Energy required to ionise 2 mole of gaseous `He^(+)` ion present in its ground state is :

Options

A. `54.4` eV
B. `108.8 N_(A)` eV
C. `54.4N_(A)` eV
D. `108.8`eV

Answer

To find the energy required to ionize 2 moles of gaseous `He^(+)` ions in their ground state, we can follow these steps:

### Step 1: Understand the Ionization Energy Formula
The ionization energy (IE) for a hydrogen-like atom or ion is given by the formula:
\[ IE = \frac{13.6 \, \text{eV} \cdot Z^2}{n^2} \]
where:
- \( Z \) is the atomic number,
- \( n \) is the principal quantum number (for ground state, \( n = 1 \)).

### Step 2: Identify the Values for Helium Ion
For the helium ion `He^(+)`:
- \( Z = 2 \) (since helium has an atomic number of 2),
- \( n = 1 \) (ground state).

### Step 3: Substitute the Values into the Formula
Substituting the values into the ionization energy formula:
\[ IE = \frac{13.6 \, \text{eV} \cdot (2)^2}{(1)^2} = \frac{13.6 \, \text{eV} \cdot 4}{1} = 54.4 \, \text{eV} \]

### Step 4: Calculate the Energy for 2 Moles
Now, we need to calculate the energy required for 2 moles of `He^(+)` ions. We know that 1 mole contains Avogadro's number (\( N_A \)) of ions:
\[ \text{Energy for 2 moles} = 54.4 \, \text{eV} \times 2 \times N_A \]
\[ = 108.8 \, N_A \, \text{eV} \]

### Final Answer
The energy required to ionize 2 moles of gaseous `He^(+)` ions in their ground state is:
\[ \text{Energy} = 108.8 \, N_A \, \text{eV} \]