If energy of electron in ground state is `-13.6` then find out speed of electron in fourth orbit of H-atom

Options

A. `5.45xx10^(6)m//s`
B. `5.45xx10^(5)m//s`
C. `5.45xx10^(4)m//s`
D. `4.45xx10^(5)m//s`

Explanation

To find the speed of an electron in the fourth orbit of a hydrogen atom, we can use the formula derived from Bohr's model of the hydrogen atom. The speed of an electron in the nth orbit is given by:

\[ v_n = \frac{2.18 \times 10^6 \, \text{m/s} \cdot Z}{n} \]

Where:
- \( v_n \) is the speed of the electron in the nth orbit,
- \( Z \) is the atomic number of the element (for hydrogen, \( Z = 1 \)),
- \( n \) is the principal quantum number (the orbit number).

### Step-by-step Solution:

1. **Identify the values**:
- For hydrogen, the atomic number \( Z = 1 \).
- We need to find the speed in the fourth orbit, so \( n = 4 \).

2. **Substitute the values into the formula**:
\[ v_4 = \frac{2.18 \times 10^6 \, \text{m/s} \cdot 1}{4} \]

3. **Calculate the speed**:
\[ v_4 = \frac{2.18 \times 10^6}{4} = 0.545 \times 10^6 \, \text{m/s} \]
\[ v_4 = 5.45 \times 10^5 \, \text{m/s} \]

4. **Final answer**:
The speed of the electron in the fourth orbit of the hydrogen atom is \( 5.45 \times 10^5 \, \text{m/s} \).