In a sample of H atom , make transition from `n = 5 to n = 1 ` If all the spectral lines are observed , then the line having the third highest energy will corresponding to

Options

A. `5 rarr 3`
B. `4 rarr 1`
C. `3 rarr 1`
D. `5 rarr 4`

Explanation

To solve the problem of determining which transition corresponds to the third highest energy when a hydrogen atom transitions from \( n = 5 \) to \( n = 1 \), we will follow these steps:

### Step 1: Identify Possible Transitions
When an electron in a hydrogen atom transitions from a higher energy level to a lower one, it emits energy in the form of light. The possible transitions from \( n = 5 \) to lower energy levels are:
- \( n = 5 \) to \( n = 4 \)
- \( n = 5 \) to \( n = 3 \)
- \( n = 5 \) to \( n = 2 \)
- \( n = 5 \) to \( n = 1 \)
- \( n = 4 \) to \( n = 3 \)
- \( n = 4 \) to \( n = 2 \)
- \( n = 4 \) to \( n = 1 \)
- \( n = 3 \) to \( n = 2 \)
- \( n = 3 \) to \( n = 1 \)
- \( n = 2 \) to \( n = 1 \)

### Step 2: Calculate Energy Levels
The energy of each level can be calculated using the formula:
\[ E_n = -\frac{13.6 \, \text{eV}}{n^2} \]
Calculating the energies for \( n = 1, 2, 3, 4, 5 \):
- \( E_1 = -13.6 \, \text{eV} \)
- \( E_2 = -3.4 \, \text{eV} \)
- \( E_3 = -1.51 \, \text{eV} \)
- \( E_4 = -0.85 \, \text{eV} \)
- \( E_5 = -0.54 \, \text{eV} \)

### Step 3: Calculate Energy Differences for Each Transition
Now we will calculate the energy differences for each transition:
1. \( n = 5 \) to \( n = 1 \):
\[ E = E_1 - E_5 = (-13.6) - (-0.54) = -13.06 \, \text{eV} \]
2. \( n = 5 \) to \( n = 2 \):
\[ E = E_2 - E_5 = (-3.4) - (-0.54) = -2.86 \, \text{eV} \]
3. \( n = 5 \) to \( n = 3 \):
\[ E = E_3 - E_5 = (-1.51) - (-0.54) = -0.97 \, \text{eV} \]
4. \( n = 4 \) to \( n = 1 \):
\[ E = E_1 - E_4 = (-13.6) - (-0.85) = -12.75 \, \text{eV} \]
5. \( n = 4 \) to \( n = 2 \):
\[ E = E_2 - E_4 = (-3.4) - (-0.85) = -2.55 \, \text{eV} \]
6. \( n = 4 \) to \( n = 3 \):
\[ E = E_3 - E_4 = (-1.51) - (-0.85) = -0.66 \, \text{eV} \]
7. \( n = 3 \) to \( n = 1 \):
\[ E = E_1 - E_3 = (-13.6) - (-1.51) = -12.09 \, \text{eV} \]
8. \( n = 3 \) to \( n = 2 \):
\[ E = E_2 - E_3 = (-3.4) - (-1.51) = -1.89 \, \text{eV} \]
9. \( n = 2 \) to \( n = 1 \):
\[ E = E_1 - E_2 = (-13.6) - (-3.4) = -10.2 \, \text{eV} \]

### Step 4: Rank the Energy Differences
Now we will rank the energy differences from highest to lowest:
1. \( n = 5 \) to \( n = 1 \): \( 13.06 \, \text{eV} \)
2. \( n = 4 \) to \( n = 1 \): \( 12.75 \, \text{eV} \)
3. \( n = 3 \) to \( n = 1 \): \( 12.09 \, \text{eV} \)
4. \( n = 5 \) to \( n = 2 \): \( 2.86 \, \text{eV} \)
5. \( n = 4 \) to \( n = 2 \): \( 2.55 \, \text{eV} \)
6. \( n = 3 \) to \( n = 2 \): \( 1.89 \, \text{eV} \)
7. \( n = 5 \) to \( n = 3 \): \( 0.97 \, \text{eV} \)
8. \( n = 4 \) to \( n = 3 \): \( 0.66 \, \text{eV} \)
9. \( n = 2 \) to \( n = 1 \): \( 10.2 \, \text{eV} \)

### Conclusion
The transition with the third highest energy corresponds to \( n = 3 \) to \( n = 1 \).