CBSE · Class 12 · Chemistry · Coordination CompoundsCalculate the crystal field stabilization energy (CFSE) for a $d^4$ octahedral coordination complex in terms of $\Deltao$ and pairing energy $P$ under both weak field and strong field ligand environments. Explain how ligand field strength dictates this energy.

Crystal Field Stabilization Energy (CFSE) Calculation for a $d^4$ Octahedral Complex

\nIn an octahedral crystal field, the degeneracy of the five $d$ orbitals is lifted, splitting them into two sets:

1. $t_{2g}$ set: Consisting of $d_{xy}, d_{yz}, d_{zx}$ orbitals, having lower energy by $-0.4\Delta_o$ relative to the barycenter.
2. $e_g$ set: Consisting of $d_{x^2-y^2}, d_{z^2}$ orbitals, having higher energy by $+0.6\Delta_o$ relative to the barycenter. \nThe distribution of $d$ electrons between $t_{2g}$ and $e_g$ orbitals depends on the relative magnitudes of the crystal field splitting energy ($\Delta_o$) and the pairing energy ($P$).

Case 1: Weak Field Ligand Environment ($\Delta_o < P$)\nWhen the ligand is weak, the energy required to pair electrons in the same orbital ($P$) is greater than the energy gap between $t_{2g}$ and $e_g$ sets ($\Delta_o$). Consequently, the fourth electron enters one of the higher-energy $e_g$ orbitals rather than pairing up in the $t_{2g}$ level.

• Electronic Configuration: $t_{2g}^3 e_g^1$
• CFSE Formula: $$\text{CFSE} = [-0.4 \times n(t_{2g}) + 0.6 \times n(e_g)]\Delta_o + mP$$ where $n(t_{2g})$ and $n(e_g)$ are the number of electrons in $t_{2g}$ and $e_g$ orbitals respectively, and $m$ is the number of electron pairs formed.
• Step-by-step Calculation:
  • Number of electrons in $t_{2g}$ ($n_{t2g}$) = $3$
  • Number of electrons in $e_g$ ($n_{eg}$) = $1$
  • Number of electron pairs formed ($m$) = $0$ (since all four electrons are currently unpaired relative to the ground state splitting, meaning no additional intra-orbital pairings beyond the spherically symmetric state occurred).
  • Substituting values: $$\text{CFSE} = [(-0.4 \times 3) + (0.6 \times 1)]\Delta_o + 0 \times P$$ $$\text{CFSE} = [-1.2 + 0.6]\Delta_o = -0.6\Delta_o$$

Case 2: Strong Field Ligand Environment ($\Delta_o > P$)\nWhen the ligand is strong, the splitting energy ($\Delta_o$) is greater than the pairing energy ($P$). It is energetically more favorable for the fourth electron to pair up in one of the lower-energy $t_{2g}$ orbitals rather than occupying the higher-energy $e_g$ orbital.

• Electronic Configuration: $t_{2g}^4 e_g^0$
• Step-by-step Calculation:
  • Number of electrons in $t_{2g}$ ($n_{t2g}$) = $4$
  • Number of electrons in $e_g$ ($n_{eg}$) = $0$
  • Number of electron pairs formed ($m$) = $1$ (one extra pair is formed in the $t_{2g}$ set compared to the spherically symmetrical uncomplexed metal ion state).
  • Substituting values into the CFSE formula: $$\text{CFSE} = [(-0.4 \times 4) + (0.6 \times 0)]\Delta_o + 1 \times P$$ $$\text{CFSE} = [-1.6]\Delta_o + P = -1.6\Delta_o + P$$

Summary of Results:

• Weak field $d^4$: $\text{CFSE} = -0.6\Delta_o$
• Strong field $d^4$: $\text{CFSE} = -1.6\Delta_o + P$