As the n (number of orbit) increases, the difference of energy between the consecutive energy levels

Options

A. Remain the same
B. Increases
C. Decreases
D. Sometimes increases and sometimes decreases

Answer

To solve the question regarding how the difference of energy between consecutive energy levels changes as the principal quantum number \( n \) increases, we will follow these steps:

### Step 1: Understand the Energy Formula
The energy of an electron in the nth orbit of a hydrogen-like atom is given by the formula:
\[ E_n = -\frac{13.6 Z^2}{n^2} \text{ eV} \]
where \( Z \) is the atomic number (for hydrogen, \( Z = 1 \)).

### Step 2: Calculate Energy Levels
We will calculate the energy for the first four energy levels (orbits) using the formula.

- For \( n = 1 \):
\[ E_1 = -\frac{13.6 \times 1^2}{1^2} = -13.6 \text{ eV} \]

- For \( n = 2 \):
\[ E_2 = -\frac{13.6 \times 1^2}{2^2} = -\frac{13.6}{4} = -3.4 \text{ eV} \]

- For \( n = 3 \):
\[ E_3 = -\frac{13.6 \times 1^2}{3^2} = -\frac{13.6}{9} \approx -1.51 \text{ eV} \]

- For \( n = 4 \):
\[ E_4 = -\frac{13.6 \times 1^2}{4^2} = -\frac{13.6}{16} \approx -0.85 \text{ eV} \]

### Step 3: Calculate Differences in Energy Levels
Now, we will calculate the differences in energy between consecutive levels:

- Difference between \( E_1 \) and \( E_2 \):
\[ \Delta E_{1 \to 2} = E_2 - E_1 = -3.4 - (-13.6) = 10.2 \text{ eV} \]

- Difference between \( E_2 \) and \( E_3 \):
\[ \Delta E_{2 \to 3} = E_3 - E_2 = -1.51 - (-3.4) \approx 1.89 \text{ eV} \]

- Difference between \( E_3 \) and \( E_4 \):
\[ \Delta E_{3 \to 4} = E_4 - E_3 = -0.85 - (-1.51) \approx 0.66 \text{ eV} \]

### Step 4: Analyze the Differences
From the calculations:
- \( \Delta E_{1 \to 2} = 10.2 \text{ eV} \)
- \( \Delta E_{2 \to 3} \approx 1.89 \text{ eV} \)
- \( \Delta E_{3 \to 4} \approx 0.66 \text{ eV} \)

As \( n \) increases, the differences in energy between consecutive levels decrease.

### Conclusion
Thus, as the principal quantum number \( n \) increases, the difference of energy between consecutive energy levels decreases.

### Final Answer
The correct answer is that the difference of energy between consecutive energy levels decreases as \( n \) increases.

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