Which of the following suggested de-Broglie wavelength(s) is `//` are possible for electron in a Bohr orbit of Hydrogen atom ?

Options

A. `19.92Å`
B. `9.96Å`
C. `4.98Å`
D. `3.32Å`

Answer

To determine which de Broglie wavelengths are possible for an electron in a Bohr orbit of a hydrogen atom, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Bohr Model**: According to the Bohr model, electrons move in circular orbits around the nucleus, and the circumference of the orbit is quantized. The relationship is given by:
\[ 2\pi r = n\lambda \]
where \( r \) is the radius of the orbit, \( n \) is the principal quantum number (1, 2, 3,...), and \( \lambda \) is the de Broglie wavelength.

2. **Identify the Radius of the First Bohr Orbit**: For the hydrogen atom, the radius of the first Bohr orbit (\( r_1 \)) is approximately:
\[ r_1 = 0.529 \text{ Å} = 0.529 \times 10^{-10} \text{ m} \]

3. **Calculate the Circumference of the Orbit**: The circumference of the first orbit can be calculated as:
\[ C = 2\pi r_1 = 2\pi \times 0.529 \text{ Å} \]

4. **Substitute the Values**: Now substituting the radius into the circumference formula:
\[ C = 2 \times 3.14 \times 0.529 \text{ Å} \]
\[ C \approx 3.32 \text{ Å} \]

5. **Determine Possible Wavelengths**: Since the circumference must be an integer multiple of the de Broglie wavelength, we can equate:
\[ n\lambda = C \]
For \( n=1 \):
\[ \lambda = \frac{C}{n} = 3.32 \text{ Å} \]
For \( n=2 \):
\[ \lambda = \frac{C}{2} \approx 1.66 \text{ Å} \]
For \( n=3 \):
\[ \lambda = \frac{C}{3} \approx 1.11 \text{ Å} \]
And so on.

6. **Conclusion**: The possible de Broglie wavelengths for the electron in a Bohr orbit of a hydrogen atom are integer fractions of the circumference. Thus, wavelengths like 3.32 Å, 1.66 Å, and 1.11 Å are possible.