Determine wavelength of electron in `4th` Bohr's orbit of hydrogen atom

Options

A. `4pia_0`
B. `pia_0`
C. `8pia_0`
D. `6pia_0`

Explanation

To determine the wavelength of an electron in the 4th Bohr orbit of a hydrogen atom, we can follow these steps:

### Step 1: Understand the Bohr Model
In the Bohr model of the hydrogen atom, electrons revolve in circular orbits around the nucleus. The radius of these orbits is quantized and given by the formula:

\[ r_n = \frac{n^2 a_0}{Z} \]

where:
- \( r_n \) is the radius of the nth orbit,
- \( n \) is the principal quantum number,
- \( a_0 \) is the Bohr radius (\( \approx 0.529 \, \text{Å} \)),
- \( Z \) is the atomic number (for hydrogen, \( Z = 1 \)).

### Step 2: Calculate the Radius for the 4th Orbit
For the 4th orbit (\( n = 4 \)):

\[ r_4 = \frac{4^2 \cdot a_0}{1} = 16 a_0 \]

### Step 3: Use the de Broglie Wavelength Formula
The de Broglie wavelength (\( \lambda \)) of an electron is given by:

\[ \lambda = \frac{h}{mv} \]

where:
- \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \, \text{Js} \)),
- \( m \) is the mass of the electron (\( \approx 9.11 \times 10^{-31} \, \text{kg} \)),
- \( v \) is the velocity of the electron.

### Step 4: Relate the Wavelength to the Orbit
From the Bohr model, the circumference of the orbit must be an integral multiple of the wavelength:

\[ 2 \pi r_n = n \lambda \]

Substituting \( r_n \) for the 4th orbit:

\[ 2 \pi (16 a_0) = 4 \lambda \]

### Step 5: Solve for the Wavelength
Rearranging the equation gives:

\[ \lambda = \frac{2 \pi (16 a_0)}{4} = 8 \pi a_0 \]

### Final Result
Thus, the wavelength of the electron in the 4th Bohr orbit of a hydrogen atom is:

\[ \lambda = 8 \pi a_0 \]