No of spectral line obtained when electron is provided will a photoon of energy of 11 ev (for H atom)

Options

A. 1
B. 2
C. 0
D. 6

Answer

To solve the problem of how many spectral lines are obtained when an electron in a hydrogen atom is provided with a photon of energy 11 eV, we can follow these steps:

### Step 1: Understand the Energy Levels of Hydrogen
According to the Bohr model, the energy levels of a hydrogen atom are given by the formula:
\[ E_n = -\frac{13.6 \, \text{eV}}{n^2} \]
where \( n \) is the principal quantum number (1, 2, 3, ...).

### Step 2: Calculate the Energy for Different Levels
- For \( n = 1 \):
\[ E_1 = -\frac{13.6}{1^2} = -13.6 \, \text{eV} \]
- For \( n = 2 \):
\[ E_2 = -\frac{13.6}{2^2} = -3.4 \, \text{eV} \]
- For \( n = 3 \):
\[ E_3 = -\frac{13.6}{3^2} \approx -1.51 \, \text{eV} \]
- For \( n = 4 \):
\[ E_4 = -\frac{13.6}{4^2} = -0.85 \, \text{eV} \]

### Step 3: Determine Energy Gaps Between Levels
Next, we calculate the energy gaps between the levels:
- Between \( n = 1 \) and \( n = 2 \):
\[ \Delta E_{1 \to 2} = E_2 - E_1 = (-3.4) - (-13.6) = 10.2 \, \text{eV} \]
- Between \( n = 2 \) and \( n = 3 \):
\[ \Delta E_{2 \to 3} = E_3 - E_2 = (-1.51) - (-3.4) = 1.89 \, \text{eV} \]
- Between \( n = 3 \) and \( n = 4 \):
\[ \Delta E_{3 \to 4} = E_4 - E_3 = (-0.85) - (-1.51) = 0.66 \, \text{eV} \]

### Step 4: Compare the Photon Energy with Energy Gaps
The photon energy provided is 11 eV. We need to see if this energy can be used to excite the electron to a higher energy level:
- The energy gap from \( n = 1 \) to \( n = 2 \) is 10.2 eV.
- The energy gap from \( n = 2 \) to \( n = 3 \) is 1.89 eV.
- The energy gap from \( n = 3 \) to \( n = 4 \) is 0.66 eV.

### Step 5: Determine if the Electron Absorbs the Photon
The photon energy (11 eV) is greater than the gap between \( n = 1 \) and \( n = 2 \) (10.2 eV) but less than the gap to \( n = 3 \) (12.09 eV). Since 11 eV does not match any specific energy gap exactly, the electron cannot transition to any higher energy state.

### Step 6: Conclusion on Spectral Lines
Since the electron cannot absorb the 11 eV photon to transition to any higher energy level, there will be no spectral lines produced.

### Final Answer
The number of spectral lines obtained is **0**.

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