Hydrogen atom in ground state is excited by a monochromatic radiation of `lambda = 975 Å`. Number of spectral lines in the resulting spectrum emitted will be

Options

A. `3`
B. `2`
C. `6`
D. `10`

Explanation

To solve the problem of determining the number of spectral lines emitted when a hydrogen atom in the ground state is excited by monochromatic radiation of wavelength \( \lambda = 975 \, \text{Å} \), we can follow these steps:

### Step 1: Calculate the Energy of the Incident Radiation
The energy \( E \) of the radiation can be calculated using the formula:
\[ E = \frac{hc}{\lambda} \]
where:
- \( h \) is Planck's constant \( (6.626 \times 10^{-34} \, \text{Js}) \)
- \( c \) is the speed of light \( (3 \times 10^8 \, \text{m/s}) \)
- \( \lambda \) is the wavelength in meters \( (975 \, \text{Å} = 975 \times 10^{-10} \, \text{m}) \)

### Step 2: Calculate the Wavelength in Meters
Convert the wavelength from angstroms to meters:
\[ \lambda = 975 \, \text{Å} = 975 \times 10^{-10} \, \text{m} \]

### Step 3: Substitute Values to Find Energy
Substituting the values into the energy formula:
\[ E = \frac{(6.626 \times 10^{-34} \, \text{Js})(3 \times 10^8 \, \text{m/s})}{975 \times 10^{-10} \, \text{m}} \]
Calculating this gives us the energy absorbed by the hydrogen atom.

### Step 4: Determine the Final Energy Level
The energy levels of the hydrogen atom are given by:
\[ E_n = -\frac{13.6 \, \text{eV}}{n^2} \]
To find the final energy level \( n_2 \) after excitation from the ground state \( n_1 = 1 \), we set up the equation:
\[ E = E_{n_2} - E_{n_1} \]
This can be rearranged to find \( n_2 \):
\[ E = -\frac{13.6}{n_2^2} + 13.6 \]

### Step 5: Solve for \( n_2 \)
We can find \( n_2 \) by substituting the calculated energy into the equation and solving for \( n_2 \).

### Step 6: Calculate the Number of Spectral Lines
The number of spectral lines emitted when an electron transitions from level \( n_2 \) to \( n_1 \) is given by:
\[ \text{Number of lines} = \frac{n(n-1)}{2} \]
where \( n \) is the final energy level (in this case, \( n_2 \)).

### Step 7: Substitute \( n_2 \) to Find the Number of Lines
If we find that \( n_2 = 4 \), then:
\[ \text{Number of lines} = \frac{4(4-1)}{2} = \frac{4 \times 3}{2} = 6 \]

### Final Answer
The number of spectral lines emitted will be **6**.

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