In H-atom if `r1` is the radius fo first Bohr orbit is x then de-Broglie wavelength of an elecrton in ` 3^(rd)` orbit is :

Options

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

Explanation

To find the de Broglie wavelength of an electron in the third Bohr orbit of a hydrogen atom, given that the radius of the first Bohr orbit (r1) is x, we can follow these steps:

### Step 1: Determine the relationship between the radii of the Bohr orbits
The radius of the nth Bohr orbit (rn) is given by the formula:
\[ r_n \propto n^2 \]
This means that the radius of the nth orbit is proportional to the square of the principal quantum number (n).

### Step 2: Establish the ratio of the radii
For the first orbit (n=1) and the third orbit (n=3), we can write:
\[ \frac{r_1}{r_3} = \frac{1^2}{3^2} = \frac{1}{9} \]

### Step 3: Express r3 in terms of r1
Since we know that \( r_1 = x \), we can express \( r_3 \) as:
\[ r_3 = 9 \cdot r_1 = 9x \]

### Step 4: Use the de Broglie wavelength formula
The de Broglie wavelength (\( \lambda \)) is related to the radius and the principal quantum number by the formula:
\[ 2\pi r_n = n \lambda \]
For the third orbit (n=3), we can substitute \( r_3 \):
\[ 2\pi r_3 = 3\lambda \]

### Step 5: Substitute r3 into the equation
Now substituting \( r_3 = 9x \) into the equation:
\[ 2\pi (9x) = 3\lambda \]

### Step 6: Solve for the de Broglie wavelength (\( \lambda \))
Now, we can solve for \( \lambda \):
\[ 18\pi x = 3\lambda \]
\[ \lambda = \frac{18\pi x}{3} \]
\[ \lambda = 6\pi x \]

### Final Answer
The de Broglie wavelength of an electron in the third Bohr orbit is:
\[ \lambda = 6\pi x \]

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