Correct!

Chrome alum contains the hexaquo cation: [Cr(H2O)6]3+

Chromium(III) chloride contains the dichloro cation: [CrCl2(H2O)4]+

According to ligand field theory (LFT), the 5-fold degeneracy of the metal d orbitals is lifted when a metal atom is surrounded by a set of negative charges or the negative end of dipoles. That is to say, the five d orbitals no longer have the same energy and split. The pattern of splitting depends on the arrangement of the ligands. In an octahedral complex, the d orbitals split into eg and t2g orbitals.

The different ligand environments cause a different amount of splitting between the chromium ion's eg and t2g orbitals and it is this that results in different colours for the two compounds.

Now try some of the experiments!

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