To determine the ratio of [tex]\( C_2H_2 \)[/tex] to [tex]\( CO_2 \)[/tex] from the given balanced chemical equation, we need to analyze the coefficients of each substance involved in the reaction. The balanced chemical equation is:
[tex]\[ 2 C_2H_2(g) + 5 O_2(g) \rightarrow 4 CO_2(g) + 2 H_2O(g) \][/tex]
From the equation, we can see the coefficients directly in front of [tex]\( C_2H_2 \)[/tex] and [tex]\( CO_2 \)[/tex]:
- The coefficient for [tex]\( C_2H_2 \)[/tex] is 2.
- The coefficient for [tex]\( CO_2 \)[/tex] is 4.
The percentage yields of gases, in terms of their volumes, are proportional to the molar coefficients in the balanced equation. This means we can express these relationships as ratios. Specifically, the volumes of [tex]\( C_2H_2 \)[/tex] to [tex]\( CO_2 \)[/tex] will be in the same ratio as their coefficients:
[tex]\[
\text{Ratio of } C_2H_2 \text{ to } CO_2 = \frac{\text{Coefficient of } C_2H_2}{\text{Coefficient of } CO_2}
\][/tex]
Plugging the values in:
[tex]\[
\text{Ratio of } C_2H_2 \text{ to } CO_2 = \frac{2}{4}
\][/tex]
Next, we simplify this ratio by dividing both the numerator and the denominator by their greatest common divisor, which is 2:
[tex]\[
\frac{2}{4} = \frac{2 \div 2}{4 \div 2} = \frac{1}{2}
\][/tex]
Therefore, the simplified ratio of [tex]\( C_2H_2 \)[/tex] to [tex]\( CO_2 \)[/tex] is:
[tex]\[
1 : 2
\][/tex]
In conclusion, assuming the reaction occurs at STP, the ratio of [tex]\( C_2H_2 \)[/tex] to [tex]\( CO_2 \)[/tex] from the balanced chemical equation is:
[tex]\[ 1 \text{ L } C_2H_2 : 2 \text{ L } CO_2 \][/tex]
