The expression for Bohr radius of nth orbit of an atoms is

Options

A. `r=(n^(2)h^(2))/(4pi^(2)mZe^(2))`
B. `r=(nh)/(4pi^(2)mZe^(2))`
C. `r=(n^(2)h^(2))/(4pi^(2)mZ)`
D. `r=(n^(2)h^(2))/(4pi^(2)me^(2))`

Explanation

To derive the expression for the Bohr radius of the nth orbit of an atom, we will follow a systematic approach using the principles of classical mechanics and electrostatics.

### Step-by-Step Solution:

1. **Identify Forces Acting on the Electron:**
- The electron in the nth orbit is influenced by two forces: the centripetal force required to keep it in circular motion and the electrostatic force of attraction between the positively charged nucleus and the negatively charged electron.
- The centripetal force \( F_c \) is given by:
\[ F_c = \frac{mv^2}{r} \]
where \( m \) is the mass of the electron, \( v \) is its velocity, and \( r \) is the radius of the orbit.

2. **Electrostatic Force:**
- The electrostatic force \( F_e \) between the nucleus (with charge \( +zE \)) and the electron (with charge \( -E \)) is given by Coulomb's law:
\[ F_e = \frac{k \cdot zE \cdot E}{r^2} = \frac{kzE^2}{r^2} \]
where \( k \) is Coulomb's constant.

3. **Set the Forces Equal:**
- For the electron to remain in a stable orbit, the centripetal force must equal the electrostatic force:
\[ \frac{mv^2}{r} = \frac{kzE^2}{r^2} \]
- Rearranging gives:
\[ mv^2 r = kzE^2 \quad \text{(Equation 1)} \]

4. **Quantization of Angular Momentum:**
- According to Bohr's postulate, the angular momentum \( L \) of the electron is quantized:
\[ L = mvr = n\frac{h}{2\pi} \]
where \( n \) is the principal quantum number and \( h \) is Planck's constant.
- From this, we can express \( v \) as:
\[ v = \frac{n h}{2\pi m r} \quad \text{(Equation 2)} \]

5. **Substitute \( v \) into Equation 1:**
- Substitute the expression for \( v \) from Equation 2 into Equation 1:
\[ m\left(\frac{n h}{2\pi m r}\right)^2 r = kzE^2 \]
- Simplifying this gives:
\[ \frac{n^2 h^2}{4\pi^2 m r} = kzE^2 \]

6. **Rearranging to Find \( r \):**
- Rearranging the equation to solve for \( r \):
\[ r = \frac{n^2 h^2}{4\pi^2 k m zE^2} \]
- This is the expression for the Bohr radius of the nth orbit.

### Final Expression:
The expression for the Bohr radius \( r_n \) of the nth orbit is:
\[ r_n = \frac{n^2 h^2}{4\pi^2 k m zE^2} \]