To solve this problem, let's break it down step by step.
1. Understanding the Reaction Equation:
The formation of sodium chloride from sodium and chlorine gas is represented by the balanced chemical equation:
[tex]\[
2 \text{Na}(s) + \text{Cl}_2(g) \rightarrow 2 \text{NaCl}(s)
\][/tex]
This means that two moles of sodium (Na) react with one mole of chlorine gas (Cl_2) to produce two moles of sodium chloride (NaCl).
2. Given Information:
- The mass of sodium chloride (NaCl) recovered is 0.1 kg, which needs to be converted to grams since molar masses are typically in grams per mole.
[tex]\[
\text{Recovered NaCl mass} = 0.1 \text{ kg} \times 1000 \frac{\text{g}}{\text{kg}} = 100 \text{ g}
\][/tex]
3. Molar Masses:
- The molar mass of NaCl is 58.44 g/mol.
- The molar mass of Cl_2 is 70.90 g/mol.
4. Calculating Moles of NaCl:
Using the molar mass of NaCl, we find the number of moles of NaCl:
[tex]\[
\text{Moles of NaCl} = \frac{\text{Recovered NaCl mass}}{\text{Molar mass of NaCl}} = \frac{100 \text{ g}}{58.44 \text{ g/mol}} \approx 1.711 \text{ mol}
\][/tex]
5. Stoichiometric Relationship:
From the balanced chemical equation, 2 moles of NaCl are produced with 1 mole of Cl_2. Therefore, the moles of Cl_2 required are half the moles of NaCl:
[tex]\[
\text{Moles of Cl}_2 = \frac{\text{Moles of NaCl}}{2} \approx \frac{1.711}{2} \approx 0.856 \text{ mol}
\][/tex]
6. Calculating Mass of Cl_2:
Using the molar mass of Cl_2, we convert the moles of Cl_2 to grams:
[tex]\[
\text{Mass of Cl}_2 = \text{Moles of Cl}_2 \times \text{Molar mass of Cl}_2 \approx 0.856 \text{ mol} \times 70.90 \text{ g/mol} \approx 60.661 \text{ g}
\][/tex]
7. Converting to Kilograms:
Finally, convert the mass of Cl_2 from grams to kilograms:
[tex]\[
\text{Mass of Cl}_2 \text{ in kg} = \frac{60.661 \text{ g}}{1000} \approx 0.061 \text{ kg}
\][/tex]
Therefore, the minimum amount of chlorine gas consumed in the process is approximately [tex]\(0.061 \text{ kg}\)[/tex].
