As the quantum number increases, the difference of energy between consecutive energy leveles

Options

A. Decreases
B. Remains the same
C. Increases
D. Some tines increases and sometines decreases

Answer

To solve the question regarding the behavior of energy differences between consecutive energy levels as the quantum number increases, we can follow these steps:

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

### Step 2: Calculate Energy Levels
Let's calculate the energy levels for \(n = 1, 2, 3, 4\):
- For \(n = 1\):
\[ E_1 = -\frac{13.6 \times 1^2}{1^2} = -13.6 \, \text{eV} \]
- For \(n = 2\):
\[ E_2 = -\frac{13.6 \times 1^2}{2^2} = -3.4 \, \text{eV} \]
- For \(n = 3\):
\[ E_3 = -\frac{13.6 \times 1^2}{3^2} = -1.51 \, \text{eV} \]
- For \(n = 4\):
\[ E_4 = -\frac{13.6 \times 1^2}{4^2} = -0.85 \, \text{eV} \]

### Step 3: Calculate Differences in Energy Levels
Now, we will calculate the differences in energy between consecutive energy 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 \(E_3\):
\[ \Delta E_{2 \to 3} = E_3 - E_2 = -1.51 - (-3.4) = 1.89 \, \text{eV} \]
- Difference between \(E_3\) and \(E_4\):
\[ \Delta E_{3 \to 4} = E_4 - E_3 = -0.85 - (-1.51) = 0.66 \, \text{eV} \]

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

We can observe that as the quantum number \(n\) increases, the energy differences between consecutive levels decrease.

### Conclusion
Therefore, as the quantum number increases, the difference of energy between consecutive energy levels decreases.