Atomic number of Mn, Fe, Co and Ni are 25, 26, 27 and 28 respectively. Which of the following outer orbital oc

NCERT Exemplar · Q33

Q. Atomic number of Mn, Fe, Co and Ni are 25, 26, 27 and 28 respectively. Which of the following outer orbital octahedral complexes have same number of unpaired electrons?

(i)

[MnCl_6]^{3-}

(ii)

[FeF_6]^{3-}

(iii)

[CoF_6]^{3-}

(iv)

[Ni(NH_3)_6]^{2+}

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Concept understanding — Crystal Field Splitting

Crystal Field Splitting: From Intuition to Precision

Imagine you are a negatively charged electron sitting on a metal ion. All around you, the space is perfectly spherical — every direction feels the same. Your energy depends only on how far you are from the nucleus, not on which way you face.

Now imagine that six negative ions (or the negative ends of polar molecules) march in from the $x$ , $y$ , and $z$ axes and stop close to you. Suddenly, the space around you is no longer uniform. If you try to move straight toward one of these approaching ions, you feel a strong repulsion — that path costs extra energy. If you move between the axes (say, along a diagonal), you feel less repulsion because you are farther from the incoming charges.

This is the core intuition: when ligands approach a metal ion, they break the spherical symmetry of the space around the metal. Different directions in space are no longer equivalent. Electrons in orbitals that point directly at the ligands get pushed up in energy; electrons in orbitals that point between the ligands stay lower.

The Precise Statement

d

For the most common geometry — octahedral — here is what happens:

Six ligands sit at the corners of an octahedron, along the $+x$ , $-x$ , $+y$ , $-y$ , $+z$ , $-z$ axes.
The $d_{x^2-y^2}$ and $d_{z^2}$ orbitals point their lobes directly along these axes. These are the $e_g$ set. They feel maximum repulsion → higher energy.
The $d_{xy}$ , $d_{xz}$ , and $d_{yz}$ orbitals point their lobes between the axes (into the octahedral faces). These are the $t_{2g}$ set. They feel less repulsion → lower energy.

The energy gap between these two sets is denoted by $\Delta_o$ (or $10\,Dq$ ). The $t_{2g}$ set drops by $0.4\Delta_o$ and the $e_g$ set rises by $0.6\Delta_o$ , keeping the average energy unchanged (the "barycentre" rule).

Note

The labels $e_g$ and $t_{2g}$ come from group theory — they describe how the orbitals transform under the symmetry operations of an octahedron. You do not need to memorise the derivation, but the notation is standard in every exam.

Why This Matters

Crystal field splitting explains three things you will see repeatedly:

Colour — electrons can jump from $t_{2g}$ to $e_g$ by absorbing visible light. The gap $\Delta_o$ determines the colour you see.
Magnetism — if $\Delta_o$ is large, electrons pair up in the lower $t_{2g}$ set (low spin). If $\Delta_o$ is small, electrons spread out (high spin). This changes the number of unpaired electrons. …

Why this formula?

Crystal Field Splitting: Why the Energy Splitting Occurs

Crystal Field Theory (CFT) explains how the d-orbitals of a transition metal ion split in energy when placed in an electrostatic field created by surrounding ligands (anions or polar molecules). The key result is that five degenerate d-orbitals split into two or more sets with different energies. Let's understand why this happens.

1. The Starting Point: Degenerate d-Orbitals

In a free transition metal ion (no ligands), all five d-orbitals have the same energy (degenerate). Their shapes are:

$d_{xy}$ , $d_{xz}$ , $d_{yz}$ — lobes lie between the x, y, z axes (called $t_{2g}$ set in octahedral symmetry)
$d_{x^2-y^2}$ , $d_{z^2}$ — lobes point directly along the x, y, z axes (called $e_g$ set)

Key idea: The spatial orientation of each orbital determines how it interacts with approaching ligands.

2. The Octahedral Case: Why $e_g$ Orbitals Are Higher in Energy

Imagine six ligands approaching along the +x, –x, +y, –y, +z, –z axes (octahedral geometry).

What happens to $d_{x^2-y^2}$ and $d_{z^2}$ ?

Their lobes point directly at the ligands.
The negatively charged ligands repel the electron density in these orbitals.
This repulsion raises the energy of these orbitals — they become less stable (higher energy).

What happens to $d_{xy}$ , $d_{xz}$ , $d_{yz}$ ?

Their lobes point between the axes (e.g., $d_{xy}$ lobes lie in the xy-plane but at 45° to x and y).
They avoid the ligands — less repulsion.
Their energy is lower than the $e_g$ set.

The Splitting Pattern

\Delta_{\text{oct}} = E(e_g) - E(t_{2g})

Where:

$E(e_g)$ = energy of $d_{x^2-y^2}$ and $d_{z^2}$ (higher)
$E(t_{2g})$ = energy of $d_{xy}$ , $d_{xz}$ , $d_{yz}$ (lower)
$\Delta_{\text{oct}}$ is called the crystal field splitting energy (CFSE)

Why the name? The $e_g$ orbitals are "doubly degenerate" (2 orbitals), $t_{2g}$ are "triply degenerate" (3 orbitals). The letters come from group theory symmetry labels.

3. The Energy Conservation Rule

The total energy of all five d-orbitals must remain constant (no energy is created or destroyed). So:

The center of gravity (average energy) of the split set equals the original degenerate energy.
For octahedral splitting:
- 2 $e_g$ orbitals go up by $+0.6\Delta_{\text{oct}}$ each
- 3 $t_{2g}$ orbitals go down by $-0.4\Delta_{\text{oct}}$ each

Check:

2 \times (+0.6\Delta) + 3 \times (-0.4\Delta) = 1.2\Delta - 1.2\Delta = 0

This conservation of energy is a fundamental constraint — the splitting is not arbitrary.

4. The Tetrahedral Case: Why It's Opposite and Smaller

In a tetrahedral complex, four ligands approach from alternate corners of a cube. The axes are different:

The $d_{xy}$ , $d_{xz}$ , $d_{yz}$ orbitals now point closer to the ligands (more repulsion).
The $d_{x^2-y^2}$ and $d_{z^2}$ orbitals point away from ligands (less repulsion).

Result:

$e$ set ( $d_{x^2-y^2}$ , $d_{z^2}$ ) — lower energy
$t_2$ set ( $d_{xy}$ , $d_{xz}$ , $d_{yz}$ ) — higher energy

The splitting is inverted compared to octahedral.

Magnitude:

\Delta_{\text{tet}} \approx \frac{4}{9} \Delta_{\text{oct}}

Why smaller?

Only 4 ligands (vs. 6) → less total repulsion.
Ligands are not directly along axes → weaker interaction. …

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The key is to identify which complexes are outer orbital (high-spin) and then count their unpaired electrons. $[MnCl_6]^{3-}$ (4 unpaired) and $[CoF_6]^{3-}$ (4 unpaired) have the same number, so the correct pair is (i) and (iii).

1. What does "outer orbital" mean?

In octahedral complexes, ligands split the d-orbitals into two sets: the lower-energy $t_{2g}$ and the higher-energy $e_g$ . The size of this splitting ( $\Delta_o$ ) depends on the ligand.

Strong field ligands (like $NH_3$ , $CN^-$ ) cause a large $\Delta_o$ — electrons pair up in the $t_{2g}$ before occupying $e_g$ . This gives low-spin (or inner orbital) complexes.
Weak field ligands (like $F^-$ , $Cl^-$ , $H_2O$ for most 3d metals) cause a small $\Delta_o$ — electrons occupy all five d-orbitals singly first (Hund's rule). This gives high-spin (or outer orbital) complexes.

The phrase "outer orbital" specifically means the complex uses the outer $nd$ orbitals (i.e., $3d$ for first-row transition metals) and follows high-spin configuration. So we must first check the ligand: $Cl^-$ and $F^-$ are weak field; $NH_3$ is strong field.

Watch out

$NH_3$ is a strong field ligand for most 3d metals (especially Co, Ni). So $[Ni(NH_3)_6]^{2+}$ will be low-spin, not outer orbital. Option (iv) is automatically disqualified.

2. Determine the oxidation state and d-electron count for each complex

We need the number of d-electrons on the central metal ion.

Complex	Metal	Oxidation state	d-electron count
$[MnCl_6]^{3-}$	Mn (Z=25)	+3 (since 6×(-1) = -6, overall -3 → Mn = +3)	$d^4$
$[FeF_6]^{3-}$	Fe (Z=26)	+3 (6×(-1) = -6, overall -3 → Fe = +3)	$d^5$
$[CoF_6]^{3-}$	Co (Z=27)	+3 (6×(-1) = -6, overall -3 → Co = +3)	$d^6$
$[Ni(NH_3)_6]^{2+}$	Ni (Z=28)	+2 (6×0 = 0, overall +2 → Ni = +2)	$d^8$

3. Apply the outer orbital (high-spin) configuration for each

For weak-field ligands, fill the d-orbitals according to Hund's rule: each orbital gets one electron before pairing.

$[MnCl_6]^{3-}$ — $d^4$ , high-spin:

$t_{2g}$ : ↑ ↑ ↑ (three electrons)

$e_g$ : ↑ (one electron)

Unpaired electrons = 4
$[FeF_6]^{3-}$ — $d^5$ , high-spin:

$t_{2g}$ : ↑ ↑ ↑

$e_g$ : ↑ ↑

Unpaired electrons = 5
$[CoF_6]^{3-}$ — $d^6$ , high-spin:

$t_{2g}$ : ↑ ↑ ↑

$e_g$ : ↑ ↑

Then the sixth electron must pair in $t_{2g}$ :

$t_{2g}$ : ↑↓ ↑ ↑

$e_g$ : ↑ ↑

Unpaired electrons = 4 …

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Common Mistakes in Crystal Field Splitting (Outer Orbital Complexes)

Mistake 1: Confusing "Outer Orbital" with "Inner Orbital"

The Error: Students often forget that outer orbital means high spin (weak field ligand, $sp^3d^2$ hybridization). They incorrectly assume all complexes are low spin.

How to Avoid:

Outer orbital = weak field ligand = high spin = maximum unpaired electrons
Inner orbital = strong field ligand = low spin = minimum unpaired electrons
For $[MnCl_6]^{3-}$ , $[FeF_6]^{3-}$ , $[CoF_6]^{3-}$ — all have weak field ligands ( $Cl^-$ , $F^-$ ), so they are high spin

Mistake 2: Wrong Oxidation State Calculation

The Error: Students misassign oxidation states, leading to wrong $d^n$ configurations.

How to Avoid:

For $[MnCl_6]^{3-}$ : $Mn + 6(-1) = -3 \Rightarrow Mn = +3$ → $d^4$ (Mn atomic no. 25, $Mn^{3+}$ = 22 electrons = $d^4$ )
For $[FeF_6]^{3-}$ : $Fe + 6(-1) = -3 \Rightarrow Fe = +3$ → $d^5$
For $[CoF_6]^{3-}$ : $Co + 6(-1) = -3 \Rightarrow Co = +3$ → $d^6$
For $[Ni(NH_3)_6]^{2+}$ : $Ni + 6(0) = +2 \Rightarrow Ni = +2$ → $d^8$

Mistake 3: Forgetting $NH_3$ is a Strong Field Ligand

The Error: Students treat $NH_3$ as weak field, making $[Ni(NH_3)_6]^{2+}$ high spin — but $NH_3$ is strong field for $3d$ metals.

How to Avoid:

$NH_3$ is a strong field ligand → causes low spin (inner orbital) for $d^6$ and below
For $d^8$ ( $Ni^{2+}$ ), strong field still gives 2 unpaired electrons (no pairing possible in $t_{2g}^6 e_g^2$ )

Mistake 4: Incorrect Electron Filling in High Spin vs Low Spin

The Error: Students fill $d$ orbitals without considering crystal field splitting energy ( $\Delta_o$ ) vs pairing energy ( $P$ ).

How to Avoid:

High spin (weak field): Fill all $t_{2g}$ singly first, then $e_g$ singly (Hund's rule)
Low spin (strong field): Pair in $t_{2g}$ before filling $e_g$

Complex	$d^n$	Ligand	Spin	$t_{2g}$	$e_g$	Unpaired e⁻
$[MnCl_6]^{3-}$	$d^4$	Weak	High	↑ ↑ ↑	↑	4

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Q.Atomic number of Mn, Fe, Co and Ni are 25, 26, 27 and 28 respectively. Which of the following outer orbital octahedral complexes have same number of unpaired electrons?

Concept understanding — Crystal Field Splitting

Crystal Field Splitting: From Intuition to Precision

The Precise Statement

Why This Matters

Why this formula?

Crystal Field Splitting: Why the Energy Splitting Occurs

1. The Starting Point: Degenerate d-Orbitals

2. The Octahedral Case: Why $e_g$ Orbitals Are Higher in Energy

What happens to $d_{x^2-y^2}$ and $d_{z^2}$ ?

What happens to $d_{xy}$ , $d_{xz}$ , $d_{yz}$ ?

The Splitting Pattern

3. The Energy Conservation Rule

4. The Tetrahedral Case: Why It's Opposite and Smaller

Result:

Magnitude:

1. What does "outer orbital" mean?

2. Determine the oxidation state and d-electron count for each complex

3. Apply the outer orbital (high-spin) configuration for each

Method: Crystal Field Theory (CFT) — Outer Orbital (High Spin) Complexes

Step 1: Identify the oxidation state and electron configuration of the central metal ion

Step 2: Determine the ligand field strength and spin state

Step 3: Fill electrons in the $t_{2g}$ and $e_g$ orbitals (high spin)

Common Mistakes in Crystal Field Splitting (Outer Orbital Complexes)

Mistake 1: Confusing "Outer Orbital" with "Inner Orbital"

Mistake 2: Wrong Oxidation State Calculation

Mistake 3: Forgetting $NH_3$ is a Strong Field Ligand

Mistake 4: Incorrect Electron Filling in High Spin vs Low Spin

Q.Atomic number of Mn, Fe, Co and Ni are 25, 26, 27 and 28 respectively. Which of the following outer orbital octahedral complexes have same number of unpaired electrons?

1. What does "outer orbital" mean?

2. Determine the oxidation state and d-electron count for each complex

3. Apply the outer orbital (high-spin) configuration for each

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