The correct order of energy difference between adjacent energy levels in H - atom .

Options

A. `E_(2) - E_(1) gt E_(3) - E_(2) gt E_(4) - E_(3)`
B. `E_(2)-E_(1) gt E_(4) - E_3 gt - E_(2)`
C. `E_(4) - E_(3) gt E_(3) - E_(2) gt E_(2) - E_(1)`
D. `E_(3) - E_(2) gt E_(4) - E_(3) gt E_(2) - E_(1)`

Answer

To determine the correct order of energy differences between adjacent energy levels in a hydrogen atom, we will 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 \( Z \) is the atomic number (for hydrogen, \( Z = 1 \)) and \( n \) is the principal quantum number.

### Step 2: Calculate Energy Levels
We will calculate the energy levels for \( n = 1, 2, 3, \) and \( 4 \):

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

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

- 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 Energy Differences
Now, we will calculate the energy differences between adjacent levels:

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

- Difference between \( E_3 \) and \( E_2 \):
\[ E_3 - E_2 = -1.51 - (-3.4) = 1.89 \text{ eV} \]

- Difference between \( E_4 \) and \( E_3 \):
\[ E_4 - E_3 = -0.85 - (-1.51) = 0.66 \text{ eV} \]

### Step 4: Order the Energy Differences
Now we can order the calculated energy differences:
1. \( E_2 - E_1 = 10.2 \text{ eV} \)
2. \( E_3 - E_2 = 1.89 \text{ eV} \)
3. \( E_4 - E_3 = 0.66 \text{ eV} \)

Thus, the order of energy differences between adjacent energy levels in a hydrogen atom is:
\[ E_2 - E_1 > E_3 - E_2 > E_4 - E_3 \]

### Final Answer
The correct order of energy differences between adjacent energy levels in the hydrogen atom is:
\[ 10.2 \text{ eV} > 1.89 \text{ eV} > 0.66 \text{ eV} \]