To determine the formula of the hydrated salt, we need to follow a step-by-step approach to find the mole ratio of the elements and water in the compound. Here's a detailed solution:
1. Determine the Mass Percent of Water:
Given:
- Mass percent of Iron (Fe) = 20.2%
- Mass percent of Oxygen (O) = 23.0%
- Mass percent of Sulfur (S) = 11.5%
The mass percent of water (H₂O) can be calculated as:
[tex]\[
\text{Mass percent of Water} = 100\% - (\text{Mass percent of Iron} + \text{Mass percent of Oxygen} + \text{Mass percent of Sulfur})
\][/tex]
[tex]\[
\text{Mass percent of Water} = 100\% - (20.2\% + 23.0\% + 11.5\%) = 45.3\%
\][/tex]
2. Convert Mass Percent to Moles:
Using the molar masses:
- Molar mass of Fe = 56 g/mol
- Molar mass of S = 32 g/mol
- Molar mass of O = 16 g/mol
- Molar mass of H₂O (water) = 2 * 1 (H) + 16 (O) = 18 g/mol
Now calculate the moles of each element based on their mass percentages:
Moles of Fe:
[tex]\[
\text{Moles of Fe} = \frac{\text{Mass percent of Fe}}{\text{Molar mass of Fe}} = \frac{20.2}{56} \approx 0.3607 \text{ moles}
\][/tex]
Moles of S:
[tex]\[
\text{Moles of S} = \frac{\text{Mass percent of S}}{\text{Molar mass of S}} = \frac{11.5}{32} \approx 0.3594 \text{ moles}
\][/tex]
Moles of O:
[tex]\[
\text{Moles of O} = \frac{\text{Mass percent of O}}{\text{Molar mass of O}} = \frac{23.0}{16} = 1.4375 \text{ moles}
\][/tex]
Moles of H₂O:
[tex]\[
\text{Moles of H₂O} = \frac{\text{Mass percent of Water}}{\text{Molar mass of H₂O}} = \frac{45.3}{18} \approx 2.5167 \text{ moles}
\][/tex]
3. Determine the Simplest Whole Number Ratio:
Calculate the mole ratios by dividing each by the smallest number of moles (approximately 0.3594 moles of S):
Ratio of Fe:
[tex]\[
\text{Ratio of Fe} = \frac{0.3607}{0.3594} \approx 1.00
\][/tex]
Ratio of S:
[tex]\[
\text{Ratio of S} = \frac{0.3594}{0.3594} = 1.00
\][/tex]
Ratio of O:
[tex]\[
\text{Ratio of O} = \frac{1.4375}{0.3594} \approx 4.00
\][/tex]
Ratio of H₂O:
[tex]\[
\text{Ratio of H₂O} = \frac{2.5167}{0.3594} \approx 7.00
\][/tex]
4. Write the Empirical Formula:
The mole ratios indicate that the compound consists of Fe, S, O in whole numbers of 1:1:4, and water in a whole number ratio of 7.
Therefore, the formula of the hydrated salt is:
[tex]\[
\text{FeSO}_4 \cdot 7\text{H}_2\text{O}
\][/tex]
So, the formula of the hydrated salt is [tex]\(\text{FeSO}_4 \cdot 7\text{H}_2\text{O}\)[/tex].
