# Phase Equilibria Lectures GIK Institute Pakistan.

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### Transcript of Phase Equilibria Lectures GIK Institute Pakistan.

The Gibbs free energy of a system is defined by the equation

G=H-TS

Thermodynamic Function

Enthalpy is a measure of the heat content of the system and is given

H=E+PV

The internal energy arises from the total kinetic and potential energies of the atoms within the system.

Kinetic energy can arise from atomic vibration in solids or liquids and from translational and rotational energies for the atoms and molecules within a liquid or gas whereas potential energy arises from the interactions, or bonds. between the atoms within the system.

If a transformation or reaction occurs the heat that is absorbed or evolved will depend on

the change in the internal energy of the system.

However it will also depend on changes in the volume of the system and the term PV

takes this into account, so at constant pressure the heat absorbed or evolved is given by

the change in H

Thermodynamic Function

When dealing with condensed phases (solids

and liquids) the PV term is usually very small in

comparison to E, that is H = E.

G includes entropy (S) which is a measure of the randomness/disorder of the system.

A system is said to be in equilibrium when it is in the most stable state. i.e. shows no

desire to change. An important consequence of the laws of classical thermodynamics is

that at constant temperature and pressure a closed system (i.e. one of fixed mass and

composition) will be in stable equilibrium if it has the lowest possible value of the Gibbs

free energy, or in mathematical terms

dG=0

G, that the state with the highest stability will be that with the best compromise

between low enthalpy and high entropy.

Equilibrium and Free Energy

Thus at low temperatures solid phases are most stable since they have the strongest atomic binding and therefore the lowest internal energy (enthalpy). At high temperatures however the - TS term dominates and phases with more freedom of atom movement, liquids and gases, become most stable.

Equilibrium and Free Energy

dG=0

Graphite and diamond at room temperature and pressure are examples of stable and

metastable equilibrium states. Given time. therefore. all diamond under these conditions

will transform to graphite.

Equilibrium and Free Energy

Any transformation that results in a decrease in Gibbs free energy is possible. Therefore

a necessary criterion for any phase transformation is

G=G2-G1

Intensive & Extensive Properties

All thermodynamics function can be divided into two types of properties:

Intensive and Extensive Properties

Intensive Properties

Independent of size of system

Example : T and P

Extensive Properties Are directly proportion to the quantity of material in a system Example: E,H,V,G and S

Single Component System

Phase changes can be induced in a single component system by changes in temperature

at a fixed pressure, say 1 atm.

A single component system could be one containing a pure element or one type of

molecule that does not dissociate over the range of temperature of interest.

To predict the phases that are stable or mixtures that are in equilibrium at different

temperatures ,It is necessary to be able to calculate the variation of G with T

Gibbs Free Energy as a Function of Temperature

Variation of Cp (Sp. heat) with temperature

Variation of Enthalpy(H) with temperature Variation of Entropy (S) with temperature

Gibbs Free Energy as a Function of Temperature

Variation of enthalpy and free energy for liquid and solid

phases of pure metal .L is the latent heat of melting and Tm is

the equilibrium melting temperature

At low temperatures GL > GS.

However, the liquid phase has a

higher entropy than the solid phase

and the Gibbs free energy of the

liquid therefore decreases more

rapidly with increasing temperature

than that of the solid.

Enthalpy & Free energy for liquid & solid phases of Pure Metal

Tm is the melting Temperature.

Free energy for liquid & solid phases of Pure Metal

The phase with the lowest free energy at

a given temperature will be the most

stable.

It shows that below the melting

temperature the solid phase is most

stable, and above this temperature the

liquid phase is stable.

At the melting temperature, where the

two curves cross, the solid and liquid

phases are in equilibrium.

Free energy and isomorphous diagram

Molar quantities: g = G/mole h = H/mole s = S/mole

Xi = mole fraction of i = Ni / Ni

If the (molar) gibbs free energy of pure A is gA, and that of pure B is gB, then the (molar) gibbs free energy for the combination of pure components is

g (pure, combined) = gAXA + gBXB

Free energy and isomorphous diagram

As a function of composition, the

gibbs free energy for the combination

of pure A and pure B is a straight line

connecting gA and gB

Free energy and isomorphous diagram

Now, lets remove the imaginary partition and let

the A and B atoms mix. There should be some

change in g due to this mixing is:

gmix = hmix - Tsmix

The enthalpy term, hmix, represents the

nature of the chemical bonding, or the extent

to which A prefers B, or A prefers A as a

neighbor.

Free energy and isomorphous diagram

gmix = hmix - Tsmix

The entropy term, smix, signifies the increase in

disorder in the system as we let the A and B atoms

mix. It is independent of the nature of the chemical

bonding.

Isomorphous, binary system as an ideal solution.

That is, A couldnt care less whether it is sitting next

to another A to a B atom. Like Ni-Cu system. Under

these circumstances,

hmix = 0.

Free energy and isomorphous diagram

Let us qualitatively examine the entropy of mixing. If

we have a system of pure A, and let the A atoms mix

with one another, there is no increase in entropy

because they were mixed to begin with. The same

holds true for a system of pure B.

If we have just one atom of B in a mole of A, removing

the invisible partition hardly changes the amount of

disorder. But, as we get to a 50:50 composition of A:B,

the increase in entropy as we allow the system to mix

is enormous. Therefore, smix is:

Free energy and isomorphous diagram

From that we can easily get the change in gibbs

free energy due to mixing:

gmix = hmix - Tsmix

hmix =0

gmix = - Tsmix

Free energy and isomorphous diagram

sum g(pure, combined) and

gmix to get the total gibbs

free energy of the solution

as a function of composition:

At this point we have the general shape of the

g(XB) curves for phases in a binary system. This

overall shape holds for both the solid and liquid

phases, so long as both make ideal (or close to

ideal) solutions.

Now compare gsol(XB) with gliq(XB) for various

temperatures, and examine how these correlate

to the phase diagram.

Free energy and isomorphous diagram

Free energy and isomorphous diagram

(1) High temperature:

At temperatures above the melting points of both pure A and pure B, the liquid is the stable phase for all compositions.

Therefore, gsol(XB) > gliq(XB) and the

gibbs free energy curves look like:

(2) Low temperature:

At temperatures below the melting points of both pure A and pure B, the solid is the stable phase for all compositions. Therefore, gsol(XB) < gliq(XB) and the gibbs free energy curves look like:

Free energy and isomorphous diagram

(3) Intermediate Temperature:

As the temperature is brought down from high

to low, the gsol(XB) starts to move below that

of gliq(XB). The minima in the two curves, in

general, do not occur at the same point, so for

some compositions, gsol(XB) > gliq(XB), while

for others gsol(XB) < gliq(XB). Therefore, at

temperatures between the melting points of

pure A and pure B, the solid and liquid curves

look like:

Free energy and isomorphous diagram

Lets say we start out with a liquid of

composition XBO and cool it to To.

The gibbs free energy of the liquid would be

given by point (1). The system realizes it

could lower its gibbs free energy by

transforming to a solid.

The gibbs free energy of that solid would be

given by point (2) on the g(XB) diagram.

But, how low can you go?

Free energy and isomorphous diagram

The system can, in fact, lower its free energy

even further by splitting up into a solid of

composition XBS and a liquid of composition

XBL.

The gibbs free energy of the solid is given by

point (4) on the g(XB) diagram and that of

the liquid by point (5) on the same diagram.

Free energy and isomorphous diagram

A system of overall composition XBo (at To). Then,

we only have solid phase present, and the gibbs

free energy is given by point (4) If system has of

overall composition XBL, then we only have liquid

phase present and the total gibbs free energy is

given by point (5).

If a system of a composition exactly in the

middle of XBS and XBL , then half the moles of

our system would be in the solid phase and the

other half in the liquid phase.

The gibbs free energy would be given by a point

half-way between points (4) and (5), sitting

precisely on the line that connects them.

Free energy and isomorphous diagram

Isomorphous system with Free energy

We have 2 phases liquid and solid. Lets consider Gibbs free energy curves for the two phases at

different T

T1 is above the equilibrium melting

temperatures of both pure components:

T1 > Tm(A) > Tm(B) the liquid phase will be the stable phase for any

composition

Isomorphous system with Free energy

Decreasing the temperature below T1 will have two effects:

1. GA Liquid and G B

liquid will increase more rapidly than GA solid and G B

Solid

2. The curvature of the G(XB) curves will decrease.

Eventually we will reach T2 melting point of pure component A, where

GA Liquid = G A

Solid

Isomorphous system with Free energy

For even lower temperature T3 < T2 = Tm(A) the Gibbs free energy curves for the

liquid and solid phases will cross.

The common tangent construction

can be used to show that for

compositions near cross-over of G

solid and G liquid, the total Gibbs free

energy can be minimized by

separation into two phases.

As temperature decreases below T3 GA Liquid and G B

liquid continue

to increase more rapidly than GA solid and G B

Solid

Therefore, the intersection of the Gibbs free

energy curves, as well as points X1 and X2 are

shifting to the right, until, at T4 = Tm(B) the

curves will intersect at X1 = X2 = 1

At T4 and below this temperature the Gibbs free

energy of the solid phase is lower than the G of

the liquid phase in the whole range of

compositions the solid phase is the only stable

phase.

Isomorphous system with Free energy

Based on the Gibbs free energy curves we can now

construct a phase diagram for a binary isomorphous

systems

Isomorphous system with Free energy

Binary Solutions with Miscibility Gap

Lets consider a system in which the liquid phase

is approximately ideal, but for the solid phase we

have Hmix > 0

i:e. the A and B atoms 'dislike' each other.

Binary Solutions with Miscibility Gap

Therefore at low temperatures (T3) the free energy

curve for the solid assumes a negative curvature in

the middle.

Binary Solutions with Miscibility Gap