A sample consisting of Hydrogen atoms in the ground state is excited by monochromatic radiation of energy 12.75 eV. If we were to observe the emission spectrum of this sample, then the number of spectral lines observed, will be

Options

A. 3
B. 6
C. 10
D. 15

Answer

To solve the problem of how many spectral lines will be observed when a sample of hydrogen atoms in the ground state is excited by monochromatic radiation of energy 12.75 eV, we can follow these steps:

### Step 1: Determine the energy levels of the hydrogen atom
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.

### Step 2: Calculate the energy of the ground state
For the ground state (\( n = 1 \)):
\[ E_1 = -\frac{13.6 \, \text{eV}}{1^2} = -13.6 \, \text{eV} \]

### Step 3: Calculate the energy of the excited state
The energy of the radiation used for excitation is 12.75 eV. Therefore, the total energy of the hydrogen atom after excitation is:
\[ E_{\text{excited}} = E_1 + 12.75 \, \text{eV} = -13.6 \, \text{eV} + 12.75 \, \text{eV} = -0.85 \, \text{eV} \]

### Step 4: Identify the principal quantum number of the excited state
Now, we need to find the principal quantum number \( n \) for which the energy is -0.85 eV:
\[ -\frac{13.6 \, \text{eV}}{n^2} = -0.85 \, \text{eV} \]
\[ \frac{13.6}{n^2} = 0.85 \]
\[ n^2 = \frac{13.6}{0.85} \]
\[ n^2 \approx 16 \]
\[ n \approx 4 \]

Thus, the hydrogen atom is excited to the \( n = 4 \) state.

### Step 5: Determine possible transitions
The possible transitions from the excited state (n=4) to lower energy states are:
- \( n = 4 \) 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 6: Count the number of spectral lines
The number of spectral lines corresponds to the number of unique transitions. The transitions can be counted as follows:
- From \( n = 4 \): 3 transitions (to \( n = 3, 2, 1 \))
- From \( n = 3 \): 2 transitions (to \( n = 2, 1 \))
- From \( n = 2 \): 1 transition (to \( n = 1 \))

The total number of transitions is:
\[ 3 + 2 + 1 = 6 \]

### Conclusion
Thus, the number of spectral lines observed will be **6**.

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