Crystallization material balance:\nAn aqueous solution (total 330 kg) contains 80 kg Na2SO4 (M = 142). Upon cooling, 80 kg of Na2SO4·10H2O crystals form and are removed. Determine the weight fraction of Na2SO4 remaining in the mother liquor.

Introduction / Context:
Crystallization problems require careful component balances, especially when hydrates form. Here, sodium sulfate crystallizes as the decahydrate (Na2SO4·10H2O). We must track how much anhydrous salt leaves in the crystals and what remains dissolved, then compute the composition of the remaining mother liquor.

Given Data / Assumptions:

• Initial solution mass: 330 kg, containing 80 kg Na2SO4 (anhydrous basis).
• Crystals removed: 80 kg Na2SO4·10H2O (molar mass 142 + 10*18 = 322 kg/kmol).
• No other losses; mother liquor is what remains after crystal removal.

Concept / Approach:
First determine how much anhydrous Na2SO4 is contained within the crystals using stoichiometric ratios from molecular weights. Subtract that from the initial Na2SO4 to find what remains dissolved. The mass of mother liquor equals initial total mass minus mass of crystals removed. Finally, compute the weight fraction of Na2SO4 in the mother liquor.

Step-by-Step Solution:

Verification / Alternative check:
The water tied up in crystals is (180/322)*80 ≈ 44.7 kg; plausibly removed from the initial solution’s water balance, leaving 250 kg mother liquor as computed.

Why Other Options Are Wrong:

• 0.00 and 1.00 are extreme and contradict mass balance.
• 0.24 overestimates the remaining solute fraction compared with the stoichiometric removal into crystals.

Common Pitfalls:
Forgetting that decahydrate contains only a fraction of Na2SO4 by mass; subtracting 80 kg directly from the initial Na2SO4; ignoring that the mother liquor mass is the initial total minus crystal mass.

Final Answer:
0.18