Consider the following ionization steps :
`M(g) rarr M^(+)(g) +e^(-), DeltaH=100eV`
`M(g) rarr M^(2+)(g)+2e^(-),DeltaH=250eV`
Select correct statement(s) a)`I.E._(1) " of "M(g) ` is 100eV b)`M^(+)(g) rarr M^(2+)(g)+e^(-),DeltaH=150eV` c)`I.E._(2)` of `M(g)` is 250eV d)`I.E._(2)` of M(g) is 14=150eV

Options

A. `I.E_(1)` of M(g) is 100 eV
B. `I.E_(1)` of `Mg^(+)(g)` is 150 eV
C. `I.E_(2)` of M(g) is 250 eV
D. `I.E_(2)` of M(g) is 150 eV

Answer

To solve the problem, we need to analyze the given ionization steps and determine the correct statements based on the provided data.

### Given Ionization Steps:
1. **First Ionization**:
\[ M(g) \rightarrow M^+(g) + e^-, \quad \Delta H = 100 \text{ eV} \]
This indicates that the first ionization energy (I.E.₁) of M is 100 eV.

2. **Second Ionization**:
\[ M(g) \rightarrow M^{2+}(g) + 2e^-, \quad \Delta H = 250 \text{ eV} \]
This indicates that the total energy required to remove two electrons from M is 250 eV.

### Step-by-Step Solution:

1. **Determine I.E.₁**:
- From the first ionization step, we can directly conclude that:
\[ \text{I.E.₁} = 100 \text{ eV} \]
- **Hint**: The first ionization energy is the energy required to remove the first electron from a neutral atom.

2. **Determine I.E.₂**:
- The total energy for removing two electrons is given as 250 eV.
- The second ionization energy (I.E.₂) can be calculated using the relationship:
\[ \text{I.E.₂} = \text{Total energy for two electrons} - \text{I.E.₁} \]
- Substituting the values:
\[ \text{I.E.₂} = 250 \text{ eV} - 100 \text{ eV} = 150 \text{ eV} \]
- **Hint**: The second ionization energy is the energy required to remove the second electron after the first has already been removed.

3. **Evaluate the Statements**:
- **Statement a**: I.E.₁ of M(g) is 100 eV.
- **True**: This is correct as derived from the first ionization step.
- **Statement b**: \( M^+(g) \rightarrow M^{2+}(g) + e^-, \Delta H = 150 \text{ eV} \)
- **True**: This is the definition of the second ionization energy, which we calculated as 150 eV.
- **Statement c**: I.E.₂ of M(g) is 250 eV.
- **False**: We calculated I.E.₂ to be 150 eV, not 250 eV.
- **Statement d**: I.E.₂ of M is 150 eV.
- **True**: This is correct as we calculated I.E.₂ to be 150 eV.

### Final Conclusion:
The correct statements are:
- a) True
- b) True
- c) False
- d) True

Thus, the correct options are a, b, and d.

### Summary of Correct Statements:
- a) I.E.₁ of M(g) is 100 eV.
- b) The ionization step from \( M^+ \) to \( M^{2+} \) releases 150 eV.
- d) I.E.₂ of M is 150 eV.