If `R_1` is the radius of the first of hydrogen atom then the radii of second , third and fourth orbitals in term of `r_1 ` are

Options

A. ` r_1^2 , r_(1) ^3 , r_1^4`
B. ` 4r_1 , 9r _1, 16 r_1`
C. ` 8 r_1 , 27 r_1 , 64 r_1`
D. `2r_1 , 6 r_1 , 8 r_1`

Answer

To find the radii of the second, third, and fourth orbitals of a hydrogen atom in terms of the radius of the first orbital \( R_1 \), we can use the formula for the radius of the nth orbit:

\[ R_n = \frac{0.53 \, n^2}{Z} \text{ angstroms} \]

where \( Z \) is the atomic number and for hydrogen, \( Z = 1 \).

### Step-by-Step Solution:

1. **Identify the radius of the first orbital**:
- The radius of the first orbital \( R_1 \) is given as \( 0.53 \) angstroms.

2. **Calculate the radius of the second orbital \( R_2 \)**:
- Using the formula:
\[ R_2 = \frac{0.53 \times 2^2}{1} = 0.53 \times 4 = 2.12 \text{ angstroms} \]
- In terms of \( R_1 \):
\[ R_2 = 4R_1 \]

3. **Calculate the radius of the third orbital \( R_3 \)**:
- Using the formula:
\[ R_3 = \frac{0.53 \times 3^2}{1} = 0.53 \times 9 = 4.77 \text{ angstroms} \]
- In terms of \( R_1 \):
\[ R_3 = 9R_1 \]

4. **Calculate the radius of the fourth orbital \( R_4 \)**:
- Using the formula:
\[ R_4 = \frac{0.53 \times 4^2}{1} = 0.53 \times 16 = 8.48 \text{ angstroms} \]
- In terms of \( R_1 \):
\[ R_4 = 16R_1 \]

### Final Results:
- The radius of the second orbital \( R_2 = 4R_1 \)
- The radius of the third orbital \( R_3 = 9R_1 \)
- The radius of the fourth orbital \( R_4 = 16R_1 \)

### Summary:
Thus, the radii of the second, third, and fourth orbitals in terms of \( R_1 \) are:
- \( R_2 = 4R_1 \)
- \( R_3 = 9R_1 \)
- \( R_4 = 16R_1 \)