How many times is the velocity of the electron in the first shell of `He^(+)` ion as compared to that in the first shell of hydrogen atom ?

Explanation

To determine how many times the velocity of the electron in the first shell of the He⁺ ion is compared to that in the first shell of a hydrogen atom, we can use Bohr's model of the atom. According to this model, the velocity of an electron in the nth orbit is given by the formula:

\[ v_n = v_0 \cdot \frac{Z}{n} \]

where:
- \( v_n \) is the velocity of the electron in the nth orbit,
- \( v_0 \) is a constant (approximately \( 2.18 \times 10^6 \) m/s),
- \( Z \) is the atomic number of the element,
- \( n \) is the principal quantum number (the shell number).

### Step-by-Step Solution:

1. **Identify the parameters for He⁺ ion:**
- For the He⁺ ion, the atomic number \( Z = 2 \) (since helium has 2 protons).
- The electron is in the first shell, so \( n = 1 \).

Using the formula for the velocity in the first shell of He⁺:
\[ v_{1, \text{He}^+} = v_0 \cdot \frac{Z}{n} = v_0 \cdot \frac{2}{1} = 2 v_0 \]

2. **Identify the parameters for hydrogen atom:**
- For hydrogen, the atomic number \( Z = 1 \).
- The electron is also in the first shell, so \( n = 1 \).

Using the formula for the velocity in the first shell of hydrogen:
\[ v_{1, \text{H}} = v_0 \cdot \frac{Z}{n} = v_0 \cdot \frac{1}{1} = v_0 \]

3. **Calculate the ratio of velocities:**
Now, we can find the ratio of the velocities of the electron in the first shell of He⁺ to that in hydrogen:
\[ \frac{v_{1, \text{He}^+}}{v_{1, \text{H}}} = \frac{2 v_0}{v_0} = 2 \]

### Conclusion:
The velocity of the electron in the first shell of the He⁺ ion is **2 times** that of the electron in the first shell of a hydrogen atom.