Name the member of the lanthanide series which is well known to exhibit a \( +4 \) oxidation state.
Step 1: Understanding the Lanthanide Series
The lanthanides (or rare earth elements) consist of 15 elements from lanthanum (La) to lutetium (Lu) in the periodic table.
Step 2: Oxidation States of Lanthanides
- The common oxidation state of lanthanides is \( +3 \).
- However, some lanthanides can exhibit other oxidation states, such as:
- \( +2 \) (e.g., Eu, Yb, Sm)
- \( +4 \) (e.g., Ce, Tb)
Step 3: Why Cerium Exhibits \( +4 \) Oxidation State
- Electronic configuration of cerium: \[ [{Xe}] 4f^1 5d^1 6s^2 \] - In the \( +4 \) oxidation state, cerium loses all its valence electrons: \[ [{Xe}] \] - This stable xenon-like configuration makes Ce\(^4+\) stable.
Step 4: Conclusion
Among lanthanides, cerium (Ce) is the most common element to exhibit a \( +4 \) oxidation state.
If the value of \( \cos \alpha \) is \( \frac{\sqrt{3}}{2} \), then \( A + A = I \), where \[ A = \begin{bmatrix} \sin\alpha & -\cos\alpha \\ \cos\alpha & \sin\alpha \end{bmatrix}. \]