Let's go step by step through your question about the **Jahn-Teller theorem** and its application to specific transition metal complexes. --- # (a) State the Jahn–Teller Theorem **Jahn-Teller Theorem:** > "Any non-linear molecule in a degenerate electronic state will undergo a geometrical distortion that removes this degeneracy and lowers the overall energy of the molecule." **Explanation:** If a molecule/ion has an uneven distribution of electrons in degenerate (same energy) orbitals, it will distort (change shape) to remove the degeneracy and become more stable. --- # (b) Apply the Jahn–Teller Theorem to Rank the Complex Ions ## Step 1: Write the Electronic Configurations Let's consider the electron configuration of each metal ion in an **octahedral field**: ### (i) \([Cu(OH_2)_6]^{2+}\) - **Cu(II):** Atomic number = 29 → Cu²⁺ = [Ar] 3d⁹ - In octahedral field: \( t_{2g}^6 e_g^3 \) ### (ii) \([Cr(NH_3)_6]^{3+}\) - **Cr(III):** Atomic number = 24 → Cr³⁺ = [Ar] 3d³ - In octahedral field: \( t_{2g}^3 e_g^ \) ### (iii) \([Re(CO)_6]^3\) - **Re(III):** Atomic number = 75 → Re³⁺ = [Xe] 5d⁴ - In octahedral field: \( t_{2g}^4 e_g^ \) --- ## Step 2: Predict Jahn–Teller Distortion **Jahn-Teller distortion** is most significant when the \( e_g \) or \( t_{2g} \) levels are unevenly filled (degenerate and unsymmetrical). ### Analyze Each Complex #### (i) \([Cu(OH_2)_6]^{2+}\): \( d^9 \) - **Electronic configuration:** \( t_{2g}^6 e_g^3 \) - **Degeneracy:** The \( e_g \) set is unevenly filled (3 electrons). - **Jahn-Teller Effect:** **Strong**, as the unevenly filled \( e_g \) orbitals (especially for d⁹) cause significant distortion. #### (ii) \([Cr(NH_3)_6]^{3+}\): \( d^3 \) - **Electronic configuration:** \( t_{2g}^3 e_g^ \) - **Degeneracy:** \( t_{2g} \) is half-filled (symmetrical), \( e_g \) is empty. - **Jahn-Teller Effect:** **Negligible/none**, as there is no degenerate occupancy in \( e_g \) orbitals. #### (iii) \([Re(CO)_6]^3\): \( d^4 \) - **Electronic configuration:** \( t_{2g}^4 e_g^ \) - **Degeneracy:** For strong field ligands like CO, electrons pair in \( t_{2g} \), so configuration is \( t_{2g}^4 \) (low-spin, paired). - **Jahn-Teller Effect:** **Weak to moderate**. In a strong field (low spin), the \( t_{2g} \) orbitals are unevenly filled. However, the effect is less pronounced than for \( e_g \) degeneracy. --- ## Step 3: Rank by Degree of Jahn-Teller Distortion 1. **[Cu(OH₂)₆]²⁺** — **Strongest Jahn-Teller effect** (d⁹, uneven \( e_g \)) 2. **[Re(CO)₆]³⁺** — **Moderate/Weak Jahn-Teller effect** (low-spin d⁴, uneven \( t_{2g} \)) 3. **[Cr(NH₃)₆]³⁺** — **No significant Jahn-Teller effect** (d³, symmetrical \( t_{2g} \)) --- ## Step 4: Type of Distortion Expected - For \( d^9 \) ([Cu(OH₂)₆]²⁺): **Elongation** along one axis (usually z-axis), leading to tetragonal distortion. - For \( d^4 \) low spin ([Re(CO)₆]³⁺): **Weak distortion**, possibly a slight elongation or compression. - For \( d^3 \) ([Cr(NH₃)₆]³⁺): **No significant distortion**; remains nearly octahedral. --- ## Step 5: Diagram Illustration ### Typical Jahn-Teller Distortion (for d⁹) ```plaintext Before distortion (octahedral): O | O----Cu----O | O After distortion (elongated along z-axis): O | O----Cu----O | O (longer Cu–O bonds along z) ``` --- # **Final Summary Table** | Complex | d-electrons | Jahn-Teller Effect | Type of Distortion | |--------------------------|-------------|--------------------|--------------------------| | [Cu(OH₂)₆]²⁺ | d⁹ | Strong | Tetragonal elongation | | [Cr(NH₃)₆]³⁺ | d³ | None/Negligible | None (octahedral) | | [Re(CO)₆]³⁺ | d⁴ (low spin)| Weak/Moderate | Slight distortion | --- ## **Summary** 1. **Jahn-Teller theorem**: Nonlinear molecules with degenerate electronic states will distort to remove the degeneracy and lower the energy. 2. **Ranking Jahn-Teller distortion**: - [Cu(OH₂)₆]²⁺ (**strongest**) - [Re(CO)₆]³⁺ (**moderate/weak**) - [Cr(NH₃)₆]³⁺ (**none**) 3. **Distortion type**: Mainly tetragonal elongation for d⁹ case. Let me know if you want the diagrams sketched or have more questions!