If velocity of an electron in 1st orbit of H atoms is V , what will be the velocity in 3rd orbit of `Li^(2+)` ?

Options

A. v
B. `v//3`
C. `3v`
D. `9v`

Explanation

To find the velocity of an electron in the third orbit of the lithium ion \( \text{Li}^{2+} \) given that the velocity of an electron in the first orbit of hydrogen is \( V \), we can use the formula derived from Bohr's model of the atom.

### Step-by-Step Solution:

1. **Identify the Formula for Velocity**:
The velocity \( v_n \) of an electron in the nth orbit of a hydrogen-like atom is given by the formula:
\[ v_n = 2.19 \times 10^6 \frac{Z}{n} \text{ m/s} \]
where \( Z \) is the atomic number and \( n \) is the principal quantum number (orbit number).

2. **Calculate the Velocity for Hydrogen**:
For hydrogen (\( Z = 1 \)) in the first orbit (\( n = 1 \)):
\[ v_{H_1} = 2.19 \times 10^6 \frac{1}{1} = 2.19 \times 10^6 \text{ m/s} \]
According to the question, this velocity is equal to \( V \):
\[ V = 2.19 \times 10^6 \text{ m/s} \]

3. **Calculate the Velocity for Lithium Ion \( \text{Li}^{2+} \)**:
For lithium (\( Z = 3 \)) in the third orbit (\( n = 3 \)):
\[ v_{Li^{2+}, 3} = 2.19 \times 10^6 \frac{3}{3} = 2.19 \times 10^6 \text{ m/s} \]
Simplifying this gives:
\[ v_{Li^{2+}, 3} = 2.19 \times 10^6 \text{ m/s} \]

4. **Relate the Velocity to \( V \)**:
Since we found that:
\[ v_{Li^{2+}, 3} = V \]
Therefore, the velocity of the electron in the third orbit of \( \text{Li}^{2+} \) is also \( V \).

### Final Answer:
The velocity of the electron in the third orbit of \( \text{Li}^{2+} \) is \( V \).