To match each molecular formula to its corresponding empirical formula, we need to simplify the molecular formula by reducing the subscripts to their smallest whole number ratio. Let's walk through each molecular formula step-by-step.
1. Molecular Formula: [tex]\( C_4H_6O_2 \)[/tex]
- The subscripts are 4, 6, and 2.
- We find the greatest common divisor (GCD) of these numbers, which is 2.
- Divide each subscript by 2:
- [tex]\( C: \frac{4}{2} = 2 \)[/tex]
- [tex]\( H: \frac{6}{2} = 3 \)[/tex]
- [tex]\( O: \frac{2}{2} = 1 \)[/tex]
- Therefore, the empirical formula is [tex]\( C_2H_3O \)[/tex].
2. Molecular Formula: [tex]\( C_4H_{12}O_4 \)[/tex]
- The subscripts are 4, 12, and 4.
- We find the GCD of these numbers, which is 4.
- Divide each subscript by 4:
- [tex]\( C: \frac{4}{4} = 1 \)[/tex]
- [tex]\( H: \frac{12}{4} = 3 \)[/tex]
- [tex]\( O: \frac{4}{4} = 1 \)[/tex]
- Therefore, the empirical formula is [tex]\( CH_3O \)[/tex].
3. Molecular Formula: [tex]\( C_2H_{10}O_2 \)[/tex]
- The subscripts are 2, 10, and 2.
- We find the GCD of these numbers, which is 2.
- Divide each subscript by 2:
- [tex]\( C: \frac{2}{2} = 1 \)[/tex]
- [tex]\( H: \frac{10}{2} = 5 \)[/tex]
- [tex]\( O: \frac{2}{2} = 1 \)[/tex]
- Therefore, the empirical formula is [tex]\( CH_5O \)[/tex].
Now, let's match the empirical formulas to the molecular formulas:
- [tex]\( C_2H_{10}O_2 \)[/tex] corresponds to the empirical formula [tex]\( CH_5O \)[/tex]. Hence:
- [tex]\( C_2H_{10}O_2 \rightarrow CH_5O \)[/tex]
- [tex]\( C_4H_6O_2 \)[/tex] corresponds to the empirical formula [tex]\( C_2H_3O \)[/tex]. Hence:
- [tex]\( C_4H_6O_2 \rightarrow C_2H_3O \)[/tex]
- [tex]\( C_4H_{12}O_4 \)[/tex] corresponds to the empirical formula [tex]\( CH_3O \)[/tex]. Hence:
- [tex]\( C_4H_{12}O_4 \rightarrow CH_3O \)[/tex]
So, the final matches are:
[tex]\[
\begin{align*}
C_2H_{10}O_2 & \rightarrow CH_5O \\
C_4H_6O_2 & \rightarrow C_2H_3O \\
C_4H_{12}O_4 & \rightarrow CH_3O \\
\end{align*}
\][/tex]
