The angular speed of electron in the nth orbit of hydrogen atom is

Options

A. Directly proportional to `n^(2)`
B. Directly proportional to n
C. Inversely proportional to `n^(3)`
D. Inversely proportional to n

Answer

To find the angular speed of an electron in the nth orbit of a hydrogen atom, we can follow these steps:

### Step 1: Understand the forces acting on the electron
The electron in a hydrogen atom revolves around the nucleus in a circular orbit. The forces acting on the electron are:
- Centripetal force (Fc) required to keep the electron in circular motion.
- Electrostatic force (Fe) of attraction between the positively charged nucleus and the negatively charged electron.

### Step 2: Set up the equation for centripetal force
The centripetal force can be expressed as:
\[ F_c = \frac{mv^2}{r} \]
where:
- \( m \) is the mass of the electron,
- \( v \) is the linear speed of the electron,
- \( r \) is the radius of the orbit.

### Step 3: Set up the equation for electrostatic force
The electrostatic force can be expressed using Coulomb's law:
\[ F_e = \frac{k \cdot Ze \cdot e}{r^2} \]
where:
- \( k \) is Coulomb's constant,
- \( Z \) is the atomic number (for hydrogen, \( Z = 1 \)),
- \( e \) is the charge of the electron.

### Step 4: Equate the centripetal force and electrostatic force
Since the electron is in a stable orbit, the centripetal force equals the electrostatic force:
\[ \frac{mv^2}{r} = \frac{k \cdot e^2}{r^2} \]

### Step 5: Rearranging the equation
Rearranging gives:
\[ mv^2 = \frac{k \cdot e^2}{r} \]

### Step 6: Relate linear speed to angular speed
The linear speed \( v \) can be related to angular speed \( \omega \) by:
\[ v = r \omega \]
Substituting this into the previous equation gives:
\[ m(r \omega)^2 = \frac{k \cdot e^2}{r} \]
This simplifies to:
\[ mr^2 \omega^2 = \frac{k \cdot e^2}{r} \]

### Step 7: Solve for angular speed \( \omega \)
Rearranging gives:
\[ \omega^2 = \frac{k \cdot e^2}{m r^3} \]
Taking the square root:
\[ \omega = \sqrt{\frac{k \cdot e^2}{m r^3}} \]

### Step 8: Substitute the expression for radius \( r \)
For the hydrogen atom, the radius of the nth orbit is given by:
\[ r_n \propto n^2 \]
Thus, we can write:
\[ \omega_n \propto \frac{1}{r_n^{3/2}} \propto \frac{1}{(n^2)^{3/2}} = \frac{1}{n^3} \]

### Conclusion
The angular speed of the electron in the nth orbit of a hydrogen atom is inversely proportional to \( n^3 \):
\[ \omega_n \propto \frac{1}{n^3} \]