If you are given Avogadro's number of atoms of a gas X . If half of the atoms are converted into `X_(g)^(+)` by energy `Delta H` . The IE of X is

Options

A. `(2DeltaH)/(N_(A))`
B. `(2N_(A))/(DeltaH)`
C. `(DeltaH)/(2N_(A))`
D. `(N_(A))/(DeltaH)`

Answer

To solve the problem, we need to determine the ionization energy (IE) of the gas X given that half of Avogadro's number of atoms are converted into cations (X⁺) by the energy ΔH.

### Step-by-Step Solution:

1. **Understanding Avogadro's Number**:
Avogadro's number (NA) is approximately \(6.022 \times 10^{23}\). If we have Avogadro's number of atoms of gas X, it means we have \(6.022 \times 10^{23}\) atoms of X.

2. **Conversion of Atoms**:
The question states that half of these atoms are converted into cations (X⁺). Therefore, the number of atoms converted is:
\[ \frac{NA}{2} = \frac{6.022 \times 10^{23}}{2} = 3.011 \times 10^{23} \text{ atoms} \]

3. **Energy Required for Ionization**:
The energy ΔH is provided to convert these atoms into cations. This energy is used to remove electrons from the atoms. The total energy required to ionize half of the atoms is ΔH.

4. **Calculating Ionization Energy for One Atom**:
The ionization energy (IE) is defined as the energy required to remove an electron from one mole of gaseous atoms. Since ΔH is the energy required for \(\frac{NA}{2}\) atoms, we can express the ionization energy for one atom as follows:
\[ \text{Energy per atom} = \frac{\Delta H}{\frac{NA}{2}} = \frac{2 \Delta H}{NA} \]

5. **Final Expression for Ionization Energy**:
Thus, the ionization energy (IE) of gas X can be expressed as:
\[ \text{IE} = \frac{2 \Delta H}{NA} \]

### Conclusion:
The ionization energy (IE) of gas X is given by:
\[ \text{IE} = \frac{2 \Delta H}{NA} \]