Ionisation potential of hydrogen atom is `13.6 eV`. Hydrogen atom in ground state is excited by monochromatic light of energy `12.1 eV`. The spectral lines emitted by hydrogen according to Bohr's theory will be

Options

A. one
B. two
C. three
D. four

Answer

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

### Step 1: Understand Ionization Potential
The ionization potential of the hydrogen atom is given as 13.6 eV. This value represents the energy required to remove an electron from the ground state of the hydrogen atom.

### Step 2: Determine Initial Energy Level
The energy of the ground state of hydrogen (n=1) can be expressed as:
\[ E_1 = -13.6 \, \text{eV} \]

### Step 3: Calculate Energy After Excitation
The hydrogen atom is excited by monochromatic light of energy 12.1 eV. The energy of the excited state can be calculated as follows:
\[ E_{\text{final}} - E_{\text{initial}} = 12.1 \, \text{eV} \]
Substituting the initial energy:
\[ E_{\text{final}} - (-13.6) = 12.1 \]
\[ E_{\text{final}} + 13.6 = 12.1 \]
\[ E_{\text{final}} = 12.1 - 13.6 = -1.5 \, \text{eV} \]

### Step 4: Relate Final Energy to Principal Quantum Number
According to Bohr's theory, the energy of an electron in the nth level is given by:
\[ E_n = -\frac{13.6}{n^2} \, \text{eV} \]
Setting this equal to the final energy:
\[ -\frac{13.6}{n^2} = -1.5 \]
Removing the negative signs:
\[ \frac{13.6}{n^2} = 1.5 \]
Now, solving for \( n^2 \):
\[ n^2 = \frac{13.6}{1.5} \approx 9.06 \]
Taking the square root:
\[ n \approx 3 \]

### Step 5: Identify Excited State
The excited state of the hydrogen atom is therefore \( n = 3 \).

### Step 6: Calculate Number of Spectral Lines
To find the number of spectral lines emitted when the electron transitions from the excited state (n=3) to the ground state (n=1), we can use the formula:
\[ \text{Number of spectral lines} = \frac{(n_2 - n_1)(n_2 - n_1 + 1)}{2} \]
Where \( n_2 = 3 \) and \( n_1 = 1 \):
\[ \text{Number of spectral lines} = \frac{(3 - 1)(3 - 1 + 1)}{2} = \frac{(2)(3)}{2} = 3 \]

### Conclusion
The number of spectral lines emitted by the hydrogen atom according to Bohr's theory will be **3**.

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