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 a hydrogen atom (H) and the fourth shell of a helium ion (He⁺), we can use the formula for the radius of an electron shell in a hydrogen-like atom:

\[ R_n = \frac{a_0 n^2}{Z} \]

where:
- \( R_n \) is the radius of the nth shell,
- \( a_0 \) is the Bohr radius (a constant),
- \( n \) is the principal quantum number (shell number),
- \( Z \) is the atomic number.

### Step 1: Identify the values for hydrogen (H) and helium ion (He⁺)

For hydrogen:
- Atomic number \( Z_1 = 1 \)
- Principal quantum number for the first shell \( n_1 = 1 \)

For helium ion:
- Atomic number \( Z_2 = 2 \)
- Principal quantum number for the fourth shell \( n_2 = 4 \)

### Step 2: Write the formula for the radii

Using the formula for the radius of the shells, we can express the radii for hydrogen and helium ion:

- Radius of the first shell of hydrogen:
\[ R_1 = \frac{a_0 n_1^2}{Z_1} = \frac{a_0 (1)^2}{1} = a_0 \]

- Radius of the fourth shell of helium ion:
\[ R_2 = \frac{a_0 n_2^2}{Z_2} = \frac{a_0 (4)^2}{2} = \frac{16 a_0}{2} = 8 a_0 \]

### Step 3: Calculate the ratio of the radii

Now, we can find the ratio of the radius of the first shell of hydrogen to the radius of the fourth shell of helium ion:

\[ \text{Ratio} = \frac{R_1}{R_2} = \frac{a_0}{8 a_0} = \frac{1}{8} \]

### Final Answer

Thus, the ratio of the radii of the first shell of the hydrogen atom to that of the fourth shell of the helium ion is:

\[ \text{Ratio} = 1 : 8 \]

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