I am doing research on the complex perovskite (1-x)Ba(Zr0.2Ti0.8)O3-x(Ba0.7Ca0.3)TiO3. This solid solution is normally represented in a pseudo-binary phase diagram. By pseudo I mean that BZT and BCT are already solid solutions, altough used as end members of a phase diagram. So considering actually the components of the system we have CaTiO3, BaTiO3, and BaZrO3. In this pseudo-phase diagram in literature sometimes appear a triple point or a quadruple point depending on the author. Now considering the rule of gibbs on a standard binary phase diagram a quadruple point (at P=constant) is forbidden. In the case of this pseudo-phase diagram, is there the same constraint? I guess the question resumes into a proper definition of independent thermodynamic components of the gibbs phase rule.
Gibbs' phase rule is F = C - P + 2 - R, where
C is the number of components,
P the number of phases,
and R the number of additional constraints.
Most problems with the phase rule arise from forgetting the “R”. In your case, you can consider your perovskite either as a binary solution as indicated in your question, with 4 - P degrees of freedom, or as a ternary mixture with 1 restraint on the composition (you do not allow all compositions, but only those that you reach by varying “x” in your perovskite formula). In both cases, there are 4 - P degrees of freedom. Consequently, a quadruple state is possible, but nothing higher.
In fact, quadruple points are a common feature in binary phase diagrams, if pressure variation is allowed. If you insist on working at 1 bar, you add another restraint, and then you are down to triple points only.
problems of this type are very interesting. we have to analyse them deeply. You may read the following paper: . S.Ranganathan, Phase Rule – A View, Trans. Indian Institute of Metals, 55(6), 551-555, 2003.
Dear Mr. Ulrich. Thank you for your answer. I did not forget the R term and I considered a standard binary phase diagram in which P=constant. The thing is that this is not a standard binary phase diagram but a pseudo-phase diagram because the end member are themselves solid solutions. So basically I am seeing a plane of the ternary system. My question refer more to as components
Thank you very much Mr. Ranganathan for your answer. The paper seems very interesting but I cannot download it unfortunately. If it is possible for you to send it to me it would be great!
Matias Acosta,
Can you send me your e-mail i.d so that I can forward the paper to you. You can contact me at [email protected]
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