Question

The energy of second Bohr orbit of the hydrogen atom is \(-328 \, \text{kJ mol}^{-1}\), hence the energy of fourth Bohr orbit would be

• A. \(-41 \, \text{kJ mol}^{-1}\)
• B. \(-1312 \, \text{kJ mol}^{-1}\)
• C. \(-164 \, \text{kJ mol}^{-1}\)
• D. \(-82 \, \text{kJ mol}^{-1}\)

Hint:

The energy of the nth orbit is inversely proportional to n\(^2\). As n increases, the energy becomes less negative (increases).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The energy of an electron in the nth orbit of a hydrogen atom is given by \(E_n = -\frac{1312}{n^2} \, \text{kJ mol}^{-1}\) or generally \(E_n \propto -\frac{1}{n^2}\).
Step 2: Detailed Explanation:}}
Given: \(E_2 = -328 \, \text{kJ mol}^{-1}\). For hydrogen, \(E_n = \frac{E_1}{n^2}\). So, \(E_2 = \frac{E_1}{4} = -328 \Rightarrow E_1 = -1312 \, \text{kJ mol}^{-1}\). Then, \(E_4 = \frac{E_1}{16} = \frac{-1312}{16} = -82 \, \text{kJ mol}^{-1}\). Alternatively, since \(E_n \propto \frac{1}{n^2}\), \(\frac{E_4}{E_2} = \frac{2^2}{4^2} = \frac{4}{16} = \frac{1}{4}\). \(E_4 = \frac{1}{4} \times (-328) = -82 \, \text{kJ mol}^{-1}\).
Step 3: Final Answer:
The energy of the fourth Bohr orbit is \(-82 \, \text{kJ mol}^{-1}\), option (D).