To determine how many liters of [tex]\( C_2H_2 \)[/tex] (acetylene) are required to produce 8 liters of [tex]\( CO_2 \)[/tex] (carbon dioxide) under standard temperature and pressure (STP), we'll use the stoichiometric relationship from the balanced chemical equation:
[tex]\[ 2C_2H_2(g) + 5O_2(g) \rightarrow 4CO_2(g) + 2H_2O(g) \][/tex]
From the balanced equation, we can see that:
- 2 liters of [tex]\( C_2H_2 \)[/tex] produce 4 liters of [tex]\( CO_2 \)[/tex].
This gives us a stoichiometric ratio:
[tex]\[ \frac{2 \, \text{L} \, C_2H_2}{4 \, \text{L} \, CO_2} \][/tex]
Let's simplify this ratio:
[tex]\[ \frac{2 \, \text{L} \, C_2H_2}{4 \, \text{L} \, CO_2} = \frac{1 \, \text{L} \, C_2H_2}{2 \, \text{L} \, CO_2} \][/tex]
This tells us that 1 liter of [tex]\( C_2H_2 \)[/tex] produces 2 liters of [tex]\( CO_2 \)[/tex].
Now, we need to calculate the amount of [tex]\( C_2H_2 \)[/tex] required to produce 8 liters of [tex]\( CO_2 \)[/tex]:
Given that 1 liter of [tex]\( C_2H_2 \)[/tex] produces 2 liters of [tex]\( CO_2 \)[/tex], we set up the proportion:
[tex]\[ \frac{1 \, \text{L} \, C_2H_2}{2 \, \text{L} \, CO_2} = \frac{x \, \text{L} \, C_2H_2}{8 \, \text{L} \, CO_2} \][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{1 \, \text{L} \, C_2H_2}{2} \times 8 \, \text{L} \, CO_2 \][/tex]
[tex]\[ x = 4 \, \text{L} \, C_2H_2 \][/tex]
Therefore, 4 liters of [tex]\( C_2H_2 \)[/tex] are required to produce 8 liters of [tex]\( CO_2 \)[/tex] under standard temperature and pressure conditions.
