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

Options

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

Answer

To solve the problem regarding the spectral lines emitted by hydrogen atoms after being excited by monochromatic radiation, we can follow these steps:

### Step 1: Understand the Ionization Potential
The ionization potential of a hydrogen atom is given as 13.6 eV. This means that to completely remove the electron from the ground state (n=1), we need to provide at least 13.6 eV of energy.

### Step 2: Determine the Energy Levels
According to Bohr's model, the energy of an electron in the nth orbit of a hydrogen atom is given by the formula:
\[ E_n = -\frac{13.6 \, \text{eV}}{n^2} \]
For the ground state (n=1):
\[ E_1 = -13.6 \, \text{eV} \]
For the first excited state (n=2):
\[ E_2 = -\frac{13.6}{2^2} = -3.4 \, \text{eV} \]
For the second excited state (n=3):
\[ E_3 = -\frac{13.6}{3^2} \approx -1.51 \, \text{eV} \]

### Step 3: Calculate the Final Energy State After Excitation
The hydrogen atom is excited by a photon with energy 12.1 eV. The initial energy of the atom in the ground state is -13.6 eV. After absorbing the photon, the total energy becomes:
\[ E_{\text{final}} = E_1 + \text{Photon Energy} = -13.6 \, \text{eV} + 12.1 \, \text{eV} = -1.5 \, \text{eV} \]
This energy corresponds to the energy level of n=3, since:
\[ E_3 \approx -1.51 \, \text{eV} \]

### Step 4: Identify Possible Transitions
The electron is now in the n=3 state. It can transition to lower energy levels (n=2 or n=1) and emit photons. The possible transitions are:
1. From n=3 to n=2
2. From n=3 to n=1

### Step 5: Calculate the Number of Spectral Lines
The number of spectral lines emitted when an electron transitions from level n to lower levels can be calculated using the formula:
\[ \text{Number of lines} = \frac{n(n-1)}{2} \]
For n=3:
\[ \text{Number of lines} = \frac{3(3-1)}{2} = \frac{3 \cdot 2}{2} = 3 \]

### Conclusion
Thus, the number of spectral lines emitted by hydrogen atoms according to Bohr's theory after being excited by the photon energy of 12.1 eV is **3**.

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