What is the energy associated with `3^(rd)` energy shell of hydrgoen atom ?

Options

A. `- 2.18 xx 10^(-18) J`
B. ` - 0.342 xx 10^(-19) J`
C. ` -0. 726 xx 10^(-18) J`
D. ` - 2.42 xx 10^(-19) J`

Answer

To find the energy associated with the third energy shell of a hydrogen atom, we can use the formula for the energy levels of hydrogen-like atoms:

### Step-by-Step Solution:

1. **Identify the Formula**: The energy \( E \) associated with an electron in a hydrogen-like atom is given by:
\[ E = -\frac{13.6 \, Z^2}{n^2} \, \text{eV} \]
or in joules:
\[ E = -2.18 \times 10^{-18} \frac{Z^2}{n^2} \, \text{J} \]
where \( Z \) is the atomic number and \( n \) is the principal quantum number (energy shell).

2. **Substitute Values for Hydrogen**: For a hydrogen atom, the atomic number \( Z = 1 \). We are interested in the third energy shell, so \( n = 3 \).

3. **Plug in the Values**:
\[ E = -2.18 \times 10^{-18} \frac{1^2}{3^2} \, \text{J} \]

4. **Calculate \( n^2 \)**:
\[ n^2 = 3^2 = 9 \]

5. **Calculate the Energy**:
\[ E = -2.18 \times 10^{-18} \frac{1}{9} \]
\[ E = -2.18 \times 10^{-18} \times 0.1111 \, \text{J} \]
\[ E = -0.242 \times 10^{-18} \, \text{J} \]

6. **Convert to Standard Form**:
\[ E = -2.42 \times 10^{-19} \, \text{J} \]

7. **Final Result**: The energy associated with the third energy shell of the hydrogen atom is:
\[ E = -2.42 \times 10^{-19} \, \text{J} \]