The ratio of radii of first shell of H atom and that of fourth shell of `He^(+)` ion is

Options

A. `1:8`
B. `1:4`
C. `1:sqrt8`
D. `1:3`

Answer

To find the ratio of the radii of the first shell of the hydrogen atom (H) and the fourth shell of the helium ion (He⁺), we will use the formula for the radii of hydrogen-like atoms derived from the Bohr model.

### Step-by-Step Solution:

1. **Understand the Formula for Radius**:
The radius of the nth orbit (shell) of a hydrogen-like atom is given by the formula:
\[ R_n = \frac{n^2}{Z} R_0 \]
where:
- \( R_n \) is the radius of the nth shell,
- \( n \) is the principal quantum number (shell number),
- \( Z \) is the atomic number of the atom,
- \( R_0 \) is the radius of the first orbit of the hydrogen atom, approximately \( 0.529 \) Å.

2. **Calculate the Radius for Hydrogen (H)**:
For the hydrogen atom (H):
- \( n = 1 \)
- \( Z = 1 \)
- Therefore, the radius \( R_1 \) is:
\[ R_1 = \frac{1^2}{1} R_0 = R_0 \]

3. **Calculate the Radius for Helium Ion (He⁺)**:
For the helium ion (He⁺):
- \( n = 4 \)
- \( Z = 2 \)
- Therefore, the radius \( R_4 \) is:
\[ R_4 = \frac{4^2}{2} R_0 = \frac{16}{2} R_0 = 8 R_0 \]

4. **Find the Ratio of the Radii**:
Now, we need to find the ratio of \( R_1 \) to \( R_4 \):
\[ \text{Ratio} = \frac{R_1}{R_4} = \frac{R_0}{8 R_0} \]
Simplifying this gives:
\[ \text{Ratio} = \frac{1}{8} \]
Thus, the ratio of the radii of the first shell of hydrogen to the fourth shell of helium ion is:
\[ 1 : 8 \]

### Final Answer:
The ratio of the radii of the first shell of H atom to that of the fourth shell of He⁺ ion is \( 1 : 8 \).

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