Hello. I have a question about ligands.

How do you figure out how many ligands a central metal cation can have?

For example, Cu^2+ and NH3. When they combine, it forms Cu(NH3)4^2+

Why does it form 4 ligands oppose to 6?

A general rule, which isn't always followed, is to double the charge to find the number of ligands. One reason, for Cu^+2, is the s, p, and d orbitals available.
29Cu is 1s2 2s2 2p6 3s2 3p6 3d9 4s2.
The Cu^+2 loses those two 4s electrons, promotes the 1 unpaired 3d electron so it has available a 3d, a 4s, and 4p orbitals and it forms dsp2 which is a square planar complex. A 6-coordinate compound would be d2sp3 and there is only 1d available in the 3d shell. So the d2sp3 is not a good option.

Ah thank you for the speedy response!

So... let me apply what I learned from you.

My next problem just so happens to be Zn^2+

e- config for Zn: [Ar] 3d10 4s2
therefore, Zn^2+: [Ar] 3d10

now i'm stuck. What gets promoted?

I could apply that unofficial rule of yours and say that because it's Zn^2+, it should have 4 ligands.

There is nothing to promote. All of the 3d orbitals are filled. My first educated guess is to use the sp3 and a coordination number of 4. You will find this in Zn(NH3)4^+2, Zn(OH)4^-2, etc. I would guess sp3 and tetrahedral. By the way, it has been found in studies that Cu^+2, does in fact, often add two more ligands to make a coordination number of 6 and the complex is octahedral. To do this it uses the 4s, 4p, 5d for an "outer" complex of d2sp3, or more accurately, sp3d2.

You're correct that for Zn^2+, the electron configuration is [Ar] 3d^10. Since all of the 3d orbitals are filled, there is no promotion of electrons to higher energy levels. In this case, the most likely arrangement of ligands would be an sp3 hybridization and a coordination number of 4. Ligands such as NH3 or OH^- have been observed to form tetrahedral complexes with Zn^2+.

It's important to note that while the "double the charge" rule can be a helpful guideline, it is not always accurate. The actual arrangement of ligands around a central metal cation is determined by a combination of factors including the electron configuration of the cation and the nature of the ligands themselves. Experimental evidence and theoretical calculations are often needed to determine the correct arrangement.