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 wavelengths 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 formula
The de Broglie wavelength (λ) of a particle is given by the formula:
\[ \lambda = \frac{h}{mv} \]
where \( h \) is Planck's constant, \( m \) is the mass of the particle, and \( v \) is its velocity.

### Step 2: Write the expression for the ratio of wavelengths
We need to find the ratio of the de Broglie wavelengths in the second (\( \lambda_2 \)) and third (\( \lambda_3 \)) orbits:
\[ \frac{\lambda_2}{\lambda_3} = \frac{h/mv_2}{h/mv_3} = \frac{v_3}{v_2} \]

### Step 3: Determine the velocities in the Bohr model
In the Bohr model, the velocity of an electron in the nth orbit is given by:
\[ v_n = \frac{e^2}{2 \epsilon_0 n h} \]
where \( e \) is the charge of the electron, \( \epsilon_0 \) is the permittivity of free space, and \( n \) is the principal quantum number.

### Step 4: Calculate the velocities for n=2 and n=3
For the second orbit (\( n=2 \)):
\[ v_2 = \frac{e^2}{2 \epsilon_0 \cdot 2 h} = \frac{e^2}{4 \epsilon_0 h} \]

For the third orbit (\( n=3 \)):
\[ v_3 = \frac{e^2}{2 \epsilon_0 \cdot 3 h} = \frac{e^2}{6 \epsilon_0 h} \]

### Step 5: Find the ratio of velocities
Now, we can find the ratio of the velocities:
\[ \frac{v_2}{v_3} = \frac{\frac{e^2}{4 \epsilon_0 h}}{\frac{e^2}{6 \epsilon_0 h}} = \frac{6}{4} = \frac{3}{2} \]

### Step 6: Substitute back to find the ratio of wavelengths
Now substituting back into the wavelength ratio:
\[ \frac{\lambda_2}{\lambda_3} = \frac{v_3}{v_2} = \frac{2}{3} \]

### Conclusion
Thus, the ratio of the de Broglie wavelengths of the electron in the second and third Bohr orbits is:
\[ \frac{\lambda_2}{\lambda_3} = \frac{2}{3} \]

### Final Answer
The ratio of the de Broglie wavelength of the electron in the second and third Bohr orbits is \( \frac{2}{3} \).

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