If 9.9 eV energy is supplied to a hydrogen atom then the number of spectral lines emitted will be

Options

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

Answer

To solve the problem of how many spectral lines will be emitted when 9.9 eV of energy is supplied to a hydrogen atom, we can follow these steps:

### Step 1: Determine the Ground State Energy of Hydrogen
The energy of an electron in a hydrogen atom at any energy level \( n \) is given by the formula:
\[ E_n = -\frac{13.6 \, \text{eV}}{n^2} \]
For the ground state (where \( n = 1 \)):
\[ E_1 = -\frac{13.6 \, \text{eV}}{1^2} = -13.6 \, \text{eV} \]

### Step 2: Calculate the Total Energy After Supplying 9.9 eV
When we supply 9.9 eV of energy to the hydrogen atom, the total energy becomes:
\[ E_{\text{total}} = E_1 + 9.9 \, \text{eV} = -13.6 \, \text{eV} + 9.9 \, \text{eV} = -3.7 \, \text{eV} \]

### Step 3: Determine the New Energy Level
To find the new energy level \( n \), we set the total energy equal to the energy formula:
\[ -\frac{13.6 \, \text{eV}}{n^2} = -3.7 \, \text{eV} \]
This simplifies to:
\[ \frac{13.6}{n^2} = 3.7 \]
Now, we can solve for \( n^2 \):
\[ n^2 = \frac{13.6}{3.7} \approx 3.68 \]
Taking the square root gives:
\[ n \approx \sqrt{3.68} \approx 1.92 \]

### Step 4: Analyze the Value of \( n \)
The value \( n \approx 1.92 \) indicates that the electron is not in a defined energy level, as \( n \) must be a whole number (1, 2, 3, ...). Since 1.92 is between 1 and 2, it means the electron is in a state that is not stable and cannot emit spectral lines.

### Step 5: Conclusion on Spectral Lines
Since the electron does not transition to a defined energy level, it cannot emit any spectral lines. Therefore, the number of spectral lines emitted is:
\[ \text{Number of spectral lines} = 0 \]

### Final Answer
The number of spectral lines emitted will be **0**.

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