On the Slopes of Phase Boundaries

1. On the Slopes of Phase Boundaries By: Ahmad Mostafa Supervisor: Dr. Mamoun Medraj

2. 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

3. Introduction • The phase diagram is not only a graphical interpretation of a system. • Each line is constructed as a result of thermodynamic calculations

4. 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

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

6. 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

7. 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.

8. 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

9. Ratios of invariants for solidus and liquidus lines ~ Experimentally Δh°f(K)= 2.3 kJ/mol From Calculations Δh°f(K)= 2.6 kJ/mol Experimental limiting liquidus and solidus slopes at Xk=1 At 336.34 k Y=1

10. 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

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

12. 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

13. 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 σγαand σγβ: are the slopes of the γ-phase boundaries at the invariant temperature. • S: is the standard molar entropies of pure substance.

14. 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.

15. 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:

16. 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.

17. Thank You