On the basis of crystal field theory explain why Co(III) forms paramagnetic octahedral complex with weak field

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Q. On the basis of crystal field theory explain why Co(III) forms paramagnetic octahedral complex with weak field ligands whereas it forms diamagnetic octahedral complex with strong field ligands.

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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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Crystal field theory explains that the splitting of $d$ -orbitals in an octahedral field ( $\Delta_o$ ) relative to the pairing energy ( $P$ ) determines the electron configuration. For Co(III), weak field ligands give a high-spin, paramagnetic $t_{2g}^4 e_g^2$ configuration, while strong field ligands give a low-spin, diamagnetic $t_{2g}^6 e_g^0$ configuration.

The Core Idea: Crystal Field Splitting and Electron Configuration

Crystal field theory (CFT) is a model that explains the electronic structure of transition metal complexes. In an octahedral complex, the five $d$ -orbitals are no longer degenerate. The $d_{x^2-y^2}$ and $d_{z^2}$ orbitals (the $e_g$ set) point directly at the ligands and experience strong repulsion, raising their energy. The $d_{xy}$ , $d_{xz}$ , and $d_{yz}$ orbitals (the $t_{2g}$ set) point between the ligands and experience less repulsion, lowering their energy. The energy gap between these two sets is called the crystal field splitting energy, denoted by $\Delta_o$ (or $10 Dq$ ).

The key to understanding the magnetic behaviour of a $d^6$ ion like Co(III) lies in a simple competition: the energy cost of pairing electrons ( $P$ ) versus the energy gain from occupying the lower-energy $t_{2g}$ orbitals ( $\Delta_o$ ).

Identify the metal ion and its $d$ -electron count. Cobalt in the +3 oxidation state, Co(III), has an electronic configuration of $[Ar] 3d^6$ . This means we have six electrons to place in the $d$ -orbitals of the octahedral complex.
Understand the two competing factors. When placing electrons into the split $d$ -orbitals, two rules apply:
- Hund's rule: Electrons prefer to occupy different orbitals with parallel spins to minimise electron-electron repulsion.
- Aufbau principle: Electrons will first fill the lower-energy $t_{2g}$ orbitals. The conflict arises because placing an electron in a higher-energy $e_g$ orbital (following Hund's rule) costs energy $\Delta_o$ , while pairing two electrons in the same $t_{2g}$ orbital costs the pairing energy, $P$ .
The decisive factor: $\Delta_o$ vs. $P$ . The actual configuration adopted is the one that minimises the total energy of the system.
- If $\Delta_o < P$ (weak field): The energy cost of promoting an electron to the $e_g$ level is less than the cost of pairing. The system will maximise the number of unpaired electrons.
- If $\Delta_o > P$ (strong field): The energy cost of promoting an electron is greater than the cost of pairing. The system will minimise the number of unpaired electrons by pairing them in the $t_{2g}$ orbitals.

Watch out

A common mistake is to think that the $t_{2g}$ set can hold a maximum of 6 electrons. This is true, but the order in which they are filled depends entirely on the ligand field strength. Don't just fill the $t_{2g}$ set first; always check if it's energetically favourable to promote an electron to $e_g$ instead.

Apply to the weak field case (paramagnetic). With a weak field ligand (e.g., $\text{H}_2\text{O}$ , $\text{F}^-$ ), $\Delta_o$ is small. The first three electrons go into the three $t_{2g}$ orbitals with parallel spins ( $t_{2g}^3$ ). The fourth electron has a choice: pair in a $t_{2g}$ orbital (cost $P$ ) or go into an $e_g$ orbital (cost $\Delta_o$ ). Since $\Delta_o < P$ , it is cheaper to promote the electron. So, the fourth electron goes into an $e_g$ orbital. The fifth and sixth electrons also follow Hund's rule, each occupying a separate orbital before pairing occurs. The final configuration is $t_{2g}^3 e_g^3$ . However, this is not the most stable arrangement. A more accurate filling is $t_{2g}^4 e_g^2$ , where the fourth electron pairs in the $t_{2g}$ set, and the fifth and sixth go to the $e_g$ set. This gives four unpaired electrons (two in $t_{2g}$ and two in $e_g$ ). A substance with unpaired electrons is paramagnetic. …

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Here is a breakdown of the common mistakes students make on this exact Crystal Field Theory (CFT) question, along with how to avoid them.

The Core Concept (The "Why")

Before listing mistakes, let's lock in the correct logic. This question tests your understanding of pairing energy (P) vs. crystal field splitting energy ( $\Delta_o$ ) .

Identify the metal ion: Co(III) is $d^6$ (Cobalt atomic number 27, Co $^{3+}$ = 27 - 3 = 24 electrons, configuration $[Ar]3d^6$ ).
Weak field ligands (e.g., $\text{F}^-, \text{H}_2\text{O}$ ): $\Delta_o$ is small. $\Delta_o < P$ . Electrons follow Hund's rule (maximum multiplicity). They occupy all five d-orbitals singly before pairing. This gives 4 unpaired electrons → Paramagnetic.
Strong field ligands (e.g., $\text{CN}^-, \text{NH}_3$ ): $\Delta_o$ is large. $\Delta_o > P$ . Electrons pair up in the lower energy $t_{2g}$ orbitals first. This gives 0 unpaired electrons → Diamagnetic.

Common Mistake #1: Confusing the Electron Count ( $d^6$ vs. $d^5$ )

The Error: Students often treat Co(III) as $d^5$ or $d^7$ , or forget that Co is in the +3 oxidation state. They might draw the diagram for Fe(III) or Co(II) instead.
Why it happens: Rushing the electronic configuration. Co (atomic number 27) is tricky because it's near Fe (26) and Ni (28).
How to Avoid:
- Write it down step-by-step:
  1. Co (atom) = $[Ar] 4s^2 3d^7$
  2. Co $^{3+}$ = Remove 3 electrons (2 from 4s, 1 from 3d) = $[Ar] 3d^6$
- Memorize the common $d^n$ configurations for 3d series: Cr $^{3+}$ ( $d^3$ ), Mn $^{2+}$ ( $d^5$ ), Fe $^{3+}$ ( $d^5$ ), Co $^{3+}$ ( $d^6$ ), Ni $^{2+}$ ( $d^8$ ), Cu $^{2+}$ ( $d^9$ ).

Common Mistake #2: Forgetting the "Pairing Energy" Condition

The Error: Students simply state "weak field = high spin, strong field = low spin" without mentioning the comparison between $\Delta_o$ and P. They might say "strong field ligands cause pairing" but don't explain why.
Why it happens: Memorizing the result without the reasoning. The exam specifically asks "on the basis of crystal field theory".
How to Avoid:
- Always state the rule: "The electronic configuration depends on whether the crystal field splitting energy ( $\Delta_o$ ) is greater than or less than the pairing energy (P) ."
- Write the inequality:
  - Weak field: $\Delta_o < P$ → High spin.
  - Strong field: $\Delta_o > P$ → Low spin.

Common Mistake #3: Drawing the $t_{2g}$ and $e_g$ Orbitals Incorrectly

The Error: Drawing the wrong number of electrons in the orbitals, or placing electrons in the $e_g$ set before filling the $t_{2g}$ set in the weak field case.
Why it happens: Misunderstanding the order of filling. In an octahedral field, the $t_{2g}$ set ( $d_{xy}, d_{xz}, d_{yz}$ ) is always lower in energy than the $e_g$ set ( $d_{x^2-y^2}, d_{z^2}$ ).
How to Avoid:
- Draw the energy level diagram carefully.
- For $d^6$ weak field:
  - Fill $t_{2g}$ with 3 electrons (one each, all parallel spins).
  - Next electron goes to $e_g$ (Hund's rule).
  - Next two electrons go to $t_{2g}$ and $e_g$ (one each, parallel spins).
  - Result: $t_{2g}^3 e_g^3$ (4 unpaired electrons).
- For $d^6$ strong field:
  - Fill $t_{2g}$ with 3 electrons (one each, parallel).
  - Next 3 electrons pair up in $t_{2g}$ .
  - Result: $t_{2g}^6 e_g^0$ (0 unpaired electrons).

Common Mistake #4: Confusing "Paramagnetic" and "Diamagnetic" …

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Q.On the basis of crystal field theory explain why Co(III) forms paramagnetic octahedral complex with weak field ligands whereas it forms diamagnetic octahedral complex with strong field ligands.

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:

The Core Idea: Crystal Field Splitting and Electron Configuration

Crystal Field Splitting — Method: Crystal Field Theory (CFT) for Octahedral Complexes

Why This Happens — The Core Idea

Step-by-Step Reasoning

Step 1: Identify the metal ion and its electron configuration

Step 2: Recall the octahedral splitting diagram

Step 3: Apply Hund's rule and pairing energy ( $P$ )

Case 1: Weak Field Ligands (e.g., $F^-$ , $H_2O$ )

Case 2: Strong Field Ligands (e.g., $CN^-$ , $CO$ )

The Core Concept (The "Why")

Common Mistake #1: Confusing the Electron Count ( $d^6$ vs. $d^5$ )

Common Mistake #2: Forgetting the "Pairing Energy" Condition

Common Mistake #3: Drawing the $t_{2g}$ and $e_g$ Orbitals Incorrectly

Common Mistake #4: Confusing "Paramagnetic" and "Diamagnetic" …

Q.On the basis of crystal field theory explain why Co(III) forms paramagnetic octahedral complex with weak field ligands whereas it forms diamagnetic octahedral complex with strong field ligands.

The Core Idea: Crystal Field Splitting and Electron Configuration

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