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Do complex ions tend to form a certain number of coordinate bonds, or does it completely depend on the ligand molecule?

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There is no way you can predict this a priori. For example, cobalt(III) forms an octahedral hexaaquacomplex $\ce{[Co(H2O)6]^3+}$. If you add chloride to the solution, you will form a tetrahedral tetrachloridocomplex $\ce{[CoCl4]-}$. This was a pure ligand exchange reaction as shown in the equation below.

$$\ce{[Co(H2O)6]^3+ + 4 Cl- <=>> [CoCl4]- + 6 H2O}$$

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