Excited atoms emits radiations consisting of only certain discrete frequencies or wavelengths. In spectroscopy it is often more convenient to use frequencies or wave numbers than are proportional to energy and spectroscopy involves transitions between energy levels. The line spectrum shown by and mono electronic excited atom (a finger pring of an atom) can be given as
`1/(lamda)=barv=R_(A)Z^(2)[1/(n_(1)^(2))-1/(n_(2)^(2))]`
When Z is atomic number of mono electronic atom and `n_(1),n_(2)` are integers and if `n_(2)gtn_(1)`, then emission spectrum is noticed and if `n_(2)ltn_(1)`, then absorptioni spectrum is noticed. Every line in spectrum can be represented as differences of two terms `(R_(A)Z^(2))/(n_(1)^(2))` and `(R_(A)Z^(2))/(n_(2)^(2))`
H-atom in ground state (13.6eV) are excited by monochromatic radiation of photon of energy 12.1 eV. The number of spectral lines emitted in H-atom will be

Options

A. 1
B. 2
C. 3
D. 6

Answer

To solve the problem of determining the number of spectral lines emitted by a hydrogen atom when it is excited, we can follow these steps:

### Step 1: Understand the Initial Conditions
The hydrogen atom is initially in the ground state, which corresponds to the energy level \( n = 1 \). The energy of the hydrogen atom in the ground state is given as \( E_1 = -13.6 \, \text{eV} \). The atom is excited by monochromatic radiation of energy \( E_{\text{photon}} = 12.1 \, \text{eV} \).

### Step 2: Calculate the Final Energy
To find the final energy of the hydrogen atom after absorption of the photon, we can use the formula:
\[ E_{\text{final}} = E_1 + E_{\text{photon}} = -13.6 \, \text{eV} + 12.1 \, \text{eV} = -1.5 \, \text{eV} \]

### Step 3: Find the Final Energy Level
The energy of the hydrogen atom at the \( n \)-th energy level is given by:
\[ E_n = -\frac{13.6}{n^2} \, \text{eV} \]
Setting this equal to the final energy we calculated:
\[ -1.5 = -\frac{13.6}{n^2} \]
This simplifies to:
\[ 1.5 = \frac{13.6}{n^2} \]
Rearranging gives:
\[ n^2 = \frac{13.6}{1.5} \approx 9.0667 \]
Taking the square root:
\[ n \approx 3 \]
Thus, the final energy level \( n_{\text{final}} = 3 \).

### Step 4: Calculate the Number of Spectral Lines
The number of spectral lines emitted when transitioning from a higher energy level to lower energy levels can be calculated using the formula:
\[ \text{Number of lines} = \frac{n(n-1)}{2} \]
Substituting \( n = 3 \):
\[ \text{Number of lines} = \frac{3(3-1)}{2} = \frac{3 \times 2}{2} = 3 \]

### Conclusion
The number of spectral lines emitted by the hydrogen atom when excited to the third energy level is **3**.

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