If the binding energy of the electron in a hydrogen atom is `13.6 eV`, the energy required to remove the electron from the first excited state of `Li^(++)` is

Options

A. `122.4 eV`
B. `30.6 eV`
C. `13.6 eV`
D. `3.4 eV`

Explanation

To solve the problem of finding the energy required to remove the electron from the first excited state of the lithium ion \( \text{Li}^{++} \), we can follow these steps:

### Step 1: Identify the Atomic Number and State
The atomic number \( Z \) for lithium is 3. The first excited state corresponds to the principal quantum number \( n = 2 \) (where \( n = 1 \) is the ground state).

### Step 2: Use the Formula for Binding Energy
The binding energy of an electron in a hydrogen-like atom is given by the formula:
\[ E_n = -\frac{13.6 \, \text{eV} \cdot Z^2}{n^2} \]
where \( E_n \) is the energy of the electron at the principal quantum number \( n \).

### Step 3: Substitute Values into the Formula
For lithium \( \text{Li}^{++} \):
- \( Z = 3 \)
- \( n = 2 \)

Substituting these values into the formula:
\[ E_2 = -\frac{13.6 \, \text{eV} \cdot (3)^2}{(2)^2} \]
\[ E_2 = -\frac{13.6 \, \text{eV} \cdot 9}{4} \]
\[ E_2 = -\frac{122.4 \, \text{eV}}{4} \]
\[ E_2 = -30.6 \, \text{eV} \]

### Step 4: Calculate the Energy Required to Remove the Electron
To remove the electron from the first excited state, we need to provide enough energy to bring it to zero energy (the energy at infinity). Thus, the energy required \( E_{\text{required}} \) is:
\[ E_{\text{required}} = 0 - E_2 \]
\[ E_{\text{required}} = 0 - (-30.6 \, \text{eV}) \]
\[ E_{\text{required}} = 30.6 \, \text{eV} \]

### Conclusion
The energy required to remove the electron from the first excited state of \( \text{Li}^{++} \) is \( 30.6 \, \text{eV} \).

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