Question

As the quantum number increases, the difference in energy between consecutive energy levels:

• A. Remains the same
• B. Increases
• C. Decreases
• D. Sometimes increases and sometimes decreases

Hint:

Energy levels get closer as quantum number increases; higher levels are more densely spaced.

Answer 1

The Correct Option is C

Step 1: Write the level spacing as a difference of squares.
For $E_n = -\dfrac{13.6}{n^2}\ \text{eV}$, the gap between consecutive levels is
\[ \Delta E_n = 13.6\left(\frac{1}{n^2}-\frac{1}{(n+1)^2}\right)\ \text{eV} \]
Step 2: Check the trend with actual numbers instead of just trusting the formula.
\[ \Delta E_1 = 13.6\left(1-\frac14\right) = 10.2\ \text{eV}, \quad \Delta E_2 \approx 1.89\ \text{eV}, \quad \Delta E_3 \approx 0.66\ \text{eV} \]
Step 3: Read off the pattern.
The gaps shrink rapidly, $10.2 \to 1.89 \to 0.66\ \text{eV}$, and keep shrinking as $n$ grows, since the levels get crowded closer and closer to the zero-energy limit at high $n$.
\[ \boxed{\text{Decreases}} \]