With the increase in peinciple quantum number, the energy difference between the two successive energy levels

Options

A. decrease
B. increase
C. first decrease and then increase
D. remain the same

Explanation

To solve the question regarding the energy difference between two successive energy levels as the principal quantum number increases, we can follow these steps:

### Step 1: Understanding Energy Levels
The energy levels of an electron in an atom are quantized and are described by the principal quantum number \( n \). The energy of an electron in a hydrogen-like atom can be calculated using the formula:
\[ E_n = -\frac{13.6 \, \text{eV} \cdot Z^2}{n^2} \]
where \( Z \) is the atomic number and \( n \) is the principal quantum number.

### Step 2: Calculate Energy for Successive Levels
Let's calculate the energy for two successive quantum levels, \( n \) and \( n+1 \):
- For \( n \):
\[ E_n = -\frac{13.6 \, \text{eV} \cdot Z^2}{n^2} \]
- For \( n+1 \):
\[ E_{n+1} = -\frac{13.6 \, \text{eV} \cdot Z^2}{(n+1)^2} \]

### Step 3: Determine the Energy Difference
The energy difference \( \Delta E \) between these two levels is given by:
\[ \Delta E = E_{n+1} - E_n \]
Substituting the values we calculated:
\[ \Delta E = -\frac{13.6 \, \text{eV} \cdot Z^2}{(n+1)^2} + \frac{13.6 \, \text{eV} \cdot Z^2}{n^2} \]
This can be simplified to:
\[ \Delta E = 13.6 \, \text{eV} \cdot Z^2 \left( \frac{1}{n^2} - \frac{1}{(n+1)^2} \right) \]

### Step 4: Analyze the Behavior as \( n \) Increases
As \( n \) increases, the difference \( \Delta E \) can be analyzed further. The term \( \frac{1}{n^2} - \frac{1}{(n+1)^2} \) can be simplified:
\[ \frac{1}{n^2} - \frac{1}{(n+1)^2} = \frac{(n+1)^2 - n^2}{n^2(n+1)^2} = \frac{2n + 1}{n^2(n+1)^2} \]
This shows that as \( n \) increases, \( \Delta E \) decreases because the denominator grows faster than the numerator.

### Conclusion
Thus, with the increase in the principal quantum number \( n \), the energy difference \( \Delta E \) between two successive energy levels decreases.

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