One hundred kilogram spice is extracted for essential oil using twice the amount of a pure organic solvent. The extracted solid mass contains 5% residual oil (oil-free solid mass basis). The liquid extracted mass contains 20% oil. Assume no solvent is retained by the extracted solid mass. Initial mass of the oil in the spice in kg is ______
Official Answer Key:
51 to 54
Step-by-Step Solution & Explanation
Solvent used = 200 kg. Let x = initial oil (kg) in the 100 kg spice; oil-free solid = (100−x) kg. Since the extracted liquid contains 20% oil by mass, and all 200 kg solvent reports to the liquid (none retained by solid): liquid mass = solvent + oil_extracted, and oil_extracted = 0.20×liquid mass. Solving: oil_extracted = 0.20×(200+oil_extracted) → 0.8×oil_extracted = 40 → oil_extracted = 50 kg. Oil retained in the solid (5% of the oil-free solid mass) = 0.05×(100−x). Overall oil balance: x = oil_extracted + oil_retained = 50 + 0.05×(100−x) → x = 50+5−0.05x → 1.05x = 55 → x = 52.4 kg, within the accepted band 51–54.
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