Step 1: Understanding the Concept:
Bohr's Atomic Theory postulates that electrons orbit the nucleus in specific energy levels or "shells".
The energy of an electron in a given shell \( n \) for a hydrogen atom is quantized.
Critically, the energy is inversely proportional to the square of the principal quantum number \( n \).
This means as an electron moves further from the nucleus (larger \( n \)), its energy increases (becomes less negative).
Step 2: Key Formula or Approach:
The formula for energy in the \( n \)-th orbit is:
\[ E_n = -\frac{13.6 Z^2}{n^2} \text{ eV} \]
For comparison between two orbits of the same atom:
\[ \frac{E_1}{E_2} = \frac{n_2^2}{n_1^2} \implies E_n \propto \frac{1}{n^2} \]
Step 3: Detailed Explanation:
We are given the energy of the second orbit \( (n = 2) \):
\[ E_2 = -328 \text{ kJ mol}^{-1} \]
We need to find the energy of the fourth orbit \( (n = 4) \).
Apply the inverse square relationship:
\[ \frac{E_4}{E_2} = \frac{2^2}{4^2} \]
\[ \frac{E_4}{E_2} = \frac{4}{16} = \frac{1}{4} \]
So, the energy of the fourth orbit is one-fourth the energy of the second orbit:
\[ E_4 = \frac{E_2}{4} \]
Substitute the value of \( E_2 \):
\[ E_4 = \frac{-328}{4} \]
\[ E_4 = -82 \text{ kJ mol}^{-1} \]
The energy is less negative than in the second orbit, which is consistent with the physics of atomic structures.
Step 4: Final Answer:
The energy of the fourth Bohr orbit is \( -82 \text{ kJ/mol} \).
This is Option (b).