Question

Assuming the validity of Bohr's atomic model for hydrogen-like ions, the radius of $ \text{Li}^{2+} $ ion in its ground state is given by $ \frac{1}{X} a_0 $, where $ a_0 $ is the first Bohr's radius.

• A. 2
• B. 1
• C. 3
• D. 9

Hint:

For hydrogen-like ions, the radius decreases as the atomic number increases, following the formula \( r = r_0 \frac{n^2}{z} \).

Answer 1

The Correct Option is C

The radius for a hydrogen-like ion is given by the formula: \[ r = r_0 \frac{n^2}{z} \] where \( r_0 \) is the radius for hydrogen, \( n \) is the principal quantum number, and \( z \) is the atomic number. For \( \text{Li}^{2+} \), we have \( n = 1 \) and \( z = 3 \), so the radius is: \[ r = r_0 \frac{1^2}{3} = \frac{r_0}{3} \] Thus, \( X = 3 \).

Answer 2

We are asked to find the radius of a \( \text{Li}^{2+} \) ion (hydrogen-like ion with atomic number \( Z=3 \)) in its ground state in terms of the Bohr radius \( a_0 \).

Concept Used:

According to Bohr’s atomic model, the radius of the \( n^{\text{th}} \) orbit for a hydrogen-like ion is given by:

\[ r_n = \frac{n^2 a_0}{Z}. \]

Here, \( a_0 = 0.529 \, \text{Å} \) is the Bohr radius for hydrogen (\( Z=1 \)).

Step-by-Step Solution:

Step 1: For \( \text{Li}^{2+} \), the atomic number is \( Z = 3 \).

Step 2: For the ground state, \( n = 1 \).

Step 3: Substitute into the formula:

\[ r_1 = \frac{1^2 \, a_0}{3} = \frac{a_0}{3}. \]

Final Computation & Result

\[ r = \frac{a_0}{3} = \frac{1}{X}a_0. \] \[ \therefore X = 3. \]

Answer: \( X = 3 \)