To determine the formula for the hydrate composed of 76.9% CaSO₃ and 23.1% H₂O, we should follow these steps:
1. Calculate the molar masses of CaSO₃ and H₂O:
- Calcium sulfite (CaSO₃):
[tex]\[
\text{Molar mass of CaSO}_3 = \text{atomic mass of Ca} + \text{atomic mass of S} + 3 \times \text{atomic mass of O}
\][/tex]
[tex]\[
\text{Molar mass of CaSO}_3 = 40.08 \, \text{g/mol} + 32.07 \, \text{g/mol} + 3 \times 16.00 \, \text{g/mol}
\][/tex]
[tex]\[
\text{Molar mass of CaSO}_3 = 120.15 \, \text{g/mol}
\][/tex]
- Water (H₂O):
[tex]\[
\text{Molar mass of H}_2\text{O} = 2 \times 1.01 \, \text{g/mol} + 16.00 \, \text{g/mol}
\][/tex]
[tex]\[
\text{Molar mass of H}_2\text{O} = 18.02 \, \text{g/mol}
\][/tex]
2. Calculate the moles of CaSO₃ and H₂O from the given percentages:
- Moles of CaSO₃:
[tex]\[
\text{Moles of CaSO}_3 = \frac{76.9 \, \text{g}}{120.15 \, \text{g/mol}}
\][/tex]
[tex]\[
\text{Moles of CaSO}_3 = 0.640 \, \text{moles}
\][/tex]
- Moles of H₂O:
[tex]\[
\text{Moles of H}_2\text{O} = \frac{23.1 \, \text{g}}{18.02 \, \text{g/mol}}
\][/tex]
[tex]\[
\text{Moles of H}_2\text{O} = 1.282 \, \text{moles}
\][/tex]
3. Calculate the mole ratio of H₂O to CaSO₃:
[tex]\[
\text{Ratio} = \frac{\text{Moles of H}_2\text{O}}{\text{Moles of CaSO}_3}
\][/tex]
[tex]\[
\text{Ratio} = \frac{1.282 \, \text{moles}}{0.640 \, \text{moles}}
\][/tex]
[tex]\[
\text{Ratio} = 2.003
\][/tex]
4. Determine the number of water molecules (H₂O) per formula unit of CaSO₃:
The mole ratio calculated is approximately 2, indicating that for every one mole of CaSO₃, there are about 2 moles of H₂O. Therefore, this most closely matches the compound [tex]\( \text{CaSO}_3 \cdot 2 \text{H}_2\text{O} \)[/tex].
Thus, the formula for the hydrate is:
[tex]\[
\boxed{\text{CaSO}_3 \cdot 2 \text{H}_2\text{O}}
\][/tex]
