if the radius of the first Bohr orbit is x, then find the de Broglie wavelength of electron in third orbit.

Options

A. (a)`2pix`
B. (b)`6pix`
C. (c)`9x`
D. (d)`x/3`

Answer

To find the de Broglie wavelength of the electron in the third Bohr orbit given that the radius of the first Bohr orbit is \( x \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between the radius of the Bohr orbits**:
The radius of the \( n \)-th Bohr orbit is given by the formula:
\[ r_n \propto n^2 \]
This means that the radius of the \( n \)-th orbit is proportional to the square of the principal quantum number \( n \).

2. **Set up the relationship for the first and third orbits**:
Let \( r_1 \) be the radius of the first orbit and \( r_3 \) be the radius of the third orbit. We know:
\[ \frac{r_1}{r_3} = \frac{1^2}{3^2} = \frac{1}{9} \]
Given that \( r_1 = x \), we can express \( r_3 \) in terms of \( x \):
\[ \frac{x}{r_3} = \frac{1}{9} \]
Rearranging gives:
\[ r_3 = 9x \]

3. **Use the de Broglie wavelength formula**:
The de Broglie wavelength \( \lambda \) of an electron in the \( n \)-th orbit is given by:
\[ \lambda = \frac{2 \pi r_n}{n} \]
For the third orbit (\( n = 3 \)):
\[ \lambda = \frac{2 \pi r_3}{3} \]

4. **Substitute \( r_3 \) into the wavelength formula**:
Now substitute \( r_3 = 9x \) into the wavelength formula:
\[ \lambda = \frac{2 \pi (9x)}{3} \]
Simplifying this gives:
\[ \lambda = \frac{18 \pi x}{3} = 6 \pi x \]

5. **Final result**:
Therefore, the de Broglie wavelength of the electron in the third orbit is:
\[ \lambda = 6 \pi x \]