By: Ahmad Mostafa
Supervisor: Dr. Mamoun Medraj
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
2
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.
ΔHfus=Tfus× ΔSfus
 The only requirement involved in the equation is that
Raoult`s law be obeyed in the limit for the liquid and solid
phases.
 Raoult`s 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
Experimentally
Δh°f(K)= 2.3 kJ/mol
Experimental limiting
liquidus and solidus slopes
at Xk=1
~
From Calculations
Δh°f(K)= 2.6 kJ/mol
At 336.34 k
Y=1
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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/mol
Which it is twice the correct value Δh°f(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 (α, β, and γ), the equation will be
derived to find the slopes in the invariant point.
 Where σ γα and σ γβ: 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, XLB=0.27
and XsB=0.175, 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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