Number of revolution per second made by electron in Bohr orbit is called orbital frequency. Which represents the orbital frequencey. If `u_(n),r_(n)` and `E_(n)` are velocity, radius and energy of nth orbit respectively?

Options

A. `(u_(n))/(2pir_(n))`
B. `-(2E)/(nh)`
C. `-(2E_(1))/(n^(3)h)`
D. Any of these

Explanation

To find the orbital frequency of an electron in a Bohr orbit, we can derive it step by step using the given parameters: velocity \( u_n \), radius \( r_n \), and energy \( E_n \) of the nth orbit.

### Step-by-Step Solution:

1. **Understanding Orbital Frequency**:
The orbital frequency \( f \) is defined as the number of revolutions per second made by an electron in its orbit. Mathematically, it can be expressed as:
\[ f = \frac{u_n}{2\pi r_n} \]
where \( u_n \) is the velocity of the electron in the nth orbit and \( r_n \) is the radius of the nth orbit.

2. **Using Known Formulas**:
In the Bohr model, the velocity \( u_n \) of the electron in the nth orbit can be expressed as:
\[ u_n = \frac{e^2}{n\hbar} \]
where \( e \) is the charge of the electron, \( n \) is the principal quantum number, and \( \hbar \) is the reduced Planck's constant.

3. **Radius of the nth Orbit**:
The radius \( r_n \) of the nth orbit is given by:
\[ r_n = \frac{n^2 \hbar^2}{k e^2 m} \]
where \( k \) is Coulomb's constant and \( m \) is the mass of the electron.

4. **Substituting Values**:
Now, substituting \( u_n \) and \( r_n \) into the formula for orbital frequency:
\[ f = \frac{\frac{e^2}{n\hbar}}{2\pi \left(\frac{n^2 \hbar^2}{k e^2 m}\right)} \]

5. **Simplifying the Expression**:
Simplifying the above expression:
\[ f = \frac{e^2 \cdot k \cdot m}{2\pi n^3 \hbar} \]

6. **Relating to Energy**:
The energy of the nth orbit \( E_n \) can be expressed as:
\[ E_n = -\frac{m e^4}{2 n^2 \hbar^2} \]
From this, we can express \( e^4 \) in terms of \( E_n \):
\[ e^4 = -2 n^2 \hbar^2 E_n / m \]

7. **Final Expression for Orbital Frequency**:
Substituting \( e^4 \) back into the frequency equation:
\[ f = \frac{-2 n^2 \hbar^2 E_n \cdot k \cdot m}{2\pi n^3 \hbar} \]
After simplification, we get:
\[ f = \frac{-k m E_n}{\pi n \hbar} \]

### Conclusion:
Thus, the orbital frequency \( f \) can be expressed in terms of the energy \( E_n \) of the nth orbit, along with constants \( k \), \( m \), and \( \hbar \).