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 key idea is Crystal Field Splitting: in an octahedral field, the $d$ orbitals split into lower-energy $t_{2g}$ and higher-energy $e_g$ sets. The magnitude of the splitting ($\Delta_o$) depends on the ligand — weak field ligands give a small $\Delta_o$, strong field ligands give a large $\Delta_o$.

For $\text{Co}^{3+}$, the electronic configuration is $3d^6$.

1. Weak field (small $\Delta_o$): The pairing energy ($P$) is larger than $\Delta_o$. Electrons occupy all five $d$ orbitals singly first (Hund's rule) before pairing. This gives the configuration $t_{2g}^4 e_g^2$ — four unpaired electrons. …

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 ($\Delta_o$).

1. 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.

2. 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, $P$.

3. 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.

4. 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. …

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

Why This Happens — The Core Idea

The difference arises because weak field ligands cause a small crystal field splitting ($\Delta_o$), while strong field ligands cause a large splitting. This determines whether electrons pair up or remain unpaired in the $t_{2g}$ and $e_g$ orbitals.

Step-by-Step Reasoning

Step 1: Identify the metal ion and its electron configuration

• Co(III) means Co$^{3+}$ (atomic number 27, loses 3 electrons).
• Electronic configuration of Co: $[Ar]\,3d^7\,4s^2$
• For Co$^{3+}$: remove 3 electrons → $[Ar]\,3d^6$

So, Co$^{3+}$ has 6 d-electrons.

Step 2: Recall the octahedral splitting diagram

• In an octahedral field, the 5 d-orbitals split into:
  • Lower energy: $t_{2g}$ (3 orbitals: $d_{xy}, d_{xz}, d_{yz}$)
  • Higher energy: $e_g$ (2 orbitals: $d_{x^2-y^2}, d_{z^2}$)
• The energy gap between them is $\Delta_o$ (or $10\,Dq$).

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

• Pairing energy ($P$) is the energy cost to put two electrons in the same orbital.
• The filling pattern depends on whether $\Delta_o > P$ or $\Delta_o < P$.

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

• $\Delta_o$ is small → $\Delta_o < P$
• Electrons prefer to occupy all five orbitals singly before pairing (Hund's rule).

Filling for 6 electrons:

1. Fill $t_{2g}$: 3 electrons (one each) → 3 unpaired
2. Next 3 electrons go into $e_g$ (one each) → 3 more unpaired

Result:

• Configuration: $t_{2g}^3\,e_g^3$
• 4 unpaired electrons → Paramagnetic

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

• $\Delta_o$ is large → $\Delta_o > P$ …

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 ($\Delta_o$) .

1. Identify the metal ion: Co(III) is $d^6$ (Cobalt atomic number 27, Co$^{3+}$ = 27 - 3 = 24 electrons, configuration $[Ar]3d^6$).
2. 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.
3. 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" …