If an iron (III) complex with the formula \[\left[ \text{Fe}(\text{NH}_3)_x (\text{CN})_y \right]^-\]has no electron in its \( e_g \) orbital, then the value of \( x + y \) is:
- 5
- 6
- 3
- 4
The Correct Option is B
Approach Solution - 1
The given complex is
\[[\text{Fe}(\text{NH}_3)_2(\text{CN})_4]^{3+}\]
Here, the oxidation state of iron is \(+3\), which corresponds to a \(d^5\) electronic configuration in the high-spin state. The ligands cyanide (\(\text{CN}^-\)) and ammonia (\(\text{NH}_3\)) are arranged such that the \(e_g\) orbitals remain unoccupied.
Given values:
\(x = 2 \, (number\ of \text{NH}_3 \, \text{ligands})\),
\(y = 4 \, (\text{number of } \text{CN}^- \, \text{ligands})\)
Thus, \(x + y = 2 + 4 = 6\).
Approach Solution -2
We are given a complex ion of the form \(\left[ \text{Fe}(\text{NH}_3)_x(\text{CN})_y \right]^-\).
We need to determine \(x + y\), given that the complex has **no electrons in its \(e_g\) orbital**. This information helps us infer the electronic configuration and the hybridization of the metal center.
Step 2: Determine the oxidation state of iron
The overall charge on the complex is \(-1\). Let the oxidation state of Fe be \(+3\) (as given in the question: iron(III) complex).
Hence, Fe(III) means the metal ion is \(\text{Fe}^{3+}\).
Step 3: Electronic configuration of Fe³⁺
The ground-state configuration of neutral iron (Z = 26) is: \[ \text{Fe} : [\text{Ar}]\,3d^6 4s^2. \] For \(\text{Fe}^{3+}\): \[ \text{Fe}^{3+} : [\text{Ar}]\,3d^5. \] So, Fe³⁺ has 5 d-electrons.
Step 4: Meaning of "no electron in \(e_g\) orbital"
In an octahedral crystal field, the five d orbitals split into two sets:
- \(t_{2g}\) orbitals (lower energy): \(d_{xy}, d_{yz}, d_{zx}\)
- \(e_g\) orbitals (higher energy): \(d_{z^2}, d_{x^2 - y^2}\)
If there are **no electrons in \(e_g\)**, it means that all 5 d-electrons are accommodated in the \(t_{2g}\) set — implying a **low-spin octahedral complex**.
Step 5: Determine nature of ligands
- \(\text{CN}^-\) is a strong field ligand (strong enough to cause pairing of electrons).
- \(\text{NH}_3\) is a weak-to-moderate field ligand (does not cause pairing in most cases).
To have a low-spin configuration (\(t_{2g}^5 e_g^0\)) for Fe³⁺, the ligand environment must include **strong field ligands** like \(\text{CN}^-\) only.
Step 6: Analyze composition and total number of ligands
For a coordination number of 6 (octahedral geometry), the total number of ligands is: \[ x + y = 6. \] Since the condition of no electrons in \(e_g\) orbitals can only be achieved with a strong field low-spin octahedral complex, the coordination number must be 6.
Step 7: Final conclusion
Therefore, the sum of the number of ligands is:
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