Whith increasing principal quantum number, the energy difference between adjacent energy levels in H-atom:

Options

A. decreases
B. increases
C. remains constant
D. decreases for low value of Z and increases for higher value of Z.

Answer

To solve the question regarding the energy difference between adjacent energy levels in a hydrogen atom as the principal quantum number increases, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Energy Levels**: The energy levels in a hydrogen atom are quantized and can be described using the principal quantum number (n). The energy of an electron in a hydrogen atom is given by the formula:
\[ E_n = -\frac{R}{n^2} \]
where \( R \) is the Rydberg constant.

2. **Calculating Energy Difference**: The energy difference (\( \Delta E \)) between two adjacent energy levels (say \( n_1 \) and \( n_2 \)) can be calculated using the formula:
\[ \Delta E = E_{n_2} - E_{n_1} = -\frac{R}{n_2^2} + \frac{R}{n_1^2} \]
For adjacent levels, we can assume \( n_2 = n_1 + 1 \).

3. **Substituting Values**: Let’s denote \( n_1 = n \) and \( n_2 = n + 1 \). The energy difference becomes:
\[ \Delta E = -\frac{R}{(n+1)^2} + \frac{R}{n^2} \]

4. **Simplifying the Expression**: To find the energy difference, we can combine the fractions:
\[ \Delta E = R \left( \frac{1}{n^2} - \frac{1}{(n+1)^2} \right) \]
This can be simplified further:
\[ \Delta E = R \left( \frac{(n+1)^2 - n^2}{n^2(n+1)^2} \right) = R \left( \frac{2n + 1}{n^2(n+1)^2} \right) \]

5. **Analyzing the Behavior as \( n \) Increases**: As the principal quantum number \( n \) increases, the term \( n^2(n+1)^2 \) increases significantly, leading to a decrease in \( \Delta E \). Thus, the energy difference between adjacent levels decreases with increasing \( n \).

6. **Conclusion**: Therefore, the energy difference between adjacent energy levels in a hydrogen atom decreases as the principal quantum number increases.

### Final Answer:
The energy difference between adjacent energy levels in a hydrogen atom decreases with increasing principal quantum number.