Number of waves in third Bohr's orbit of hydrogen will be

Options

A. 3
B. 6
C. 9
D. 12

Explanation

To find the number of waves in the third Bohr orbit of hydrogen, we can follow these steps:

### Step-by-Step Solution:

1. **Determine the Circumference of the Third Bohr Orbit**:
The circumference \( C \) of the orbit is given by the formula:
\[ C = 2 \pi r \]
where \( r \) is the radius of the third Bohr orbit.

2. **Calculate the Radius of the Third Bohr Orbit**:
The radius of the \( n \)-th Bohr orbit is given by the formula:
\[ r_n = n^2 \cdot r_1 \]
where \( r_1 \) is the radius of the first Bohr orbit (approximately \( 0.529 \, \text{Å} \)). For the third orbit (\( n = 3 \)):
\[ r_3 = 3^2 \cdot r_1 = 9 \cdot r_1 \]

3. **Determine the Number of Waves**:
The number of waves \( n \) in the orbit can be calculated using the relationship:
\[ n = \frac{C}{\lambda} \]
where \( \lambda \) is the wavelength of the electron in that orbit.

4. **Calculate Wavelength Using de Broglie's Equation**:
The wavelength \( \lambda \) can be calculated using the de Broglie wavelength formula:
\[ \lambda = \frac{h}{mv} \]
where \( h \) is Planck's constant, \( m \) is the mass of the electron, and \( v \) is the velocity of the electron.

5. **Substituting Values**:
From Bohr's model, we know that:
\[ mv r = \frac{nh}{2 \pi} \]
For the third orbit (\( n = 3 \)):
\[ mv r_3 = \frac{3h}{2 \pi} \]

6. **Combine the Equations**:
Substitute \( mv r_3 \) into the equation for the number of waves:
\[ n = \frac{2 \pi r_3}{\lambda} = \frac{2 \pi r_3}{\frac{h}{mv}} = \frac{2 \pi r_3 \cdot mv}{h} \]
Now substitute \( mv r_3 = \frac{3h}{2 \pi} \):
\[ n = \frac{2 \pi r_3 \cdot \frac{3h}{2 \pi}}{h} = 3 \]

### Final Answer:
The number of waves in the third Bohr orbit of hydrogen is **3**.