Supposing the electron of the hydrogn atom is present in the L- shell if it liberates 10.2 eV what is the energy of the system ?

Options

A. `-3.4 eV`
B. `-13.6 eV`
C. `-10.2 eV`
D. 0 eV

Answer

To solve the problem step by step, we will use the Bohr model of the hydrogen atom.

### Step 1: Identify the energy of the electron in the L-shell
According to the Bohr model, the energy of an electron in the nth orbit is given by the formula:

\[ E_n = -\frac{13.6 \, Z^2}{n^2} \, \text{eV} \]

For hydrogen, \( Z = 1 \). In the case of the L-shell, \( n = 2 \).

### Step 2: Calculate the energy of the electron in the L-shell
Substituting the values into the formula:

\[ E_2 = -\frac{13.6 \, (1)^2}{(2)^2} = -\frac{13.6}{4} = -3.4 \, \text{eV} \]

### Step 3: Determine the energy liberated by the electron
The problem states that the electron liberates 10.2 eV of energy. This energy corresponds to the transition of the electron from the L-shell to a state where it is completely free (ionization).

### Step 4: Set up the equation for energy conservation
The energy liberated can be expressed as:

\[ \Delta E = E_2 - E \]

Where \( E \) is the energy of the system after the electron has liberated the energy.

### Step 5: Substitute known values into the equation
We know:

\[ \Delta E = 10.2 \, \text{eV} \]
\[ E_2 = -3.4 \, \text{eV} \]

Substituting these values into the equation gives:

\[ 10.2 = -3.4 - E \]

### Step 6: Solve for E
Rearranging the equation to solve for \( E \):

\[ E = -3.4 - 10.2 = -13.6 \, \text{eV} \]

### Conclusion
The energy of the system after the electron liberates 10.2 eV is:

\[ E = -13.6 \, \text{eV} \]

### Final Answer
The energy of the system is \(-13.6 \, \text{eV}\).

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