Question

Which of the following statements regarding the energy of the stationary state is true in the following one-electron systems ?

• A. +8.72 × 10⁻¹⁸ J for first orbit of He⁺ ion
• B. +2.18 × 10⁻¹⁸ J for second orbit of He⁺ ion
• C. -2.18 × 10⁻¹⁸ J for third orbit of Li²⁺ ion
• D. -1.09 × 10⁻¹⁸ J for second orbit of H atom.

Hint:

If $Z = n$, the energy of that orbit is always exactly equal to the energy of the first orbit of Hydrogen ($-2.18 \times 10^{-18}$ J).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The energy of a stationary state in a hydrogen-like atom is given by Bohr's formula.
Step 2: Key Formula or Approach:
$$E_n = -2.18 \times 10^{-18} \left( \frac{Z^2}{n^2} \right) \text{ J/atom}$$ Note: Energy must be negative for a bound electron.
Step 3: Detailed Explanation:
- Option 1 \& 2: Eliminate immediately because energy is positive. - Option 3 ($Li^{2+}$): $Z = 3, n = 3$. $$E = -2.18 \times 10^{-18} \left( \frac{3^2}{3^2} \right) = -2.18 \times 10^{-18} \text{ J}$$ (True). - Option 4 ($H$): $Z = 1, n = 2$. $$E = -2.18 \times 10^{-18} \left( \frac{1^2}{2^2} \right) = -0.545 \times 10^{-18} \text{ J}$$ (False).
Step 4: Final Answer:
Statement 3 is the only true calculation.