different between nth and `(n + 1) th` Bohr's radius of hydrogen atom is equal to `(n = 1) th` Bohr's radius. The value of `n` is

Options

A. 1
B. 2
C. 3
D. 4

Explanation

To solve the problem, we need to find the value of \( n \) such that the difference between the \( n \)th and \( (n+1) \)th Bohr's radius of the hydrogen atom is equal to the \( (n=1) \)th Bohr's radius.

### Step-by-Step Solution:

1. **Understanding Bohr's Radius**:
The Bohr radius for the hydrogen atom is given by the formula:
\[ r_n = n^2 \cdot r_1 \]
where \( r_1 \) is the Bohr radius for \( n=1 \) (approximately \( 0.529 \) Å).

2. **Finding the Difference Between Radii**:
The difference between the \( n \)th and \( (n+1) \)th radii can be expressed as:
\[ r_{n+1} - r_n = (n+1)^2 \cdot r_1 - n^2 \cdot r_1 \]
Simplifying this gives:
\[ r_{n+1} - r_n = r_1 \left( (n^2 + 2n + 1) - n^2 \right) = r_1 (2n + 1) \]

3. **Setting the Equation**:
According to the problem, this difference is equal to the \( (n=1) \)th Bohr radius:
\[ r_1 (2n + 1) = r_1 \]

4. **Dividing by \( r_1 \)**:
Since \( r_1 \) is non-zero, we can divide both sides by \( r_1 \):
\[ 2n + 1 = 1 \]

5. **Solving for \( n \)**:
Rearranging the equation gives:
\[ 2n = 1 - 1 = 0 \implies n = 0 \]

6. **Conclusion**:
The value of \( n \) that satisfies the condition is:
\[ n = 0 \]