The velocity of an electron in its fifth orbit, if the velocity of an electron in the second orbit of sodium atom (atomic number =11) is v, will be :

Options

A. v
B. `(22v)/(5)`
C. `(5)/(2)v`
D. `(2)/(5)v`

Explanation

To find the velocity of an electron in its fifth orbit given that the velocity of an electron in the second orbit of a sodium atom (atomic number Z = 11) is V, we can use the formula for the velocity of an electron in a given orbit:

1. **Formula for Electron Velocity**:
The velocity \( v_n \) of an electron in the nth orbit is given by the formula:
\[ v_n = 2.18 \times 10^6 \frac{Z}{n} \text{ m/s} \]
where \( Z \) is the atomic number and \( n \) is the orbit number.

2. **Velocity in the Second Orbit**:
For the second orbit (n = 2):
\[ v_2 = 2.18 \times 10^6 \frac{Z}{2} \]
Given that \( v_2 = V \), we can express it as:
\[ V = 2.18 \times 10^6 \frac{Z}{2} \]

3. **Velocity in the Fifth Orbit**:
Now, for the fifth orbit (n = 5):
\[ v_5 = 2.18 \times 10^6 \frac{Z}{5} \]

4. **Finding the Ratio of Velocities**:
To find the relationship between \( v_5 \) and \( V \), we can take the ratio:
\[ \frac{v_5}{v_2} = \frac{2.18 \times 10^6 \frac{Z}{5}}{2.18 \times 10^6 \frac{Z}{2}} \]
Here, \( 2.18 \times 10^6 \) and \( Z \) cancel out:
\[ \frac{v_5}{v_2} = \frac{2}{5} \]

5. **Expressing \( v_5 \) in terms of \( V \)**:
Since \( v_2 = V \):
\[ v_5 = \frac{2}{5} V \]

Thus, the velocity of the electron in the fifth orbit is:
\[ v_5 = \frac{2}{5} V \]