what is the ratio of de-Broglie wavelength of electron in the second and third Bohr orbits in the hydrogen atoms?

Options

A. `2//3`
B. `3//2`
C. `4//3`
D. `3//4`

Explanation

To find the ratio of the de Broglie wavelength of an electron in the second and third Bohr orbits of a hydrogen atom, we can follow these steps:

### Step 1: Understand the de Broglie wavelength relationship
The de Broglie wavelength (\( \lambda \)) of an electron in a particular orbit is related to the circumference of that orbit. The relationship is given by:
\[ 2 \pi r_n = n \lambda_n \]
where \( r_n \) is the radius of the nth orbit, \( n \) is the principal quantum number, and \( \lambda_n \) is the de Broglie wavelength in that orbit.

### Step 2: Write the equations for the second and third orbits
For the second orbit (\( n = 2 \)):
\[ 2 \pi r_2 = 2 \lambda_2 \]
For the third orbit (\( n = 3 \)):
\[ 2 \pi r_3 = 3 \lambda_3 \]

### Step 3: Express the wavelengths in terms of the radii
From the equations above, we can express the wavelengths as:
\[ \lambda_2 = \frac{2 \pi r_2}{2} = \pi r_2 \]
\[ \lambda_3 = \frac{2 \pi r_3}{3} \]

### Step 4: Find the ratio of the wavelengths
Now, we want to find the ratio \( \frac{\lambda_2}{\lambda_3} \):
\[ \frac{\lambda_2}{\lambda_3} = \frac{\pi r_2}{\frac{2 \pi r_3}{3}} = \frac{3 r_2}{2 r_3} \]

### Step 5: Use the formula for the radius of the nth orbit
The radius of the nth orbit in a hydrogen atom is given by:
\[ r_n \propto \frac{n^2}{Z} \]
where \( Z \) is the atomic number. For hydrogen, \( Z = 1 \), so:
\[ r_2 \propto 2^2 = 4 \]
\[ r_3 \propto 3^2 = 9 \]

### Step 6: Substitute the values of the radii into the ratio
Now we substitute \( r_2 \) and \( r_3 \) into the ratio:
\[ \frac{\lambda_2}{\lambda_3} = \frac{3 \cdot 4}{2 \cdot 9} = \frac{12}{18} = \frac{2}{3} \]

### Final Answer
Thus, the ratio of the de Broglie wavelength of the electron in the second and third Bohr orbits in a hydrogen atom is:
\[ \frac{\lambda_2}{\lambda_3} = \frac{2}{3} \]