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 are emitted when 9.9 eV of energy is supplied to a hydrogen atom, we can follow these steps:

### Step 1: Understand the Energy Levels of the Hydrogen Atom
The energy of an electron in the nth orbit of a hydrogen atom is given by the formula:
\[ E_n = -\frac{13.6 \, \text{eV}}{n^2} \]
where \( n \) is the principal quantum number.

### Step 2: Calculate the Energy of the Ground State
For the ground state (where \( n = 1 \)):
\[ E_1 = -\frac{13.6 \, \text{eV}}{1^2} = -13.6 \, \text{eV} \]

### Step 3: Determine the Total Energy After Supplying 9.9 eV
When 9.9 eV of energy is supplied to the hydrogen atom, the total energy of the electron becomes:
\[ E_a = E_1 + 9.9 \, \text{eV} = -13.6 \, \text{eV} + 9.9 \, \text{eV} = -3.7 \, \text{eV} \]

### Step 4: Find the Corresponding n for the New Energy Level
We need to find the value of \( n \) for which the energy \( E_n \) equals -3.7 eV:
\[ -\frac{13.6 \, \text{eV}}{n^2} = -3.7 \, \text{eV} \]
Removing the negative signs gives:
\[ \frac{13.6}{n^2} = 3.7 \]
Now, rearranging for \( n^2 \):
\[ n^2 = \frac{13.6}{3.7} \approx 3.67 \]

### Step 5: Calculate n
Taking the square root of both sides:
\[ n = \sqrt{3.67} \approx 1.9 \]

### Step 6: Analyze the Value of n
Since \( n \) must be a positive integer, and we found \( n \approx 1.9 \), this indicates that the electron does not transition to an integer energy level. The electron is not able to reach a valid excited state.

### Step 7: Conclusion on Spectral Lines
Because the electron cannot transition to a valid energy level (since there is no integer value for \( n \) between 1 and 2), it means that no spectral lines will be emitted. Therefore, the number of spectral lines emitted is:
\[ \text{Number of spectral lines emitted} = 0 \]

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

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