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 \( \text{Li}^{2+} \) and express it 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 (orbit number), \( a_0 \) is the Bohr radius, and \( Z \) is the atomic number of the element.

2. **Identify the Atomic Number**:
For lithium \( \text{Li} \), the atomic number \( Z \) is 3.

3. **Determine the Orbit Number**:
We are interested in the second Bohr orbit, so \( n = 2 \).

4. **Substitute Values into the Formula**:
Now we can substitute \( n \) and \( Z \) into the formula:
\[ R_2 = \frac{2^2 a_0}{3} = \frac{4 a_0}{3} \]

5. **Express in the Given Form**:
The problem states that the radius can be expressed as \( \frac{x a_0}{y} \). From our calculation:
\[ R_2 = \frac{4 a_0}{3} \]
We can see that \( x = 4 \) and \( y = 3 \).

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

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

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