the metal d-orbitals that are directly facing the ligands in `K_(3)[CO(CN)_(6)[` are :

Options

A. `d_(xz), d_(yz) and d_(z^2)`
B. `d_(xy),d_(xz) and d_(yz)`
C. `d_(x^2-y^2) and d_(z^2)`
D. `d_(xy) and d_(x^2-y^2)`.

Explanation

To determine the metal d-orbitals that are directly facing the ligands in the complex \( K_3[Co(CN)_6] \), we can follow these steps:

### Step 1: Determine the oxidation state of cobalt in the complex
The formula of the complex is \( K_3[Co(CN)_6] \). The oxidation state of \( Co \) can be calculated as follows:

- Let the oxidation state of \( Co \) be \( x \).
- The oxidation state of \( CN \) is \(-1\) (since it is a ligand).
- There are 6 \( CN \) ligands, contributing a total of \(-6\) to the overall charge.
- The overall charge of the complex is neutral (0).

Setting up the equation:
\[ x + 6(-1) = 0 \implies x - 6 = 0 \implies x = +3 \]

### Step 2: Determine the electronic configuration of cobalt
Cobalt in the +3 oxidation state has the electronic configuration:
\[ Co^{3+}: [Ar] 3d^6 \]

### Step 3: Identify the coordination number and geometry
The complex has 6 ligands (6 \( CN \) ligands), which gives it a coordination number of 6. This coordination number corresponds to an octahedral geometry.

### Step 4: Understand the splitting of d-orbitals in an octahedral field
In an octahedral field, the five d-orbitals split into two sets:
- \( e_g \) set: \( d_{z^2} \) and \( d_{x^2-y^2} \)
- \( t_{2g} \) set: \( d_{xy} \), \( d_{yz} \), and \( d_{zx} \)

### Step 5: Identify which d-orbitals face the ligands
In an octahedral arrangement:
- The \( e_g \) orbitals (\( d_{z^2} \) and \( d_{x^2-y^2} \)) are oriented directly towards the ligands.
- The \( t_{2g} \) orbitals (\( d_{xy} \), \( d_{yz} \), and \( d_{zx} \)) are oriented between the ligands and do not face them directly.

### Conclusion
The d-orbitals that are directly facing the ligands in \( K_3[Co(CN)_6] \) are:
- \( d_{z^2} \)
- \( d_{x^2-y^2} \)

Thus, the answer is \( d_{z^2} \) and \( d_{x^2-y^2} \).

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