A photon of energy 12.09 eV is absorbed by an electron in ground state of a hydrogen atoms. What will be the energy level of electron ? The energy of electron in the ground state of hydrogen atom is `-13.6 eV`

Answer

To solve the problem step by step, we need to determine the energy level of the electron after it absorbs a photon of energy 12.09 eV while it is initially in the ground state of a hydrogen atom.

### Step 1: Understand the initial conditions
The energy of the electron in the ground state of a hydrogen atom is given as:
\[ E_1 = -13.6 \, \text{eV} \]

### Step 2: Calculate the new energy of the electron after photon absorption
When the photon is absorbed, the energy of the electron will increase by the energy of the photon. The energy of the photon is given as:
\[ E_{\text{photon}} = 12.09 \, \text{eV} \]

The new energy \( E_N \) of the electron after absorbing the photon can be calculated as:
\[ E_N = E_1 + E_{\text{photon}} \]
Substituting the values:
\[ E_N = -13.6 \, \text{eV} + 12.09 \, \text{eV} \]
\[ E_N = -1.51 \, \text{eV} \]

### Step 3: Relate the energy to the energy levels of the hydrogen atom
The energy levels of the hydrogen atom can be described by the formula:
\[ E_n = \frac{-13.6 \, \text{eV}}{n^2} \]
where \( n \) is the principal quantum number.

### Step 4: Set up the equation to find \( n \)
We know:
\[ E_N = \frac{-13.6 \, \text{eV}}{n^2} \]
Substituting \( E_N = -1.51 \, \text{eV} \):
\[ -1.51 = \frac{-13.6}{n^2} \]

### Step 5: Solve for \( n^2 \)
Rearranging the equation:
\[ n^2 = \frac{-13.6}{-1.51} \]
Calculating:
\[ n^2 = \frac{13.6}{1.51} \approx 9.01 \]

### Step 6: Find \( n \)
Taking the square root:
\[ n \approx 3 \]

### Conclusion
The energy level of the electron after absorbing the photon is \( n = 3 \).

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