in equilibrium.
Recall that therefore
For a transition from one phase to another under equilibrium conditions at constant temperature and pressure,
Rearrangement yields
Sβ-Sα : Entropy change (ΔS) associated with the phase change of α→β Vβ-Vα : Volume change (ΔV) associated with the phase change of α→β
This equation is known as the Clapeyron equation

54. 3.2.2.PhaseTransformation
The Clapeyron equation is limited to equilibria involving phases of fixed composition (e.g., one-component system) because μ has been assumed independent of composition:
For the solid-liquid phase boundary,
ΔHf : Heat of fusion > 0 ΔVf : Volume change upon melting > 0 in most cases and always small.
Therefore is positive and very large.
This indicates that the slope of the solid-liquid phase boundary is positive and steep in most cases. Integrating the clapeyron equation assuming that ΔHf and ΔHf are constant,

55. 3.2.2.PhaseTransformation For the liquid-vapor phase boundary,
ΔHf : Heat of vaporization > 0 ΔVf : Volume change upon vaporizarion = V(g) - V(l) ≒ V(g) = RT/P
This equation is known as the Clausius-Clapeyron equation.
Integration yields
Using the equation ΔHv can be estimated with a knowledge of the equilibrium vapor pressure of a liquid at two different temperature
For the solid-vapor phase boundary (sublimation), an analogous is obtained by replacing ΔHv with the heat pf sublimation ΔHs.

56. 3.2.3.Phase Equilibria and Free Energies
When a liquid solution is cooled slowly, the temperature will eventually be reached the liquidus point, and a solid phase will begin to separate from the liquid solution.
This solid phase could be a pure component, a solid solution or a compound. Now a question arises as to in which direction the system will proceed with cooling.
Precipitating a pure component? a solid sloution? a component? or something else?
The answer is To the state at which the free energy of the system is minimum under the given thermodynamic conditions.
Therefore phase changes can be predicted from thermodynamic information on the free energy-composition-temperature relationship.

57. 3.2.3.Phase Equilibria and Free Energies
Suppose that the temperature of the A-B solution is sufficiently high so that
T1 ( > Tm(A), Tm(B) )
T1 ( > Tm(A), Tm(B) )
And hence liquid is stable for both A and B.
Therefore the natural choice of the standard state for A and B will be the pure liquid A and B
Solid solution
Standard state
Liquid solution
The free energy of mixing (GM) will change with composition. A B

58. 3.2.3.Phase Equilibria and Free Energies
Recall
As the temperature T1 is higher than the melting points of A and B, Tm(A) and Tm(B), pure solid A and B will be unstable at this temperature
unstable state
The free energy of mixing (GM) of the solid solution, if it existed indeed, will vary with composition as shown in the figure
At the temperature T1, therefore, liquid solutions have free energies lower than solid solutions, and hence are stable over the entire range of composition.

59. 3.2.3.Phase Equilibria and Free Energies
Next, suppose that the temperature of the A-B solution is sufficiently high so that
T2 ( < Tm(A), Tm(B) )
T2 ( < Tm(A), Tm(B) )
And hence solid is stable for both A and B.
Therefore the natural choice of the standard state for A and B will be the pure solid A and B
Liquid solution
Standard state
Solid solution
The free energy of mixing (GM) will change with composition. A B

60. 3.2.3.Phase Equilibria and Free Energies
As the temperature T2 is higher than the melting points of A and B, Tm(A) and Tm(B), pure liquid A and B will be unstable at this temperature
unstable state
The free energy of mixing (GM) of the liquid solution, if it existed indeed, will vary with composition as shown in the figure
At the temperature T2, therefore, solid solutions are stable over the entire range of composition.

61. 3.2.3.Phase Equilibria and Free Energies
If the change in the free energy of mixing (GM) of a mixing with composition is concave downward at constant temperature and pressure, why is homogeneous solution is stable over the entire range of composition?
Consider a mixture of A and B with an average composition of “a” shown in the figure. If this mixture forms a homogeneous solution (α), the free energy of mixing of tthe solution is given by GαM. Is there any way to lower the free energy of mixing below this value? If the mixture forms two separate phases, β and γ, with the composition of b and c, respectively, the molar free energies of mixing are GβM and GγM, respectively. As the average composition of the system (sum of β and γ phases) must be of the same as the initial mixture (a), the proportion of each phase is given by
GβM
GmM
GαM
GγM A b a c B
It is clear that the phase separation has resulted in an increase in the free energy of mixing (GmM > GαM) and hence there is no way of lowering the free energy of mixing below GαM.

62. 3.2.3.Phase Equilibria and Free Energies
Now we have a question as to the shape of the free energy curve : Why is the curve of GM versus Ni concave downward?
Recall that
T, P constant
GM = HM - TSM
at constant T HM
-TSM GM
If HM < 0
GM < 0 always and concave downward. A B

63. 3.2.3.Phase Equilibria and Free Energies
If HM > 0
At high T -TSM will dominate, and hence GM (= HM – TSM) will be concave downward.
At low T HM term dominate, and hence GM (= HM – TSM) will change with composition in more complicated manner.
T, P constant
T, P constant HM HM GM GM
If this occurs, the solution will not be homogeneous, but phase separation will take place in a certain range of composition.
-TSM
-TSM A B A B

64. 3.2.3.Phase Equilibria and Free Energies
T, P constant
Consider a solution of composition “a” at constant T and P. If the solution does not dissociate, but maintains homogeneity, GM of the solution is represented by “J” in the figure. It can be seen from the figure that it is possible to lower GM below J by dissociating into two separate coexisting solutions. For instance, if the solution dissociates into two solutions of compositions “b” and “c” respectively J GM K M L N
Common tangent A B

65. 3.2.3.Phase Equilibria and Free Energies
GM for solution b : K
GM for solution c : L
Average GM of the two solutions : M
Note M<J
Minimum GM occurs when the solution dissociates into two solutions of composition “d” and “e” which are the intercepts of common tangent to free energy curve.
Average GM of the two solutions : N
Fraction of solution d = ae / de Fraction of solution e = da / de

66. 3.2.3.Phase Equilibria and Free Energies
Summary
T, P constant
Solutions in these composition ranges are stable and do not dissociate into other phases since GM does not decrease by dissociation. GM A B
Solution in this composition range, however, are not stable to maintain one-phase solutions, and hence dissociate into two coexisting phases d and e
In other words, a miscibility gap will appear in the phase diagram

67. Miscibility gap
When the solution is slowly cooled, it maintains homogeneous α phase until the temperature reaches T' at which it begins to dissociate into phase α' and α" . As the temperature is further decreased, the compositions of α' and α" are changed along the gap boundary curve. At the temperature T1, composition of α' : d composition of α" : e fraction of α' = ae / de fraction of α" = da / de When the average composition is changed, the proportions of α' and α" are changed accordingly, but the compositions of α ' and α" stay at d and e, respectively, as long as the average composition lies between d and e. Now, we examine the change of B, GBM = RTlnaB GBM is constant,so aB is constant in this composition.
d a e
d a e
d e

68. Miscibility gap
We now consider an A-B binary solution at temperature T2 which is above the melting point of A (TM,A), but the melting point of B (TM,B).
Liquid B : unstable
B : Solid → solid phase standard state
Stable Phase
A : Liquid → Liquid phase standard state
Solid A : unstable

69. Miscibility gap T2 Free energy of mixing of solid solutions
Coexistence of liquid and solid solutions
Free energy of mixing of liquid solutions
Solid solutions T
Liquid
Liquid + Solid
Liquid solutions T2
This is an example of phase diagrams when A and B have the same or similar crystal structures.
Solid A NB B

70. Miscibility gap
Consider a system which consist of components A and B that have differing crystal structure and has phase diagram as shown below ; L
There are two terminal solid solutions, namely α and β
Let’s examine free energies of mixing at the temperature T1
L+α
L+β β α
α+β A NB B
Free energy of mixing of form homogeneous α solid solution from solid A and B
Free energy of mixing of form homogeneous β solid solution from solid A and B
Free energy of mixing of form homogeneous liquid solution all over the range.
Stable phases at T1
liquid
L+β β