To find out how many molecules of sucrose ([tex]\(\text{C}_{12} \text{H}_{22} \text{O}_{11}\)[/tex]) are present in a sample containing [tex]\(3.6 \times 10^{24}\)[/tex] carbon atoms, we need to use a conversion factor that relates the number of carbon atoms to the number of sucrose molecules.
### Step-By-Step Solution:
1. Identify Given Data:
- The number of carbon atoms in the sample: [tex]\(3.6 \times 10^{24}\)[/tex] atoms of carbon (C).
2. Determine the Structure of Sucrose:
- A molecule of sucrose ([tex]\(\text{C}_{12} \text{H}_{22} \text{O}_{11}\)[/tex]) contains:
- 12 carbon atoms (C)
- 22 hydrogen atoms (H)
- 11 oxygen atoms (O)
3. Find the Conversion Factor:
- The number of carbon atoms per sucrose molecule is 12.
4. Calculate the Number of Sucrose Molecules:
- To find how many sucrose molecules correspond to the given number of carbon atoms, we divide the total number of carbon atoms by the number of carbon atoms per molecule of sucrose.
[tex]\[
\text{Number of sucrose molecules} = \frac{\text{Total number of carbon atoms}}{\text{Number of carbon atoms per sucrose molecule}}
\][/tex]
5. Perform the Calculation:
[tex]\[
\text{Number of sucrose molecules} = \frac{3.6 \times 10^{24} \text{ atoms of C}}{12 \text{ atoms of C per molecule}} = 3.0 \times 10^{23} \text{ molecules of sucrose}
\][/tex]
Therefore, the conversion factor needed to determine the number of sucrose molecules from the number of carbon atoms is the reciprocal of the number of carbon atoms per sucrose molecule.
### Result:
Given:
[tex]\[ 3.6 \times 10^{24} \text{ atoms of C} \times \left( \frac{1 \text{ molecule of sucrose}}{12 \text{ atoms of C}} \right) = 3.0 \times 10^{23} \text{ molecules of sucrose} \][/tex]
So, the missing conversion factor in the table would be:
[tex]\[
\begin{array}{l|l}
3.6 \times 10^{24} \text{ atoms C} & \left( \frac{1 \text{ molecule of sucrose}}{12 \text{ atoms of C}} \right) \\
\hline
3.0 \times 10^{23} \text{ molecules of sucrose}
\end{array}
\][/tex]
Thus, the conversion factor from carbon atoms to sucrose molecules is [tex]\(\frac{1 \text{ molecule of sucrose}}{12 \text{ atoms of C}}\)[/tex].
