Which of the following transitions of electrons in the hydrogen atom will emit maximum energy

Options

A. `n_5 to n_4`
B. `n_4 to n_3`
C. `n_3 to n_2`
D. energy is same in all A,B and C

Explanation

To determine which transition of electrons in the hydrogen atom will emit the maximum energy, we can follow these steps:

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

### Step 2: Identify the Given Transitions
We have three transitions to consider:
1. Transition from \( n = 5 \) to \( n = 4 \)
2. Transition from \( n = 4 \) to \( n = 3 \)
3. Transition from \( n = 3 \) to \( n = 2 \)

### Step 3: Calculate the Energy Change for Each Transition
We will calculate the energy emitted for each transition using the energy formula.

1. **For the transition \( n = 5 \) to \( n = 4 \)**:
\[ \Delta E_{5 \to 4} = E_4 - E_5 = \left(-\frac{13.6}{4^2}\right) - \left(-\frac{13.6}{5^2}\right) \]
\[ = -\frac{13.6}{16} + \frac{13.6}{25} = -0.85 + 0.544 = -0.306 \, \text{eV} \]

2. **For the transition \( n = 4 \) to \( n = 3 \)**:
\[ \Delta E_{4 \to 3} = E_3 - E_4 = \left(-\frac{13.6}{3^2}\right) - \left(-\frac{13.6}{4^2}\right) \]
\[ = -\frac{13.6}{9} + \frac{13.6}{16} = -1.51 + 0.85 = -0.66 \, \text{eV} \]

3. **For the transition \( n = 3 \) to \( n = 2 \)**:
\[ \Delta E_{3 \to 2} = E_2 - E_3 = \left(-\frac{13.6}{2^2}\right) - \left(-\frac{13.6}{3^2}\right) \]
\[ = -\frac{13.6}{4} + \frac{13.6}{9} = -3.4 + 1.51 = -1.89 \, \text{eV} \]

### Step 4: Compare the Energy Changes
Now we compare the magnitudes of the energy changes:
- \( |\Delta E_{5 \to 4}| = 0.306 \, \text{eV} \)
- \( |\Delta E_{4 \to 3}| = 0.66 \, \text{eV} \)
- \( |\Delta E_{3 \to 2}| = 1.89 \, \text{eV} \)

### Conclusion
The transition from \( n = 3 \) to \( n = 2 \) emits the maximum energy of \( 1.89 \, \text{eV} \). Therefore, the correct answer is the transition from \( n = 3 \) to \( n = 2 \).