For which of the following reaction `DeltaH^(@)` vlaue is equal to the first ionization energy of Ca is?

Options

A. `Ca^(+)(g) to Ca^(2+)(g) +e`
B. `Ca(g)to Ca^(+)(g)+e`
C. `Ca(s) to Ca^(+) (g)+e`
D. `Ca(g) to Ca^(2+)(g)+2e`

Answer

To determine which reaction corresponds to the first ionization energy of calcium (Ca), we need to understand what ionization energy is and analyze the given options accordingly.

### Step-by-Step Solution:

1. **Definition of Ionization Energy**:
- Ionization energy is defined as the energy required to remove an electron from an isolated neutral gaseous atom. For calcium, this means we are looking for a reaction where a neutral gaseous calcium atom loses one electron to form a positively charged ion.

2. **Identify the First Ionization Reaction**:
- The first ionization energy of calcium can be represented by the following reaction:
\[ \text{Ca (g)} \rightarrow \text{Ca}^+ (g) + e^- \]
- This equation shows that a neutral gaseous calcium atom (Ca) loses one electron (e-) to form a calcium ion (Ca+).

3. **Analyze the Given Options**:
- The options provided are:
1. \(\text{Ca} + 2 \rightarrow \text{Ca}^{2+} + e^-\)
2. \(\text{Ca (s)} \rightarrow \text{Ca}^+ (g) + e^-\)
3. \(\text{Ca (g)} \rightarrow \text{Ca}^+ (g) + e^-\)
4. \(\text{Ca (g)} \rightarrow \text{Ca}^{2+} (g) + 2e^-\)

4. **Evaluate Each Option**:
- **Option 1**: This represents the second ionization energy (removing two electrons), so it is not correct.
- **Option 2**: This involves a solid state of calcium, which is not in the gaseous state, so it is not correct.
- **Option 3**: This correctly represents the first ionization energy of calcium, where a neutral gaseous calcium atom loses one electron to form a Ca+ ion.
- **Option 4**: This represents the second ionization energy (removing two electrons), so it is also not correct.

5. **Conclusion**:
- The correct option that represents the first ionization energy of calcium is:
\[ \text{Ca (g)} \rightarrow \text{Ca}^+ (g) + e^- \]

### Final Answer:
The reaction for which \(\Delta H^\circ\) value is equal to the first ionization energy of Ca is:
\[ \text{Ca (g)} \rightarrow \text{Ca}^+ (g) + e^- \]