Given the following information
`A^(-)(g) to A^(2+)(g) + 3e^(-) "    " DeltaH_1 = 1400 kJ mol^(-1)`
`A(g) to A^(2+)(aq) + 2e^(-) "    " DeltaH_2 = 700 kJ mol^(-1)`
`DeltaH_(eg)[A^(+)(g)] = -350kJ mol^(-1)`
where `DeltaH_(eg)` is electron gain enthalpy.
`(IE_1 + IE_2)` for `A (g)=950 kJ mol^(-1)`
The value of `DeltaH_(eg)` of `A^(2+) (g) `in `kJ mol^(-1)`is

Options

A. `+600`
B. `-600`
C. `-500`
D. `+500`

Answer

To find the value of the electron gain enthalpy (ΔH_eg) of A^(2+)(g), we can follow these steps:

### Step 1: Understand the Given Information
We are provided with the following data:
1. ΔH_1 for the reaction A^(-)(g) → A^(2+)(g) + 3e^(-) is 1400 kJ/mol.
2. ΔH_2 for the reaction A(g) → A^(2+)(aq) + 2e^(-) is 700 kJ/mol.
3. ΔH_eg for A^(+)(g) is -350 kJ/mol.
4. The sum of the first and second ionization energies (IE_1 + IE_2) for A(g) is 950 kJ/mol.

### Step 2: Determine Ionization Energies
From the information given, we can express the ionization energies as follows:
- The first ionization energy (IE_1) can be calculated from the total ionization energy:
\[ IE_1 + IE_2 = 950 \text{ kJ/mol} \]
- We know that the second ionization energy (IE_2) corresponds to the process A^(+) → A^(2+) + e^(-), which is associated with ΔH_2 = 700 kJ/mol. Thus, we can find IE_2:
\[ IE_2 = 700 \text{ kJ/mol} \]
- Now, substituting IE_2 into the equation for total ionization energy:
\[ IE_1 + 700 = 950 \implies IE_1 = 950 - 700 = 250 \text{ kJ/mol} \]

### Step 3: Relate Electron Gain Enthalpy
Next, we need to find the electron gain enthalpy (ΔH_eg) for A^(2+)(g). The relationship can be established as follows:
- The electron gain enthalpy for A^(2+) can be derived from the equation:
\[ A^(+) + e^(-) → A^(2+) \]
- The energy change for this process is related to the ionization energy of A^(+) and the electron gain enthalpy of A^(2+):
\[ ΔH_eg(A^{2+}) = IE_1 - ΔH_eg(A^{+}) \]
- Substituting the known values:
\[ ΔH_eg(A^{2+}) = 250 \text{ kJ/mol} - (-350 \text{ kJ/mol}) \]

### Step 4: Calculate ΔH_eg(A^(2+))
Now we can calculate:
\[ ΔH_eg(A^{2+}) = 250 + 350 = 600 \text{ kJ/mol} \]

### Final Answer
The value of ΔH_eg for A^(2+)(g) is **600 kJ/mol**.