Ionization potential of hydrogen atom is 13.6 eV. Hydrogen atoms in the ground state are excited by monochromatic radiation of photon energy 12.1 eV. According to Bohr's theory, the spectral lines emitted by hydrogen will be

Options

A. Two
B. Three
C. Four
D. One

Answer

To solve the problem step by step, we will follow the concepts of Bohr's atomic model and the energy levels of the hydrogen atom.

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

### Step 2: Determine the Initial Energy
The energy of an electron in the nth level of a hydrogen atom is given by the formula:
\[ E_n = -\frac{13.6 \, \text{eV}}{n^2} \]
For the ground state (n=1):
\[ E_1 = -\frac{13.6 \, \text{eV}}{1^2} = -13.6 \, \text{eV} \]

### Step 3: Calculate the Final Energy After Excitation
When the hydrogen atom is excited by a photon of energy 12.1 eV, the total energy after excitation becomes:
\[ E_{\text{final}} = E_1 + \text{Photon Energy} = -13.6 \, \text{eV} + 12.1 \, \text{eV} = -1.5 \, \text{eV} \]

### Step 4: Determine the Final State (n)
We need to find the principal quantum number (n) corresponding to the final energy of -1.5 eV. Using the energy formula:
\[ E_n = -\frac{13.6 \, \text{eV}}{n^2} \]
Setting \(E_n = -1.5 \, \text{eV}\):
\[ -1.5 = -\frac{13.6}{n^2} \]
This simplifies to:
\[ 1.5n^2 = 13.6 \]
\[ n^2 = \frac{13.6}{1.5} \approx 9.0667 \]
Taking the square root:
\[ n \approx 3 \]

### Step 5: Calculate the Number of Spectral Lines
The number of spectral lines emitted when an electron transitions from a higher energy level (nf) to a lower energy level (ni) can be calculated using the formula:
\[ \text{Number of lines} = \frac{n_f - n_i (n_f - n_i + 1)}{2} \]
Here, \(n_i = 1\) (initial state) and \(n_f = 3\) (final state):
\[ \text{Number of lines} = \frac{3 - 1 (3 - 1 + 1)}{2} = \frac{2 \cdot 3}{2} = 3 \]

### Conclusion
The number of spectral lines emitted by the hydrogen atom when excited by the photon of energy 12.1 eV is **3**.