If the radius of first Bohr's orbit is x,then de-broglie wavelength of electron in 3rd orbit is nearly

Options

A. `2pi x`
B. `6pi x`
C. `9x`
D. `x//3`

Explanation

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

### Step 1: Determine the radius of the third Bohr orbit
The radius of the nth Bohr orbit is given by the formula:
\[ r_n = n^2 \cdot r_1 \]
where \( r_1 \) is the radius of the first orbit. Here, we have:
- \( n = 3 \) (for the third orbit)
- \( r_1 = x \)

Substituting these values into the formula:
\[ r_3 = 3^2 \cdot x = 9x \]

### Step 2: Use the de Broglie wavelength formula
The de Broglie wavelength \( \lambda \) is related to the radius and the principal quantum number \( n \) by the formula:
\[ 2\pi r = n\lambda \]
We can rearrange this to find \( \lambda \):
\[ \lambda = \frac{2\pi r}{n} \]

### Step 3: Substitute the values for the third orbit
Now we substitute \( r = r_3 = 9x \) and \( n = 3 \) into the wavelength formula:
\[ \lambda = \frac{2\pi (9x)}{3} \]

### Step 4: Simplify the expression
Calculating this gives:
\[ \lambda = \frac{18\pi x}{3} = 6\pi x \]

### Conclusion
Thus, the de Broglie wavelength of the electron in the third orbit is:
\[ \lambda \approx 6\pi x \]