Question

How does the energy difference between the consecutive energy level vary as the quantum number n increases?

• A. Increases
• B. Decreases
• C. Remains unchanged
• D. First increases and then decreases

Hint:

Energy levels get closer as $n$ increases.

Answer 1

The Correct Option is B

To understand how the energy difference between consecutive energy levels varies as the quantum number \( n \) increases, we need to consider the formula for the energy levels in an atom, particularly the hydrogen atom or hydrogen-like ions:

\(E_n = -\frac{R_H}{n^2}\)

Here, \( R_H \) is the Rydberg constant and \( n \) is the principal quantum number. This equation describes the energy of an electron at a particular energy level \( n \).

To understand how the energy difference changes between consecutive levels, say between level \( n \) and \( n+1 \), consider the change in energy:

\(\Delta E = E_{n+1} - E_n = -\frac{R_H}{(n+1)^2} + \frac{R_H}{n^2}\)

Let's simplify this expression:

\(\Delta E = R_H \left(\frac{1}{n^2} - \frac{1}{(n+1)^2}\right)\)

\(\Delta E = R_H \left(\frac{(n+1)^2 - n^2}{n^2(n+1)^2}\right)\)

\(\Delta E = R_H \left(\frac{(n^2 + 2n + 1) - n^2}{n^2(n+1)^2}\right)\)

\(\Delta E = R_H \left(\frac{2n + 1}{n^2(n+1)^2}\right)\)

As \( n \) increases, the denominator \( n^2(n+1)^2 \) increases faster than the numerator \( 2n + 1 \), causing the value of the energy difference \( \Delta E \) to decrease. Hence, the energy difference between consecutive energy levels decreases as the quantum number \( n \) increases.

Therefore, the correct answer is:

• Decreases

Explanation of Incorrect Options:

• Increases: As shown, the energy difference actually decreases, not increases.
• Remains unchanged: The energy difference changes with increasing \( n \), hence it does not remain unchanged.
• First increases and then decreases: From the derived formula, the energy difference continuously decreases with increasing \( n \).