On the Slopes of Phase Boundaries

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On the Slopes of Phase Boundaries

• By Ahmad Mostafa
• Supervisor Dr. Mamoun Medraj

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Contents

• Introduction
• 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
• Conclusions

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Introduction

• The phase diagram is not only a graphical
  interpretation of a system.

• Each line is constructed as a result of
  thermodynamic calculations

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Introduction

• Any phase diagram should be evaluated.

• 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
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Introduction

• A relation between the slopes of the liquidus at
  certain composition and the extent of the solid
  solution.

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Limiting Slope Equation

• The 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 Equation

• In many cases , the only thermodynamic data
  required are the entropies of fusion.

• Entropy of fusion is the increase in entropy
  when melting a substance.
  ?HfusTfus ?Sfus

• The only requirement involved in the equation is
  that Raoults law be obeyed in the limit for the
  liquid and solid phases.

• Raoults law the vapor pressure of the ideal
  solution is dependent on the vapor pressure of
  each chemical component.

• 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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Applications

• 1- Ratios of invariants for solidus and liquidus
  lines

2- Ratios of slopes at invariants

• Eutectic with no intermediate compounds

• Invariant with an allotropic transformation

• Peritectic melting of compound

• Eutectic with an intermediate compound

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

From Calculations ?hf(K) 2.6 kJ/mol
Experimentally ?hf(K) 2.3 kJ/mol
Experimental limiting liquidus and solidus slopes
at Xk1
At 336.34 k
Y1
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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 774C,
resulted in ?hf(Sr) 14.6 kJ/mol Which it is
twice the correct value ?hf(Sr) 7.4 kJ/mol
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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 invariants

• For binaries involving three phases (a, ß, and
  ?), the equation will be derived to find the
  slopes in the invariant point.

• Where s?a and s?ß are the slopes of the ?-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
  (?Gexcess) 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 -15C
  (258 K)

• From figure, XLB0.27 and XsB0.175, agreed
  within 0.005 with the measured solidus using the
  following formula

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Conclusions

• The 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.

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Thank You