How do the energy gaps between successive electron energy levels in an atom very from low to high n values ?

Options

A. All energy gaps are the same
B. The energy gap decreases as n increases
C. The energy gap increases as n increases
D. the energy gap changes unpredictably as n increases

Answer

To understand how the energy gaps between successive electron energy levels in an atom vary from low to high n values, we can break down the explanation into several steps:

### Step-by-Step Solution:

1. **Understanding Energy Levels**:
- In an atom, electrons occupy different energy levels, which are quantized. These energy levels are denoted by the principal quantum number \( n \), where \( n = 1, 2, 3, \ldots \).

**Hint**: Recall that each energy level corresponds to a specific distance from the nucleus and has a specific energy associated with it.

2. **Energy Gap Definition**:
- The energy gap refers to the difference in energy between two successive energy levels, for example, between \( n=1 \) and \( n=2 \), or \( n=2 \) and \( n=3 \).

**Hint**: Think of the energy gap as the energy required for an electron to jump from one level to the next.

3. **Trend with Increasing n**:
- As the principal quantum number \( n \) increases, the energy levels become more closely spaced. This means that the energy gap between successive levels decreases.

**Hint**: Consider how the electron's distance from the nucleus affects its energy; farther electrons experience less attraction and thus have lower energy differences.

4. **Effective Nuclear Charge (Z effective)**:
- The effective nuclear charge (\( Z_{\text{effective}} \)) is the net positive charge experienced by an electron in a multi-electron atom. As \( n \) increases, \( Z_{\text{effective}} \) for the valence shell also increases.

**Hint**: Remember that as more shells are added, the inner electrons shield the outer electrons from the full charge of the nucleus, affecting their energy levels.

5. **Relationship Between n, Z effective, and Energy Gap**:
- The relationship can be summarized as follows: as \( n \) increases, \( Z_{\text{effective}} \) increases, leading to a decrease in the energy gap. Thus, we can say that \( n \) is directly proportional to \( Z_{\text{effective}} \) and inversely proportional to the energy gap.

**Hint**: Consider how the attraction between the nucleus and the outer electrons changes as you add more energy levels.

6. **Conclusion**:
- Therefore, we conclude that the energy gap between successive electron energy levels decreases as the principal quantum number \( n \) increases.

**Hint**: This trend can be observed in the hydrogen atom and can be generalized to other atoms as well.

### Final Answer:
The energy gaps between successive electron energy levels in an atom decrease as the principal quantum number \( n \) increases.