Understanding the Problem
To find the total number of oxygen atoms in a solution containing sucrose ($C_{12}H_{22}O_{11}$) and water ($H_2O$), we must calculate the oxygen contribution from both the solute (sucrose) and the solvent (water).
Step 1: Calculate Molar Masses
First, we identify the atomic masses: $C = 12$, $H = 1$, $O = 16$ (in g/mol).
- Molar Mass of Sucrose ($C_{12}H_{22}O_{11}$): $(12 \times 12) + (22 \times 1) + (11 \times 16) = 144 + 22 + 176 = 342 \text{ g/mol}$
- Molar Mass of Water ($H_2O$): $(2 \times 1) + 16 = 18 \text{ g/mol}$
Step 2: Calculate Moles of Each Component
- Moles of Sucrose: $n_{\text{sucrose}} = \frac{\text{mass}}{\text{molar mass}} = \frac{34.2 \text{ g}}{342 \text{ g/mol}} = 0.1 \text{ mol}$
- Moles of Water: $n_{\text{water}} = \frac{180 \text{ g}}{18 \text{ g/mol}} = 10 \text{ mol}$
Step 3: Count Oxygen Atoms
Now, we determine how many moles of oxygen atoms come from each molecule type:
Oxygen from Sucrose: Each sucrose molecule has 11 oxygen atoms. $n_{O(\text{sucrose})} = 0.1 \text{ mol} \times 11 = 1.1 \text{ mol of O atoms}$
Oxygen from Water: Each water molecule has 1 oxygen atom. $n_{O(\text{water})} = 10 \text{ mol} \times 1 = 10 \text{ mol of O atoms}$
Total moles of Oxygen atoms: $n_{O(\text{total})} = 1.1 + 10 = 11.1 \text{ mol}$
Step 4: Convert to Number of Atoms
Using Avogadro's number ($N_A = 6.022 \times 10^{23} \text{ atoms/mol}$):
$\text{Total Oxygen atoms} = 11.1 \times 6.022 \times 10^{23}$ $\text{Total Oxygen atoms} \approx 6.684 \times 10^{24} \text{ atoms}$
Summary
By breaking down the mixture into its individual components and identifying the stoichiometric ratio of oxygen in each, we find that the solution contains approximately $6.684 \times 10^{24}$ oxygen atoms.