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 determine the number of spectral lines 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
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 Levels
Calculate the energy for the first three energy levels (n = 1, 2, and 3):
- For \( n = 1 \):
\[ E_1 = -\frac{13.6 \, \text{eV}}{1^2} = -13.6 \, \text{eV} \]
- For \( n = 2 \):
\[ E_2 = -\frac{13.6 \, \text{eV}}{2^2} = -3.4 \, \text{eV} \]
- For \( n = 3 \):
\[ E_3 = -\frac{13.6 \, \text{eV}}{3^2} = -1.51 \, \text{eV} \]

### Step 3: Determine Energy Differences
Calculate the energy differences between the levels:
- Difference between \( n = 1 \) and \( n = 2 \):
\[ \Delta E_{1 \to 2} = E_2 - E_1 = -3.4 \, \text{eV} - (-13.6 \, \text{eV}) = 10.2 \, \text{eV} \]
- Difference between \( n = 1 \) and \( n = 3 \):
\[ \Delta E_{1 \to 3} = E_3 - E_1 = -1.51 \, \text{eV} - (-13.6 \, \text{eV}) = 12.09 \, \text{eV} \]

### Step 4: Compare Photon Energy with Energy Differences
The photon energy provided is 11 eV. Now, we check if this energy is sufficient for any transitions:
- For the transition from \( n = 1 \) to \( n = 2 \):
- Required energy: 10.2 eV (11 eV > 10.2 eV, so transition is possible)
- For the transition from \( n = 1 \) to \( n = 3 \):
- Required energy: 12.09 eV (11 eV < 12.09 eV, so transition is not possible)

### Step 5: Determine Possible Transitions
Since the photon energy (11 eV) is sufficient for the transition from \( n = 1 \) to \( n = 2 \), but not sufficient for the transition from \( n = 1 \) to \( n = 3 \), the only possible transition is:
- \( n = 1 \) to \( n = 2 \)

### Step 6: Calculate Spectral Lines
Each transition corresponds to a spectral line. Since we have one valid transition (from \( n = 1 \) to \( n = 2 \)), we will have:
- Number of spectral lines = 1

### Final Answer
The number of spectral lines obtained is **1**.

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