If each hydrogen atom in the ground state `1.0 mol of H ` atom are excited by axeited by absorbing photon of energy `8.4 eV, 12.09 eV and 15.0 eV` of energy, then number of spectral lines emitted is equal to

Options

A. none
B. two
C. three
D. four

Answer

To solve the problem of how many spectral lines are emitted when hydrogen atoms are excited by absorbing photons of given energies, we can follow these steps:

### Step 1: Identify the Energy Levels of Hydrogen
Using the Bohr model, the energy levels of a hydrogen atom are given by the formula:
\[ E_n = -\frac{13.6 \, \text{eV}}{n^2} \]
where \( n \) is the principal quantum number.

- For \( n = 1 \):
\[ E_1 = -13.6 \, \text{eV} \]
- For \( n = 2 \):
\[ E_2 = -3.4 \, \text{eV} \]
- For \( n = 3 \):
\[ E_3 = -1.51 \, \text{eV} \]
- For \( n = 4 \):
\[ E_4 = -0.85 \, \text{eV} \]

### Step 2: Calculate the Energy Differences for Transitions
We need to find the energy differences corresponding to the transitions between these levels:

- Transition from \( n = 1 \) to \( n = 2 \):
\[ E_2 - E_1 = -3.4 - (-13.6) = 10.2 \, \text{eV} \]

- Transition from \( n = 1 \) to \( n = 3 \):
\[ E_3 - E_1 = -1.51 - (-13.6) = 12.09 \, \text{eV} \]

- Transition from \( n = 1 \) to \( n = 4 \):
\[ E_4 - E_1 = -0.85 - (-13.6) = 12.75 \, \text{eV} \]

### Step 3: Analyze the Given Energies
The energies given in the problem are:
- 8.4 eV
- 12.09 eV
- 15.0 eV

Now we analyze which of these energies correspond to the transitions:

- **8.4 eV**: This energy does not correspond to any transition between the calculated levels.
- **12.09 eV**: This energy corresponds to the transition from \( n = 1 \) to \( n = 3 \).
- **15.0 eV**: This energy is greater than 13.6 eV, which means it would ionize the hydrogen atom and no spectral lines would be emitted.

### Step 4: Determine the Number of Spectral Lines
Since only the energy of 12.09 eV corresponds to a valid transition (from \( n = 1 \) to \( n = 3 \)), we can calculate the number of spectral lines emitted from this transition.

The formula to calculate the number of spectral lines emitted for transitions between energy levels is:
\[ \text{Number of lines} = \frac{(n_2 - n_1)(n_2 - n_1 + 1)}{2} \]

Here, \( n_1 = 1 \) and \( n_2 = 3 \):
\[ \text{Number of lines} = \frac{(3 - 1)(3 - 1 + 1)}{2} = \frac{2 \times 3}{2} = 3 \]

### Final Answer
The number of spectral lines emitted is **3**.

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