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FREE ENERGY- FREE ENERGY- COMPOSITION CURVESCOMPOSITION CURVES

Free Energy-Composition CurvesFree Energy-Composition Curves 22

Equilibrium in Heterogeneous Equilibrium in Heterogeneous SystemsSystems

For many systems A and B components For many systems A and B components do not do not have the same crystal structurehave the same crystal structure in their in their pure states at a given temperature.pure states at a given temperature.

TWO free energy curvesTWO free energy curves must must be drawn, one for each structure.be drawn, one for each structure.

The The stablestable form of pure A and B form of pure A and B at a given T (and P) can be at a given T (and P) can be denoted as denoted as and and respectively. respectively.

Suppose that Suppose that is fcc is fcc and and is bcc is bcc,, Molar free energies of fcc A and Molar free energies of fcc A and

bcc B are shown as points bcc B are shown as points ‘a’‘a’ and and ‘b’.‘b’.


Equilibrium in Heterogeneous Equilibrium in Heterogeneous SystemsSystems

Molar free energy curve for the phase

Molar free energy curve for the and phases

is fccis fcc and and is bcc is bcc


The FIRST STEPFIRST STEP in drawing the free-energy curve of the fcc -phase is:

ConvertConvert the stable bccstable bcc arrangement of B atoms into an unstable fccinto an unstable fcc arrangement.

This will raise the free energy from point ‘b’ to ‘c’.

The free energy curve for The free energy curve for --phase can now be constructed phase can now be constructed as before by mixing as before by mixing fcc A and fcc A and fcc Bfcc B..

A similar procedure will produce the molar free energy curve for the --phasephase.

Points ‘a’ and ‘f’ shows the difference in free energy when fcc A is converted to bcc A.

is fccis fcc and and is bcc is bcc


It is evident from this exercise that A-rich alloys will have the A-rich alloys will have the lowest free energy as a homogeneous lowest free energy as a homogeneous -phase-phase, and, B-rich B-rich alloys as alloys as -phase-phase.

For alloys with compositions near the cross For alloys with compositions near the cross over in the G curves, the situation is not so over in the G curves, the situation is not so straight forward.straight forward.

In this case, it can be shown that the total free energy can be minimized by the atoms separating into two phasesseparating into two phases.


A general property of molar free A general property of molar free energy diagrams for phase energy diagrams for phase

mixturesmixturesSuppose an alloy consists of Suppose an alloy consists of and and phase, phase, each with a molar free energy of each with a molar free energy of GG and G and G

Overall compositionOverall composition of the phase mixture is given by

The lever rulelever rule gives the relative number of moles of and that must be present.

The molar free energy of the phase mixture G is given by the point on the straight line between and .

BX


The lengths ‘ad’ and ‘cf’lengths ‘ad’ and ‘cf’ represent the molar free energies of and phases present in the alloy.

Point ‘g’Point ‘g’ is obtained by the intersection of ‘be’ and ‘dc’

‘bcg’ and ‘acd’, as well as ‘deg’ and ‘dfc’, form similar triangles.

According to the lever rule 1

mol of the alloy will contain

bc/ac mol of and ab/ac mol

of .It follows that ‘bg’ and ‘ge’ represent the separate contributions from the and phases to the total free energy of 1 mol of alloy.

Therefore, the length ‘be’ represents Therefore, the length ‘be’ represents the molar free energy of the phase the molar free energy of the phase mixture.mixture.

Therefore, bg/ad = bg/ad = bc/acbc/ac

And ge/cf = ab/acge/cf = ab/ac.


CONSIDER ALLOY XO

If the atoms are arranged as a single single homogeneous phasehomogeneous phase, the free energy will be lowest as -phase-phase

oo GG

However, the system can However, the system can lower its free energylower its free energy, if the , if the atoms separate into two atoms separate into two phases with compositions phases with compositions 11

and and 11 for example. for example.

The free energy of the system will then be reduced to GG11


Further reductionsFurther reductions in free energy can be achieved if the phase compositions are adjusted to ee and and ee

The free energy of the The free energy of the system, system, GGee, is now a , is now a minimumminimum and there and there would be no desire for would be no desire for further change.further change.

Thus the system is now system is now in equilibriumin equilibrium and e and e are the equilibrium compositions of the and phases. Figure 1.27Figure 1.27


The result is quite general and can be applied to any alloyapplied to any alloy with an overall composition between e and e

— — only the relative amounts of only the relative amounts of the two phases change the two phases change according to the lever rule.according to the lever rule.

When the alloy composition lies outside this rangeoutside this range,

the minimum free energy the minimum free energy lies on the Glies on the G and G and G curves curves and the Equilibrium State of and the Equilibrium State of the alloy is a homogeneous the alloy is a homogeneous single phase.single phase.


It could also be seen that the equilibrium between two phases requires that the tangentstangents to to each G-curveeach G-curve at the equilibrium compositions lie on a common lie on a common line.line.

So each component must have the same chemical potential in the two phases, i.e. for heterogeneous equilibrium:

Similarly, the activitiesactivities of the components must also be equal, i.e.

AA BB and

AA aa

BB aa and


VARIATION OF ACTIVITY WITH ALLOY VARIATION OF ACTIVITY WITH ALLOY COMPOSITIONCOMPOSITION

Between A and e, and e and B, where single phasessingle phases are stable, the activities (or activities (or chemical potential) varychemical potential) vary.

Note that ideal solutions have been assumed so we see straight-line relationship between ‘a’ and ‘X’.

Between e and e the phase

compositions in compositions in equilibrium do not changeequilibrium do not change and the activities are also activities are also equalequal and are given by the points ‘q’ and ‘r’.

Figure 1.28Figure 1.28

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