In a hydrogen atom, the binding energy of the electron in the ground state is `E_(1)` then the frequency of revolution of the electron in the nth orbit is

Options

A. `2E_(1)//nh`
B. `2E_(1)n//h`
C. `E_(1)//nh`
D. `E_(1)n//h`

Answer

To find the frequency of revolution of the electron in the nth orbit of a hydrogen atom, we can follow these steps:

### Step 1: Understand the Binding Energy
The binding energy of the electron in the ground state of a hydrogen atom is denoted as \( E_1 \). In the ground state, the binding energy is equal to the kinetic energy of the electron in that orbit.

### Step 2: Write the Expression for Kinetic Energy
The kinetic energy \( K \) of the electron can be expressed as:
\[ K = \frac{1}{2} mv^2 \]
where \( m \) is the mass of the electron and \( v \) is its velocity.

### Step 3: Angular Momentum in nth Orbit
According to Bohr's model, the angular momentum \( L \) of the electron in the nth orbit is given by:
\[ L = mvr = \frac{nh}{2\pi} \]
where \( r \) is the radius of the nth orbit, \( n \) is the principal quantum number, and \( h \) is Planck's constant.

### Step 4: Relate Kinetic Energy to Angular Momentum
From the above expressions, we can relate the kinetic energy to the angular momentum. The total energy in the nth orbit can be expressed as:
\[ E_n = -\frac{K}{2} \]
Thus, we can say:
\[ E_n = -\frac{1}{2} \cdot \frac{1}{2} mv^2 = -E_1 \]

### Step 5: Find the Frequency
To find the frequency \( f \) of revolution, we can use the relationship between linear velocity and frequency:
\[ f = \frac{v}{2\pi r} \]
From the angular momentum equation, we can express \( v \) in terms of \( r \) and \( n \):
\[ v = \frac{nh}{2\pi m r} \]
Substituting this into the frequency equation gives:
\[ f = \frac{nh}{2\pi m r \cdot 2\pi r} = \frac{nh}{4\pi^2 m r^2} \]

### Step 6: Substitute for \( r \)
Using the relationship between the radius of the nth orbit and the principal quantum number, we know:
\[ r_n = n^2 \cdot r_1 \]
where \( r_1 \) is the radius of the ground state. Substituting this into the frequency equation gives:
\[ f_n = \frac{2E_1}{nh} \]

### Final Answer
The frequency of revolution of the electron in the nth orbit is given by:
\[ f_n = \frac{2E_1}{nh} \]