A hydrogen-like atom in its first excited state has excitation energy of 91.8 eV. The energy required to remove the electron from its ion is:

Options

A. 91.8 eV
B. 122.4 eV
C. 183.6 eV
D. 61.2 eV

Explanation

To solve the problem of finding the energy required to remove the electron from a hydrogen-like atom in its first excited state, we can follow these steps:

### Step 1: Understand the Excitation Energy
The excitation energy given is 91.8 eV. This energy is the amount required to move the electron from the ground state (n=1) to the first excited state (n=2).

### Step 2: Use the Formula for Energy Levels
According to the Bohr model, the total energy \( E_n \) of an electron in a hydrogen-like atom is given by the formula:
\[ E_n = -\frac{13.6 Z^2}{n^2} \text{ eV} \]
where \( Z \) is the atomic number and \( n \) is the principal quantum number.

### Step 3: Calculate the Energy for n=1 and n=2
1. For \( n=1 \):
\[ E_1 = -\frac{13.6 Z^2}{1^2} = -13.6 Z^2 \text{ eV} \]
2. For \( n=2 \):
\[ E_2 = -\frac{13.6 Z^2}{2^2} = -\frac{13.6 Z^2}{4} \text{ eV} \]

### Step 4: Relate Excitation Energy to Energy Levels
The excitation energy from \( n=1 \) to \( n=2 \) can be expressed as:
\[ E_2 - E_1 = 91.8 \text{ eV} \]
Substituting the expressions for \( E_1 \) and \( E_2 \):
\[ \left(-\frac{13.6 Z^2}{4}\right) - \left(-13.6 Z^2\right) = 91.8 \]
This simplifies to:
\[ -\frac{13.6 Z^2}{4} + 13.6 Z^2 = 91.8 \]
\[ \frac{3}{4} \cdot 13.6 Z^2 = 91.8 \]

### Step 5: Solve for Z^2
To find \( Z^2 \):
\[ 13.6 Z^2 = 91.8 \cdot \frac{4}{3} \]
Calculating the right side:
\[ Z^2 = \frac{91.8 \cdot 4}{3 \cdot 13.6} \]
Calculating this gives:
\[ Z^2 = \frac{367.2}{40.8} = 9 \]
Thus, \( Z = 3 \).

### Step 6: Calculate the Energy Required to Remove the Electron
The energy required to remove the electron from the ground state (n=1) to infinity is given by:
\[ E = E_\infty - E_1 \]
Where \( E_\infty = 0 \) (energy at infinity) and \( E_1 = -13.6 Z^2 \):
\[ E = 0 - (-13.6 \cdot 9) = 13.6 \cdot 9 = 122.4 \text{ eV} \]

### Final Answer
The energy required to remove the electron from its ion is **122.4 eV**.

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