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

Answer

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 radius of the hydrogen atom is equal to the \( (n - 1) \)th Bohr radius.

### Step-by-step Solution:

1. **Understanding the Bohr Radius Formula**:
The radius of the \( n \)th orbit of a hydrogen atom is given by:
\[ R_n = \frac{n^2}{Z} \cdot a_0 \]
where \( a_0 \) is the Bohr radius constant and \( Z \) is the atomic number. For hydrogen, \( Z = 1 \), so:
\[ R_n = n^2 \cdot a_0 \]

2. **Setting Up the Equation**:
We need to find the difference between the \( (n + 1) \)th and \( n \)th radii:
\[ R_{n+1} - R_n = R_{n-1} \]
Substituting the formula for the radii:
\[ (n + 1)^2 \cdot a_0 - n^2 \cdot a_0 = (n - 1)^2 \cdot a_0 \]

3. **Simplifying the Equation**:
Factor out \( a_0 \) from the equation:
\[ a_0 \left( (n + 1)^2 - n^2 \right) = a_0 \left( (n - 1)^2 \right) \]
Since \( a_0 \) is a constant and not zero, we can divide both sides by \( a_0 \):
\[ (n + 1)^2 - n^2 = (n - 1)^2 \]

4. **Expanding the Squares**:
Expanding both sides:
\[ (n^2 + 2n + 1) - n^2 = (n^2 - 2n + 1) \]
This simplifies to:
\[ 2n + 1 = n^2 - 2n + 1 \]

5. **Rearranging the Equation**:
Move all terms to one side:
\[ 2n + 1 - 1 = n^2 - 2n \]
Simplifying gives:
\[ 4n = n^2 \]

6. **Rearranging to Solve for \( n \)**:
Rearranging the equation:
\[ n^2 - 4n = 0 \]
Factoring out \( n \):
\[ n(n - 4) = 0 \]

7. **Finding the Possible Values for \( n \)**:
The solutions to this equation are:
\[ n = 0 \quad \text{or} \quad n = 4 \]
Since \( n \) must be a positive integer representing the orbit number, we have:
\[ n = 4 \]

### Final Answer:
Thus, the value of \( n \) is \( 4 \).