Identify the ion having 4f\(^6\) electronic configuration.
The electronic configuration of Gd (Gadolinium) is $[\text{Xe}]\,4f^7\,5d^1\,6s^2$. When Gd loses 3 electrons (one from the 6s orbital and two from the 5d and 6s orbitals), it becomes Gd$^{3+}$ with the electronic configuration $[\text{Xe}]\,4f^7$. This configuration does not match $4f^6$.
The electronic configuration of Sm (Samarium) is $[\text{Xe}]\,4f^6\,6s^2$. When Sm loses 2 electrons from the 6s orbital, it becomes Sm$^{2+}$ with the configuration $[\text{Xe}]\,4f^6$. This matches the given $4f^6$ configuration.
The configuration of Sm$^{3+}$ is $[\text{Xe}]\,4f^5$, and that of Tb$^{3+}$ (Terbium) is $[\text{Xe}]\,4f^8$. Neither of these matches the required configuration.
Therefore, the ion with a $4f^6$ configuration is Sm$^{2+}$.
Correct Answer:
Option 3: Sm2+
Explanation:
1. Gd3+ (Gadolinium(III) ion):
Gd: [Xe] 4f7 5d1 6s2
Gd3+: [Xe] 4f7
2. Sm3+ (Samarium(III) ion):
Sm: [Xe] 4f6 6s2
Sm3+: [Xe] 4f5
3. Sm2+ (Samarium(II) ion):
Sm: [Xe] 4f6 6s2
Sm2+: [Xe] 4f6
4. Tb3+ (Terbium(III) ion):
Tb: [Xe] 4f9 6s2
Tb3+: [Xe] 4f8
Therefore, Sm2+ has the 4f6 electronic configuration.
A quantity \( X \) is given by: \[ X = \frac{\epsilon_0 L \Delta V}{\Delta t} \] where:
- \( \epsilon_0 \) is the permittivity of free space,
- \( L \) is the length,
- \( \Delta V \) is the potential difference,
- \( \Delta t \) is the time interval.
The dimension of \( X \) is the same as that of: