To determine if an ion has a noble gas configuration, we examine its electron configuration and compare it with that of a nearby noble gas:
- Sr²⁺ (\(Z = 38\)) loses two electrons, resulting in the electron configuration \([Kr]\), which matches the noble gas krypton.
- Cs⁺ (\(Z = 55\)) loses one electron, resulting in the electron configuration \([Xe]\), matching xenon.
- La³⁺ (\(Z = 57\)) loses three electrons, resulting in the electron configuration \([Xe]\), also matching xenon.
- Yb²⁺ (\(Z = 70\)) loses two electrons, resulting in the electron configuration \([Xe]\), matching xenon.
On the other hand:
- Pb²⁺ does not match any noble gas configuration due to its partially filled \(d\)-orbitals.
- Fe²⁺ does not match a noble gas configuration either, as it retains electrons in the \(d\)-orbital.
Thus, only *Sr²⁺, Cs⁺, La³⁺, and Yb²⁺ have noble gas configurations, totaling four ions.
The Correct answer is: 2
Let $ P_n = \alpha^n + \beta^n $, $ n \in \mathbb{N} $. If $ P_{10} = 123,\ P_9 = 76,\ P_8 = 47 $ and $ P_1 = 1 $, then the quadratic equation having roots $ \alpha $ and $ \frac{1}{\beta} $ is: