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What causes coordination complexes to change colour as they are heated? Do the ligands detach from the central metal or do the electrons held in the d orbitals leave? If both happen, which happens first?

Specifically, I am trying to explain the colour changes observed when potassium ferrioxalate $(\ce{K3[Fe(C2O4)3]})$ was heated. The colour change is from green-yellow to brown/red.

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Many things can happen when you heat a coordination complex: ligand loss, reactions in the ligand structure, loss of hydration water, isomerization, oxidation/reduction of the metal center etc.

In the specific case of potassium iron(III) oxalate trihydrate, as reported in this paper, the thermogravimetric analysis shows that the first structural change happens at 113 ºC with the loss of three water molecules (from the trihydrate salt), yielding potassium iron(III) oxalate, a yellow-orange solid. When the temperature reaches 296 ºC (in air), iron(III) is reduced to iron(II) to form iron(II) oxalate, an orange solid.

Further heating leads to the decomposition of iron oxalate into FeO at around 470 ºC. FeO is a brown/red solid (probably the compound you mentioned in your question).

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