If the radius of the first Bohr orbit is r, then the de broglie wavelength of the electron in the 4th orbit will be

Options

A. `4pi r`
B. `6pir`
C. `8pir`
D. `(pir)/(4)`

Explanation

To find the de Broglie wavelength of the electron in the fourth Bohr orbit, we can follow these steps:

### Step-by-Step Solution:

1. **Understand Bohr's Model**: According to Bohr's model, the angular momentum of an electron in an orbit is quantized and given by the formula:
\[ L = mvr = n \frac{h}{2\pi} \]
where \( L \) is the angular momentum, \( m \) is the mass of the electron, \( v \) is its velocity, \( r \) is the radius of the orbit, \( n \) is the principal quantum number, and \( h \) is Planck's constant.

2. **Relate Radius to Principal Quantum Number**: The radius of the nth orbit in the Bohr model is given by:
\[ r_n = n^2 r \]
where \( r \) is the radius of the first orbit. For the fourth orbit (\( n = 4 \)):
\[ r_4 = 4^2 r = 16r \]

3. **Use de Broglie Wavelength Formula**: The de Broglie wavelength \( \lambda \) of a particle is given by:
\[ \lambda = \frac{h}{mv} \]

4. **Relate Wavelength to Orbit Radius**: From Bohr's model, we also know that:
\[ 2\pi r_n = n\lambda \]
For the fourth orbit:
\[ 2\pi r_4 = 4\lambda \]

5. **Substitute for \( r_4 \)**: Substitute \( r_4 = 16r \) into the equation:
\[ 2\pi (16r) = 4\lambda \]

6. **Simplify the Equation**: This simplifies to:
\[ 32\pi r = 4\lambda \]
Dividing both sides by 4 gives:
\[ \lambda = 8\pi r \]

7. **Final Result**: Therefore, the de Broglie wavelength of the electron in the fourth orbit is:
\[ \lambda = 8\pi r \]

### Conclusion:
The de Broglie wavelength of the electron in the fourth orbit is \( 8\pi r \).

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