With increasing member, the energy difference between adjacent levels in atoms.

Options

A. decreases
B. increases
C. remains constant
D. decreases for low `Z` and increases for high `Z`

Explanation

To solve the question regarding the energy difference between adjacent energy levels in atoms as the principal quantum number (n) increases, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Energy Levels**: The energy of an electron in an atom is quantized and can be described by the formula:
\[ E_n = -\frac{13.6 \, Z^2}{n^2} \]
where \(E_n\) is the energy of the level, \(Z\) is the atomic number, and \(n\) is the principal quantum number.

2. **Energy Difference Between Levels**: The energy difference (\(\Delta E\)) between two adjacent energy levels \(n_1\) and \(n_2\) can be expressed as:
\[ \Delta E = E_{n_2} - E_{n_1} = -\frac{13.6 \, Z^2}{n_2^2} + \frac{13.6 \, Z^2}{n_1^2} \]

3. **Calculating the Difference**: For adjacent levels, let’s assume \(n_2 = n\) and \(n_1 = n-1\):
\[ \Delta E = -\frac{13.6 \, Z^2}{n^2} + \frac{13.6 \, Z^2}{(n-1)^2} \]

4. **Simplifying the Expression**: We can factor out \(13.6 \, Z^2\):
\[ \Delta E = 13.6 \, Z^2 \left( \frac{1}{(n-1)^2} - \frac{1}{n^2} \right) \]

5. **Finding a Common Denominator**: To combine the fractions:
\[ \Delta E = 13.6 \, Z^2 \left( \frac{n^2 - (n-1)^2}{(n-1)^2 n^2} \right) \]
Simplifying the numerator:
\[ n^2 - (n^2 - 2n + 1) = 2n - 1 \]
Thus,
\[ \Delta E = 13.6 \, Z^2 \frac{2n - 1}{(n-1)^2 n^2} \]

6. **Analyzing the Behavior as \(n\) Increases**: As \(n\) increases, the terms \((n-1)^2\) and \(n^2\) increase, leading to a decrease in \(\Delta E\). Therefore, the energy difference between adjacent levels decreases as \(n\) increases.

### Conclusion:
With increasing principal quantum number \(n\), the energy difference between adjacent energy levels in atoms decreases.

---