According to Bohr model , radius of the first orbit of hydrogen atom is `r_1` then the radii of second , third and fouth orbitals in term of ` r_1` are

Options

A. `2.5 (h) /(pi)`
B. `6 (h) /(pi)`
C. `3 (h)/(pi)`
D. `(2.5 h ) /( 2 pi )`

Answer

To find the radii of the second, third, and fourth orbits of a hydrogen atom in terms of the radius of the first orbit \( r_1 \), we can use the formula derived from the Bohr model of the atom.

### Step-by-Step Solution:

1. **Understand the Formula**: According to the Bohr model, the radius of the \( n \)-th orbit (\( R_n \)) is given by the formula:
\[ R_n = \frac{h^2}{4 \pi^2 e^2} \cdot \frac{n^2}{Z} \]
For hydrogen (\( Z = 1 \)), this simplifies to:
\[ R_n = 0.529 \cdot n^2 \text{ angstroms} \]

2. **Calculate \( R_1 \)**: For the first orbit (\( n = 1 \)):
\[ R_1 = 0.529 \cdot 1^2 = 0.529 \text{ angstroms} \]

3. **Calculate \( R_2 \)**: For the second orbit (\( n = 2 \)):
\[ R_2 = 0.529 \cdot 2^2 = 0.529 \cdot 4 = 2.116 \text{ angstroms} \]
In terms of \( r_1 \):
\[ R_2 = 4 \cdot r_1 \]

4. **Calculate \( R_3 \)**: For the third orbit (\( n = 3 \)):
\[ R_3 = 0.529 \cdot 3^2 = 0.529 \cdot 9 = 4.761 \text{ angstroms} \]
In terms of \( r_1 \):
\[ R_3 = 9 \cdot r_1 \]

5. **Calculate \( R_4 \)**: For the fourth orbit (\( n = 4 \)):
\[ R_4 = 0.529 \cdot 4^2 = 0.529 \cdot 16 = 8.464 \text{ angstroms} \]
In terms of \( r_1 \):
\[ R_4 = 16 \cdot r_1 \]

### Final Results:
- \( R_2 = 4 \cdot r_1 \)
- \( R_3 = 9 \cdot r_1 \)
- \( R_4 = 16 \cdot r_1 \)