To find the number of moles of [tex]\( \text{Pb}( \text{NO}_3)_2 \)[/tex] required to produce 12 moles of [tex]\( \text{Al}( \text{NO}_3)_3 \)[/tex], let's follow these step-by-step instructions:
1. Identify the given data:
- The balanced chemical equation is:
[tex]\[
3 \text{Pb}( \text{NO}_3)_2 + 2 \text{AlCl}_3 \rightarrow 3 \text{PbCl}_2 + 2 \text{Al}( \text{NO}_3)_3
\][/tex]
- We are given that 12 moles of [tex]\( \text{Al}( \text{NO}_3)_3 \)[/tex] are produced.
2. Determine the stoichiometric relationship:
- From the balanced equation, 3 moles of [tex]\( \text{Pb}( \text{NO}_3)_2 \)[/tex] produce 2 moles of [tex]\( \text{Al}( \text{NO}_3)_3 \)[/tex].
3. Set up a ratio to find the moles required:
- The molar ratio between [tex]\( \text{Pb}( \text{NO}_3)_2 \)[/tex] and [tex]\( \text{Al}( \text{NO}_3)_3 \)[/tex] is:
[tex]\[
\frac{3 \text{ moles } \text{ of } \text{Pb}( \text{NO}_3)_2}{2 \text{ moles } \text{ of } \text{Al}( \text{NO}_3)_3}
\][/tex]
- This can be simplified as:
[tex]\[
\frac{3}{2} = 1.5
\][/tex]
4. Calculate the moles of [tex]\( \text{Pb}( \text{NO}_3)_2 \)[/tex] needed:
- To generate 12 moles of [tex]\( \text{Al}( \text{NO}_3)_3 \)[/tex]:
[tex]\[
\text{Moles of } \text{Pb}( \text{NO}_3)_2 = 12 \text{ moles } \text{ of } \text{Al}( \text{NO}_3)_3 \times \left( \frac{3 \text{ moles } \text{ of } \text{Pb}( \text{NO}_3)_2}{2 \text{ moles } \text{ of } \text{Al}( \text{NO}_3)_3} \right)
\][/tex]
- Simplify the calculation:
[tex]\[
\text{Moles of } \text{Pb}( \text{NO}_3)_2 = 12 \times 1.5 = 18 \text{ moles}
\][/tex]
Therefore, 18 moles of [tex]\( \text{Pb}( \text{NO}_3)_2 \)[/tex] are required to generate 12 moles of [tex]\( \text{Al}( \text{NO}_3)_3 \)[/tex].
