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. (a) ` r_1^2 , r_(1) ^3 , r_1^4`
B. (b) ` 4r_1 , 9r _1, 16 r_1`
C. (c) ` 8 r_1 , 27 r_1 , 64 r_1`
D. (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 derived from the Bohr model of the atom. The formula for the radius of the nth orbital is given by:

\[ R_n = \frac{0.529 \, n^2}{Z} \text{ Angstroms} \]

Where:
- \( R_n \) is the radius of the nth orbital,
- \( n \) is the principal quantum number (1 for the first orbital, 2 for the second, etc.),
- \( Z \) is the atomic number (which is 1 for hydrogen).

### Step-by-Step Solution:

1. **Identify the radius of the first orbital**:
- For the first orbital (\( n = 1 \)):
\[ R_1 = \frac{0.529 \, (1)^2}{1} = 0.529 \text{ Angstroms} \]

2. **Calculate the radius of the second orbital**:
- For the second orbital (\( n = 2 \)):
\[ R_2 = \frac{0.529 \, (2)^2}{1} = \frac{0.529 \times 4}{1} = 2.116 \text{ Angstroms} \]
- In terms of \( R_1 \):
\[ R_2 = 4R_1 \]

3. **Calculate the radius of the third orbital**:
- For the third orbital (\( n = 3 \)):
\[ R_3 = \frac{0.529 \, (3)^2}{1} = \frac{0.529 \times 9}{1} = 4.761 \text{ Angstroms} \]
- In terms of \( R_1 \):
\[ R_3 = 9R_1 \]

4. **Calculate the radius of the fourth orbital**:
- For the fourth orbital (\( n = 4 \)):
\[ R_4 = \frac{0.529 \, (4)^2}{1} = \frac{0.529 \times 16}{1} = 8.464 \text{ Angstroms} \]
- In terms of \( R_1 \):
\[ R_4 = 16R_1 \]

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