To determine the volume of acetylene gas [tex]\((C_2H_2)\)[/tex] required to produce 75.0 liters of carbon dioxide [tex]\((CO_2)\)[/tex] at standard temperature and pressure (STP), we will use the balanced chemical equation for the reaction between [tex]\(C_2H_2\)[/tex] and [tex]\(O_2\)[/tex]:
[tex]\[ 2 \, C_2H_2 + 5 \, O_2 \rightarrow 4 \, CO_2 + 2 \, H_2O \][/tex]
From the balanced equation, we can observe the following stoichiometric relationship between [tex]\(C_2H_2\)[/tex] and [tex]\(CO_2\)[/tex]:
- 2 volumes of [tex]\(C_2H_2\)[/tex] produce 4 volumes of [tex]\(CO_2\)[/tex].
This can be simplified to show that:
- 1 volume of [tex]\(C_2H_2\)[/tex] produces 2 volumes of [tex]\(CO_2\)[/tex].
With this relationship in mind, we can set up the following ratio to find the volume of [tex]\(C_2H_2\)[/tex] needed to produce the desired volume of [tex]\(CO_2\)[/tex]:
[tex]\[ \frac{1 \, \text{volume of} \, C_2H_2}{2 \, \text{volumes of} \, CO_2} = \frac{X \, \text{liters of} \, C_2H_2}{75.0 \, \text{liters of} \, CO_2} \][/tex]
Solving for [tex]\(X\)[/tex], we get:
[tex]\[ X = \frac{75.0 \, \text{liters of} \, CO_2}{2} \][/tex]
[tex]\[ X = 37.5 \, \text{liters of} \, C_2H_2 \][/tex]
Therefore, the volume of [tex]\(C_2H_2\)[/tex] required to produce 75.0 liters of [tex]\(CO_2\)[/tex] at STP is:
[tex]\[ \boxed{37.5} \, \text{liters} \][/tex]
