Isothermal Phase Diagrams of Binary Alloys and Two Oxidants

Abstract

An example of an isothermal thermodynamic phase diagram of a quaternary system composed of a binary alloy and two oxidants has been calculated under appropriate values of the stability of the corresponding compounds, assuming that each metal forms only one compound with each oxidant. Initially, two-dimensional sections of the complete three-dimensional diagram made along planes corresponding to constant values of one of the variables involved were examined in detail. The actual structure of this type of three-dimensional diagram was then discussed by analyzing the shape and the boundaries of the various regions corresponding to the conditions of stability of each phase taken separately or in combination with other phases.

Appendix

Using the notation introduced above, the equilibrium constants for the formation of the two oxides have the general form

$$ {{K}}_{ 1} = {1 \mathord{\left/ {\vphantom {1 {\left[ {\left( { 1-{{ x}}} \right){\text{ P}}\left( {{\text{O}}_{ 2} } \right)^{ 1/ 2} } \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {\left( { 1-{{ x}}} \right){\text{ P}}\left( {{\text{O}}_{ 2} } \right)^{ 1/ 2} } \right]}} $$

(1)

and

$$ {{K}}_{ 2} = {1 \mathord{\left/ {\vphantom {1 {\left[ {{{x}}{\text{ P}}\left( {{\text{O}}_{ 2} } \right)^{ 1/ 2} } \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {{{x}}{\text{ P}}\left( {{\text{O}}_{ 2} } \right)^{ 1/ 2} } \right]}} $$

(2)

Similarly, the equilibrium constants for the formation of the two chlorides have the form

$$ {{J}}_{ 1} = {1 \mathord{\left/ {\vphantom {1 {\left[ {\left( { 1-{\text{ x}}} \right){\text{ P}}\left( {{\text{Cl}}_{ 2} } \right)} \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {\left( { 1-{{ x}}} \right){\text{ P}}\left( {{\text{Cl}}_{ 2} } \right)} \right]}} $$

(3)

and

$$ {{J}}_{ 2} = {1 \mathord{\left/ {\vphantom {1 {\left[ {{{x}}{\text{ P}}\left( {{\text{Cl}}_{ 2} } \right)} \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {{{x}}{\text{ P}}\left( {{\text{Cl}}_{ 2} } \right)} \right]}} $$

(4)

For the calculation of point E₁, with x = x(E₁) and P(O₂) = P(O₂,E₁), one variable, e.g. P(O₂,E₁) is eliminated from Eqs. 1 and 2 and the resulting equation is solved for x(E₁). Then, substitution of x(E₁) into one of the two equations yields P(O₂,E₁). In the same way, for the calculation of point E₂, with x = x(E₂) and P(Cl₂) = P(Cl₂,E₂), the latter variable is eliminated from Eqs. 3 and 4, obtaining x(E₂). After this, substitution of x(E₂) into either Eqs. 3 or 4 yields P(Cl₂,E₂).

Other important parameters involved in the construction of the complete 3D phase diagram are P(Cl₂,E₁) and P(O₂,E₂), already defined earlier. The first of them is calculated by setting x = x(E₁) in Eq. 4 and solving for P(Cl₂). Similarly, P(O₂,E₂) is calculated by setting x = x(E₂) in Eq. 2 and solving for P(O₂). With the present choice of the various equilibrium constants, the combination of P(O₂,E₁) and P(Cl₂,E₁) corresponds to a condition of equilibrium between an alloy with x = x(E₁) and a mixture of AO, ACl₂ and BO (point η₁ of the 3D phase diagram), while the combination of P(O₂,E₂) and P(Cl₂,E₂) corresponds to a condition of equilibrium between an alloy with x = x(E₂) and a mixture of ACl₂, BO and BCl₂ (point η₂ of the 3D phase diagram).