By: Ahmad Mostafa

Supervisor: Dr. Mamoun Medraj

ContentsIntroduction

Limiting Slope equation

Applications

Ratios of invariants for solidus and liquidus lines

Ratios of slopes at invariants

Calculation of solidus composition from the liquidus

slope

Conclusions2

IntroductionThe phase diagram is not only a graphical

interpretation of a system.

3

Each line is constructed as a result of thermodynamic calculations

Introduction

4

During evaluating a phase diagram, it is important to check that the diagram is consistent with its thermodynamic properties.

A hypothetical phase diagram with common thermodynamic improbable features

Any phase diagram should be evaluated.

IntroductionA relation between the slopes of the liquidus

at certain composition and the extent of the solid solution.

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Limiting Slope EquationThe relationship can be derived

thermodynamically through the following equation:-

Where and are the slopes of the liquidus and solidus.

: is the mole fraction of component A. : is the molar enthalpy of fusion of A. : is the melting point of A in kelvins

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Limiting Slope EquationIn many cases , the only thermodynamic data

required are the entropies of fusion.Entropy of fusion: is the increase in entropy when

melting a substance. ΔHfus=Tfus× ΔSfus

Raoult`s law: the vapor pressure of the ideal solution is dependent on the vapor pressure of each chemical component.

The only requirement involved in the equation is that Raoult`s law be obeyed in the limit for the liquid and solid phases.

: Is the partial pressure of the component in the solution.

: The vapor pressure of the pure component : The mole fraction of the component in the solution.

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Applications1- Ratios of invariants for solidus and liquidus lines

2- Ratios of slopes at invariants

3- Ratios of slopes for solidus composition from the liquidus slope

• Eutectic with no intermediate compounds

• Invariant with an allotropic transformation

• Peritectic melting of compound

• Eutectic with an intermediate compound

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Ratios of invariants for solidus and liquidus lines

Experimental limiting liquidus and solidus

slopes at Xk=1

Y=1

From CalculationsΔh°f(K)= 2.6 kJ/mol

ExperimentallyΔh°f(K)= 2.3 kJ/mol ~

At 336.34 k

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Ratios of invariants for solidus and liquidus lines• An example of a diagram which does not pass the calculations test is the Na-Sr diagram

Limiting liquidus and solidus slopes at 774°C, resulted in Δh°f(Sr)= 14.6 kJ/molWhich it is twice the correct value Δh°f(Sr)= 7.4 kJ/mol 10

Ratios of invariants for solidus and liquidus lines• The recent critical evaluation of Na-Sr phase diagram

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Ratios of invariants for solidus and liquidus lines• The recent critical evaluation of Na-Sr phase diagram

The liquidus slope became much steeper The probability of loosing Na by volatilization resulted in incorrect liquidus of previous diagram

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Ratios of slopes at invariantsFor binaries involving three phases (α, β, and γ), the

equation will be derived to find the slopes in the invariant point.

Where σγα and σγβ: are the slopes of the γ-phase boundaries at the invariant temperature.

S: is the standard molar entropies of pure substance.

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Calculation of solidus composition from the liquidus slope

For many binaries, the liquidus has been measured, but data on the solidus are lacking.

To calculate the solidus composition at a given temperature, it is necessary to know the composition and the slope of the liquidus at same temperature as well as the excess free energy (ΔGexcess) of the liquid and the entropy of the solid.

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Calculation of solidus composition from the liquidus slope

• An example is the Cs-K phase diagram at -15°C (258 K)

• From figure, XL

B=0.27 and

Xs

B=0.175, agreed

within 0.005 with the measured solidus using the following formula:

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ConclusionsThe used equations were derived to test the binary phase

diagram for thermodynamic consistency.

The experimental results are the source of thermodynamic

data.

The accuracy of the phase diagram can be verified by

thermodynamic principles.

Phase diagram construction is mainly based on coupling

thermodynamic data and experimental results.

The slopes of the invariant points can also provide

valuable information on the phases.16