The radius of the second Bohr orbit , in terms of the Bohr radius `(a_0)` of `Li^(2+)` is given as `(xa_0)/(y)` . Find the sum of x + y here ?

Answer

To solve the problem, we need to find the radius of the second Bohr orbit for the lithium ion \( \text{Li}^{2+} \) in terms of the Bohr radius \( a_0 \).

### Step-by-Step Solution:

1. **Understand the Formula for Bohr Radius**:
The radius of the \( n^{th} \) Bohr orbit for a hydrogen-like atom is given by the formula:
\[ r_n = \frac{n^2 a_0}{Z} \]
where:
- \( n \) is the principal quantum number (the orbit number),
- \( a_0 \) is the Bohr radius (the radius of the first orbit of hydrogen),
- \( Z \) is the atomic number of the atom.

2. **Identify the Atomic Number and Orbit Number**:
For \( \text{Li}^{2+} \):
- The atomic number \( Z \) is 3 (since lithium has 3 protons).
- We are interested in the second Bohr orbit, so \( n = 2 \).

3. **Substitute Values into the Formula**:
Now, substituting the values into the formula:
\[ r_2 = \frac{2^2 a_0}{3} \]
Simplifying this gives:
\[ r_2 = \frac{4 a_0}{3} \]

4. **Express in Terms of \( \frac{xa_0}{y} \)**:
We can express this result in the form \( \frac{xa_0}{y} \):
\[ r_2 = \frac{4 a_0}{3} \implies x = 4 \quad \text{and} \quad y = 3 \]

5. **Calculate the Sum \( x + y \)**:
Finally, we need to find the sum of \( x \) and \( y \):
\[ x + y = 4 + 3 = 7 \]

### Final Answer:
The sum of \( x + y \) is \( 7 \).