According to the bohr's atomic model, the relation between principal quantum number (n) and radius of orbit (r) is

Options

A. `r infty 1/n`
B. `r infty 1/ (n^(2))`
C. `r infty n`
D. `r infty n^(2)`

Answer

To find the relation between the principal quantum number (n) and the radius of the orbit (r) according to Bohr's atomic model, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Forces**: According to Bohr's model, the centripetal force acting on the electron in a circular orbit is provided by the electrostatic (Coulomb) force between the positively charged nucleus and the negatively charged electron.

\[ \text{Centripetal Force} = \frac{mv^2}{r} \]
\[ \text{Electrostatic Force} = \frac{ke^2}{r^2} \]

Where:
- \( m \) is the mass of the electron,
- \( v \) is the velocity of the electron,
- \( r \) is the radius of the orbit,
- \( k = \frac{1}{4\pi\epsilon_0} \) is Coulomb's constant,
- \( e \) is the charge of the electron.

2. **Setting the Forces Equal**: From the balance of forces, we have:
\[ \frac{mv^2}{r} = \frac{ke^2}{r^2} \]

Rearranging gives:
\[ mv^2 = \frac{ke^2}{r} \]

3. **Angular Momentum Condition**: According to Bohr's model, the angular momentum of the electron is quantized and is given by:
\[ L = mvr = n \frac{h}{2\pi} \]

Where \( h \) is Planck's constant and \( n \) is the principal quantum number.

4. **Squaring the Angular Momentum Equation**: Squaring both sides of the angular momentum equation gives:
\[ (mvr)^2 = \left(n \frac{h}{2\pi}\right)^2 \]
\[ m^2v^2r^2 = \frac{n^2h^2}{4\pi^2} \]

5. **Substituting for \( mv^2 \)**: From the centripetal force equation, we can express \( mv^2 \) in terms of \( r \):
\[ mv^2 = \frac{ke^2}{r} \]

Substituting this into the squared angular momentum equation:
\[ \left(\frac{ke^2}{r}\right)r^2 = \frac{n^2h^2}{4\pi^2} \]
\[ ke^2r = \frac{n^2h^2}{4\pi^2} \]

6. **Solving for \( r \)**: Rearranging gives:
\[ r = \frac{n^2h^2}{4\pi^2ke^2} \]

7. **Identifying the Relation**: This shows that the radius \( r \) is directly proportional to the square of the principal quantum number \( n \):
\[ r \propto n^2 \]

### Conclusion:
Thus, according to Bohr's atomic model, the relation between the principal quantum number \( n \) and the radius of the orbit \( r \) is:
\[ r \propto n^2 \]