To find the number of moles of water of crystallization (x) in the hydrated sodium carbonate \(\text{Na}_2\text{CO}_3 \cdot x\text{H}_2\text{O}\), we can follow these steps:
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Determine the mass of water lost:
- Initial mass of the hydrated sample: 14.2 g
- Final mass after heating (anhydrous sodium carbonate): 5.3 g
- Mass of water lost = Initial mass - Final mass \[ \text{Mass of water lost} = 14.2 , \text{g} - 5.3 , \text{g} = 8.9 , \text{g} \]
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Calculate the number of moles of anhydrous sodium carbonate \(\text{Na}_2\text{CO}_3\):
- Molar mass of \(\text{Na}_2\text{CO}_3\) = \( (2 \times 23.0) + (12.0) + (3 \times 16.0) \) \[ = 46.0 + 12.0 + 48.0 = 106.0 , \text{g/mol} \]
- Moles of anhydrous sodium carbonate: \[ \text{Moles of } \text{Na}_2\text{CO}_3 = \frac{\text{mass}}{\text{molar mass}} = \frac{5.3 , \text{g}}{106.0 , \text{g/mol}} \approx 0.050 , \text{mol} \]
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Calculate the number of moles of water lost:
- Molar mass of water \( \text{H}_2\text{O} = (2 \times 1.0) + (16.0) = 18.0 , \text{g/mol} \)
- Moles of water lost: \[ \text{Moles of water} = \frac{8.9 , \text{g}}{18.0 , \text{g/mol}} \approx 0.494 , \text{mol} \]
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Determine x (number of moles of water of crystallization):
- Assume the number of moles of sodium carbonate corresponds to x moles of water: \[ \text{Each mole of } \text{Na}_2\text{CO}_3 \text{ contains x moles of water} \]
- Therefore, the number of moles of water of crystallization per mole of sodium carbonate is: \[ x = \frac{0.494 , \text{mol of water}}{0.050 , \text{mol of Na}_2\text{CO}_3} \approx 9.88 \approx 10 \]
Thus, the number of moles of water of crystallization is approximately 10. Hence, x is approximately 10, and the formula of hydrated sodium carbonate is \(\text{Na}_2\text{CO}_3 \cdot 10\text{H}_2\text{O}\).
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