To solve the problem, we will follow these steps:
### Step 1: Calculate the total ionization energy required to convert Mg to Mg²⁺
The first ionization energy (IE1) of magnesium is given as 7.646 eV, and the second ionization energy (IE2) is given as 15.035 eV.
Total ionization energy (IE_total) can be calculated as:
\[
IE_{\text{total}} = IE_1 + IE_2 = 7.646 \, \text{eV} + 15.035 \, \text{eV} = 22.681 \, \text{eV}
\]
### Step 2: Convert the total ionization energy from eV to kJ/mol
We are given that 1 eV = 96.5 kJ/mol. Therefore, we can convert the total ionization energy to kJ/mol:
\[
IE_{\text{total}} \, (\text{kJ/mol}) = 22.681 \, \text{eV} \times 96.5 \, \text{kJ/mol} = 2188.7 \, \text{kJ/mol}
\]
### Step 3: Calculate the number of moles of magnesium in 12 mg
The molar mass of magnesium (Mg) is approximately 24 g/mol. We need to convert 12 mg to grams:
\[
12 \, \text{mg} = 12 \times 10^{-3} \, \text{g} = 0.012 \, \text{g}
\]
Now, we calculate the number of moles of magnesium:
\[
\text{Moles of Mg} = \frac{\text{mass (g)}}{\text{molar mass (g/mol)}} = \frac{0.012 \, \text{g}}{24 \, \text{g/mol}} = 0.0005 \, \text{mol} = 5 \times 10^{-4} \, \text{mol}
\]
### Step 4: Calculate the total energy required for 12 mg of magnesium
Now, we can find the total energy required to convert 0.0005 moles of magnesium to Mg²⁺:
\[
\text{Total energy} = IE_{\text{total}} \times \text{moles of Mg} = 2188.7 \, \text{kJ/mol} \times 0.0005 \, \text{mol} = 1.09435 \, \text{kJ}
\]
### Step 5: Multiply the total energy by 10 and round to the nearest integer
Finally, we multiply the total energy by 10:
\[
\text{Final energy} = 1.09435 \, \text{kJ} \times 10 = 10.9435 \, \text{kJ}
\]
Rounding this to the nearest integer gives us:
\[
\text{Final answer} = 11 \, \text{kJ}
\]
### Summary of the Steps:
1. Calculate total ionization energy: \( IE_{\text{total}} = 22.681 \, \text{eV} \)
2. Convert to kJ/mol: \( 2188.7 \, \text{kJ/mol} \)
3. Calculate moles of Mg in 12 mg: \( 0.0005 \, \text{mol} \)
4. Calculate total energy for 12 mg: \( 1.09435 \, \text{kJ} \)
5. Multiply by 10 and round: \( 11 \, \text{kJ} \)
