In photons of energy 12.75 eV are passing through hydrogen gas in ground state then no. of lines in emission spectrum will be

Options

A. 6
B. 4
C. 3
D. 2

Explanation

To find the number of lines in the emission spectrum of hydrogen gas when photons of energy 12.75 eV are absorbed, we can follow these steps:

### Step 1: Determine the energy levels of hydrogen
The energy levels of a hydrogen atom can be calculated using the formula:
\[ E_n = -\frac{13.6 \, \text{eV}}{n^2} \]
where \( n \) is the principal quantum number.

### Step 2: Calculate the initial energy in the ground state
For the ground state (\( n = 1 \)):
\[ E_1 = -\frac{13.6 \, \text{eV}}{1^2} = -13.6 \, \text{eV} \]

### Step 3: Calculate the total energy after absorbing the photon
When the hydrogen atom absorbs a photon of energy 12.75 eV, the total energy becomes:
\[ E_{\text{total}} = E_1 + \text{Energy of photon} = -13.6 \, \text{eV} + 12.75 \, \text{eV} = -0.85 \, \text{eV} \]

### Step 4: Determine the final energy level
We set the total energy equal to the energy of the nth level:
\[ -\frac{13.6 \, \text{eV}}{n^2} = -0.85 \, \text{eV} \]
This simplifies to:
\[ \frac{13.6}{n^2} = 0.85 \]

### Step 5: Solve for \( n^2 \)
Rearranging gives:
\[ n^2 = \frac{13.6}{0.85} \approx 16 \]
Thus, \( n = 4 \).

### Step 6: Calculate the number of lines in the emission spectrum
The number of spectral lines in the emission spectrum can be calculated using the formula:
\[ \text{Number of lines} = \frac{n(n-1)}{2} \]
For \( n = 4 \):
\[ \text{Number of lines} = \frac{4(4-1)}{2} = \frac{4 \times 3}{2} = 6 \]

### Final Answer
The number of lines in the emission spectrum will be **6**.

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