To calculate the percentage atom economy for Stage 1, we need to use the formula:
[tex]\[
\text{Percentage Atom Economy} = \left( \frac{\text{Mass of Desired Product}}{\text{Total Mass of Reactants}} \right) \times 100
\][/tex]
Here, the desired product is zinc oxide (ZnO), and the reactants are zinc sulfide (ZnS) and oxygen (O₂).
Step-by-step solution:
Step 1: Write down the balanced chemical equation
The balanced equation for the roasting of zinc sulfide in air is:
[tex]\[
2 \, \text{ZnS (s)} + 3 \, \text{O}_2 \, \text{(g)} \rightarrow 2 \, \text{ZnO (s)} + 2 \, \text{SO}_2 \, \text{(g)}
\][/tex]
Step 2: Identify the relative formula masses (Mr) given:
- [tex]\( \text{ZnS} = 97 \)[/tex]
- [tex]\( \text{O}_2 = 32 \)[/tex]
- [tex]\( \text{ZnO} = 81 \)[/tex]
- [tex]\( \text{SO}_2 = 64 \)[/tex]
Step 3: Calculate the total mass of reactants
Using the stoichiometric coefficients from the balanced equation:
[tex]\[
2 \, \text{moles of ZnS} + 3 \, \text{moles of O}_2
\][/tex]
The total mass of the reactants is calculated as:
[tex]\[
\text{Total mass of reactants} = 2 \, \text{moles of ZnS} \times 97 \, \text{g/mol} + 3 \, \text{moles of O}_2 \times 32 \, \text{g/mol}
\][/tex]
\begin{align}
\text{Total mass of reactants} &= (2 \times 97) + (3 \times 32) \\
&= 194 + 96 \\
&= 290 \, \text{g}
\end{align}
Step 4: Calculate the mass of the desired product (ZnO)
Using the stoichiometric coefficients from the balanced equation:
[tex]\[
2 \, \text{moles of ZnO}
\][/tex]
The mass of the desired product is calculated as:
[tex]\[
\text{Mass of ZnO} = 2 \, \text{moles of ZnO} \times 81 \, \text{g/mol}
\][/tex]
\begin{align}
\text{Mass of ZnO} &= 2 \times 81 \\
&= 162 \, \text{g}
\end{align}
Step 5: Calculate the percentage atom economy
[tex]\[
\text{Percentage Atom Economy} = \left( \frac{\text{Mass of ZnO}}{\text{Total Mass of Reactants}} \right) \times 100
\][/tex]
\begin{align}
\text{Percentage Atom Economy} &= \left( \frac{162}{290} \right) \times 100 \\
&\approx 55.862 \%
\end{align}
So, the percentage atom economy for Stage 1 to produce zinc oxide (ZnO) is 55.862% when rounded to three significant figures.
