MSE 303_Note8_Calculating Phase Diagrams (Gaskel Chap10).pdf

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MSE 303

Thermodynamics & Equilibrium Processes

Calculating Phase Diagrams

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Temperature, T1

Calculate the Gibbs free energy G as a function of composition at T = T1 for liquid and solid phases

Liquid has lower energy over entire composition range

Hence at T1, our phase diagram must show liquid phase across entire composition

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Temperature, T2

This equivalence of Gibbs free energy indicates a phase transition from liquid to solid state

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Temperature, T3

This indicates that two phases: Liquid and solid are in equilibrium

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Temperature, T4

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Based on the free energy curves we have generated, we can now construct the phase diagram

A B

L

S

L

L+S

S S

L

L

S All temperatures between T2 and T4 are have similar curve

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Actual Example

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Binary Solution with Miscibility Gap

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Binary Solution with Miscibility Gap

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Eutectic Phase Diagram

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Eutectic Phase Diagram

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Gibbs Free Energy Curve vs Composition, GM(Xi)

Partial molar Gibbs Free Energies of component in different phases

The partial molar Gibbs free energy change due to introduction of component i into the solution

General principles of calculating phase diagrams:

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Gibbs Free Energy Curve vs Composition, GM(Xi)

The curve relating Gibbs free energy to composition if solutions exhibit ideal behavior:

To be able to calculate partial molar free energy of a component I (or A or B), It is convenient to choose a standard state at which we define Gi0 as zero

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Gibbs Free Energy Curve vs Composition, GM(Xi)

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Figure 10.2 The activities of component B obtained from lines I, II, and III in Fig. 10.1.

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Figure 10.8 (a) The phase diagram for the system AB. (b) The Gibbs free energies of mixing in the system AB at the temperature T.

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Check at XA =1, XA=0

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Check at XB =1, XB=0

liquid

Similarly at The formation of an ideal solid solution from liquid A and solid B

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At any composition the formation of a homogeneous liquid solution from pure liquid A and pure solid B can be considered as being a two-step process involving 1. The melting of XB moles of B, which involves the change in Gibbs free energy 2. The mixing of XB moles of liquid B and XA moles of liquid A to form an ideal liquid solution, which involves the change in Gibbs free energy,

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ideal liquid solution from liquid A and solid B

Similarly, at any composition, the formation of an ideal solid solution from liquid A and solid B involves a change in Gibbs free energy of

( l ) 10.5

10.6

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From (1)

( l )

( l )

From chapter 9,

10.7

10.8

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From 10.6,

10.10

(6)

Also use

We got

10.9

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Similarly,

The solidus and liquidus compositions are thus determined as follows

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Position of solidus and liquidus can be calculated at any temperature

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assuming ideal solution behavior, we can calculate G0m and the phase diagram

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Given

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Assuming ideal binary liquid and solid solutions

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In (a), liquid A and solid B are chosen as standard states located at G=0

In (b), liquid A and liquid B are chosen as standard states located at G=0

In (c), solid A and solid B are chosen as standard states located at G=0

Fig. 10.10 shows the Gibbs free energy of mixing curves for a binary system AB which forms ideal solid solutions and ideal liquid solutions, drawn at a temperature of 500 K, which is lower than Tm,(B) and higher than Tm,(A). At and

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Example 2: Gaskell page 287

Consider a phase diagram that exhibits complete miscibility in the liquid state and complete immiscibility in the solid state

At the indicated composition, at temperature T, we have pure A in equilibrium with the liquid solution having the composition of the point p

p

AlA

AlsA

aRTG

GG

ln0 )(

)(0)(

+=

=

Same as

with aA=XA for Routian liquid

Thus,

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Consider using this equation to calculate the liquidus lines in a binary eutectic system Take for example, the Cd-Bi system

If liquidus solutions are ideal, we can calculate the Bi liquidus from

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or

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