Define components of mineral assemblage?

Question

Given the a mineral assemblage with the minerals $$cp (\ce{CuFeS_2}), bn (\ce{Cu_5FeS_4}), cc (\ce{Cu_2S}), mt (\ce{Fe_3O_4}), py(\ce{FeS_2})$$ where cp, bn, mt are in contact, and py, cp, bn, cc are in contact. What are the components?

I was told #components=#constituents-#reactions -#constraints but I don't know how to define constituents or constraints. I'm not sure what the number of reactions is either. I was trying to define constituents based on oxidation states of the metals (assuming S and O are always -2), and I got $$\ce{Cu^{+1}, Fe^{+4}, Fe^{+3}, Fe^{+2}}$$ where the $$\ce{Fe^{+3}} : cp (\ce{CuFeS_2}), bn (\ce{Cu_5FeS_4})$$ and the $$\ce{Fe^{+4}}: py(\ce{FeS_2})$$ and $$\ce{Fe^{+3}, Fe^{+2}}: mt (\ce{Fe_3O_4})$$

As for the number of reactions, I'm confused because only one of the minerals has oxygen (so can't be part of a reaction with one of the others to produce another), and I was having a hard time balancing reactions between the others.

Can anyone give me any advice on how to figure out what the components are, and in general how I should define components, constituents, and reactions based on assemblages of minerals?

This question is based on a problem from Walther "Essentials of Geochemistry" here is a [Link] to the question.

Just to clarify based on the given answers, the components are cc, cp, py, and mt? Since bn is the only one that can be formed from a reactions of the others?

Answer 1

Let's start by understanding what a component is. The components of a system are the minimum amount of chemical 'species' required to describe a system. What does that mean? Let's take a familiar example. Periclase, forsterite, enstatite and quartz. That's MgO, Mg₂SiO₄, MgSiO₃ and SiO₂. So we can say we have 4 components, each of those formulas I just noted. But that's not the minimum number. We can get a smaller number. So we can say, for example that our components are Mg, Si and O. We can create all of the mineral formulas from those 3 components. But wait - we can go even lower! If we choose our components to be only MgO and SiO₂, we have 2 components that can still describe our entire system. We can combine 2MgO+1SiO₂ to describe forsterite and 1MgO+1SiO₂ to describe enstatite. Basically, the components have to be independent. If there is one component that can be described by others, then it is not actually a component.

So let's go to your problem. First of all, I would say that your assumption that the oxidation state of Cu is +1 and that of S is -2 is not entirely correct. In pyrite, S is actually -1 and Fe is +2 (Fe⁴⁺ does not exist). As an aside, the oxidation state of Cu (and others) in sulfides is a can of worms: those are not ionic compounds but rather a mixture of metallic and covalent compounds where oxidation state does not mean much, especially in the more complex cases such as bornite and chalcopyrite.

That said, if you look carefully and try to balance your reactions (try to make one mineral using the others), you can see that bn = cp + 2cc, so bn is not a component. Can you still reduce the amount of components? No. Fe₃O₄ must be there, and there is no way you can make CuFeS₂ from Cu₂S and FeS₂. You might think about using the components S-Fe-Cu-O, but that would actually be incorrect because for that to happen S has to be the same valence, which obviously is not (compare Cu₂S^2- and FeS^1-₂).