The problem involves determining the size of the chelate rings formed in the coordination compound $Co(acac)_3$. Acetylacetone (acac) acts as a bidentate ligand, and its deprotonated form, the acetylacetonate ion, binds to the metal center $Co^{3+}$.
Let's analyze how the chelation occurs:
- Acetylacetone (acac), in its anion form, can donate electrons from two oxygen atoms. These oxygens are bonded to two carbonyl groups on either end of the ligand.
- Upon coordination, each acetylacetonate ion forms two bonds with the metal center, making it a bidentate ligand. This results in the formation of a five-membered ring with the cobalt ion.
- In the resulting coordination complex, $Co(acac)_3$, each acetylacetone forms a similar structure.
Each acac ligand forms a six-membered chelate ring with the metal ion:
- One $C=C$ bond in the acetylacetonate and the $O-C-Co$ framework contribute to the ring formation.
- Counting the atoms involved, including the metal center $Co^{3+}$, illustrates six members in the ring: C-O-Co-O-C-C.
The correct answer, based on this structure, is that the rings of the chelate are six-membered.
Option Analysis:
- Five-membered: Incorrect, as the chelate rings include the cobalt ion and additional atoms from the ligand.
- Four-membered: Incorrect, as this would not include enough atoms to complete either part of the acac ligand or the metal coordination.
- Six-membered: Correct, as properly explained above.
- Three-membered: Incorrect, such a small ring would be highly strained and is not structurally feasible here.
Therefore, the correct option is "Six membered."