Hydrogen atom in its ground state is excited by means of a monochromatic radiation of wavelength `970.6 Å`. Different wavelengths are possible in the spectrum. After absorbing the energy of radiation, hydrogen atom goes to the excited state. After `10^(-8)` s, the hydrogen atom will come to the ground state by emitting the absorbed energy.
Number of different wavelengths present in the spectrum

Options

A. 4
B. 6
C. 8
D. 5

Answer

To find the number of different wavelengths present in the spectrum after a hydrogen atom is excited to a higher energy level and subsequently returns to the ground state, we can follow these steps:

### Step 1: Calculate the Energy Associated with the Wavelength
We start by using the formula for energy associated with a photon:
\[ E = \frac{hc}{\lambda} \]
Where:
- \( h \) (Planck's constant) = \( 6.67 \times 10^{-34} \, \text{J s} \)
- \( c \) (speed of light) = \( 3 \times 10^{8} \, \text{m/s} \)
- \( \lambda \) (wavelength) = \( 970.6 \, \text{Å} = 970.6 \times 10^{-10} \, \text{m} \)

Substituting the values:
\[ E = \frac{(6.67 \times 10^{-34})(3 \times 10^{8})}{970.6 \times 10^{-10}} \]

### Step 2: Convert Energy to Electron Volts
To convert the energy from joules to electron volts, we use the conversion factor \( 1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J} \):
\[ E_{\text{eV}} = \frac{E_{\text{J}}}{1.6 \times 10^{-19}} \]

### Step 3: Calculate the Energy Change for the Transition
Using the formula for energy levels in a hydrogen atom:
\[ \Delta E = -13.6 \, \text{eV} \cdot Z^2 \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right) \]
For hydrogen, \( Z = 1 \). Since the atom is initially in the ground state, \( n_i = 1 \). We need to find \( n_f \).

Setting the energy equal to the calculated energy:
\[ 12.75 \, \text{eV} = -13.6 \left( \frac{1}{n_f^2} - 1 \right) \]

### Step 4: Solve for \( n_f \)
Rearranging the equation gives:
\[ \frac{1}{n_f^2} = 1 + \frac{12.75}{13.6} \]
Calculating this will yield \( n_f \).

### Step 5: Determine the Number of Wavelengths
The number of different wavelengths (or spectral lines) emitted when the electron transitions from \( n_f \) to lower energy levels can be calculated using the formula:
\[ \text{Number of wavelengths} = \frac{n_f(n_f - 1)}{2} \]

### Step 6: Calculate and Conclude
Substituting the value of \( n_f \) obtained from the previous step into the formula will give the total number of different wavelengths.

### Final Answer
After performing the calculations, we find that the number of different wavelengths present in the spectrum is **6**.

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