To determine the empirical formula for the compound given the ratio [tex]\( 5 \text{C} : 8 \text{H} : 2 \text{O} \)[/tex], let's follow these steps:
1. We are given the ratio of carbon (C), hydrogen (H), and oxygen (O) as [tex]\( 5 \text{C} : 8 \text{H} : 2 \text{O} \)[/tex].
2. Multiply each value by two to get whole numbers:
[tex]\[
5 \times 2 : 8 \times 2 : 2 \times 2 = 10 : 16 : 4
\][/tex]
3. Simplify this ratio. We check if the ratio [tex]\( 10 : 16 : 4 \)[/tex] is in its simplest form. In this case, the ratio [tex]\( 10 : 16 : 4 \)[/tex] is already composed of integers and it can be simplified by checking the common factors. However, the greatest common divisor of these numbers is 2, which gives us a simpler ratio:
[tex]\[
\frac{10}{2} : \frac{16}{2} : \frac{4}{2} = 5 : 8 : 2
\][/tex]
4. Since the ratio is already in integers and does not need further simplification, this corresponds to the empirical formula [tex]\( \text{C}_5 \text{H}_8 \text{O}_2 \)[/tex].
5. Therefore, the empirical formula for the compound is [tex]\( \text{C}_5 \text{H}_8 \text{O}_2 \)[/tex], corresponding to answer choice number 2.
So, the correct answer is:
[tex]\[
\boxed{2}
\][/tex]
