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_(3) - 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 can follow these steps:

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

### Step 2: Calculate Energy Difference Between Levels
The energy difference between two adjacent levels \( n_2 \) and \( n_1 \) can be calculated using:
\[ \Delta E = E_{n_2} - E_{n_1} = -\frac{13.6}{n_2^2} + \frac{13.6}{n_1^2} \]
This can be simplified to:
\[ \Delta E = -13.6 \left( \frac{1}{n_2^2} - \frac{1}{n_1^2} \right) \]

### Step 3: Calculate \( E_2 - E_1 \)
For \( n_2 = 2 \) and \( n_1 = 1 \):
\[ \Delta E_{2-1} = -13.6 \left( \frac{1}{2^2} - \frac{1}{1^2} \right) \]
\[ = -13.6 \left( \frac{1}{4} - 1 \right) \]
\[ = -13.6 \left( -\frac{3}{4} \right) \]
\[ = 10.2 \text{ eV} \]

### Step 4: Calculate \( E_3 - E_2 \)
For \( n_2 = 3 \) and \( n_1 = 2 \):
\[ \Delta E_{3-2} = -13.6 \left( \frac{1}{3^2} - \frac{1}{2^2} \right) \]
\[ = -13.6 \left( \frac{1}{9} - \frac{1}{4} \right) \]
\[ = -13.6 \left( \frac{4 - 9}{36} \right) \]
\[ = -13.6 \left( -\frac{5}{36} \right) \]
\[ = 1.89 \text{ eV} \]

### Step 5: Calculate \( E_4 - E_3 \)
For \( n_2 = 4 \) and \( n_1 = 3 \):
\[ \Delta E_{4-3} = -13.6 \left( \frac{1}{4^2} - \frac{1}{3^2} \right) \]
\[ = -13.6 \left( \frac{1}{16} - \frac{1}{9} \right) \]
\[ = -13.6 \left( \frac{9 - 16}{144} \right) \]
\[ = -13.6 \left( -\frac{7}{144} \right) \]
\[ = 0.66 \text{ eV} \]

### Step 6: Compare the Energy Differences
Now we have:
- \( \Delta E_{2-1} = 10.2 \text{ eV} \)
- \( \Delta E_{3-2} = 1.89 \text{ eV} \)
- \( \Delta E_{4-3} = 0.66 \text{ eV} \)

### Step 7: Order the Energy Differences
The order of energy differences from highest to lowest is:
\[ \Delta E_{2-1} > \Delta E_{3-2} > \Delta E_{4-3} \]
Thus, the correct order is:
\[ 10.2 \text{ eV} > 1.89 \text{ eV} > 0.66 \text{ eV} \]

### Final Answer
The correct order of energy difference between adjacent energy levels in a hydrogen atom is:
\[ E_2 - E_1 > E_3 - E_2 > E_4 - E_3 \]