Question

The energy of the second bohr orbit of the hydrogen atom is $-328 \, \text{kJ mol}^{-1}$; hence the energy of the fourth bohr orbit would be:

• A. $-41 \, \text{kJ/mol}$
• B. $-82 \, \text{kJ/mol}$
• C. $-164 \, \text{kJ/mol}$
• D. $-1312 \, \text{kJ/mol}$

Hint:

To solve these quickly: if the orbit number doubles (from 2 to 4), the energy becomes $\frac{1}{2^2} = \frac{1}{4}$ of its previous value. Conversely, if the orbit number triples, the energy becomes $\frac{1}{9}$ of the original value. Always remember that energy levels get closer together as $n$ increases.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Electron energy in Bohr orbits is negative, indicating a bound state. As an electron moves further from the nucleus (higher \( n \)), the energy becomes less negative (approaching zero).
Key Formula or Approach:
For a specific atom, energy \( E_n \) is inversely proportional to the square of the orbit number:
\[ E_n \propto \frac{1}{n^2} \]
The proportionality holds for the same element across different orbits.
Step 2: Detailed Explanation:
We are given \( E_2 = -328 \, \text{kJ mol}^{-1} \).
We need to find \( E_4 \).
Using the ratio:
\[ \frac{E_4}{E_2} = \frac{2^2}{4^2} = \frac{4}{16} = \frac{1}{4} \]
\[ E_4 = E_2 \times \frac{1}{4} = \frac{-328}{4} = -82 \, \text{kJ mol}^{-1} \]
As \( n \) increases from 2 to 4, the energy increases from -328 to -82 (it becomes less negative).
Step 3: Final Answer:
The energy of the fourth orbit is \( -82 \, \text{kJ/mol} \).