What is Solubility?
Solubility is defined as the maximum amount of a solute (salt) that can be dissolved in a given quantity of solvent (usually 100g of water) at a specific temperature to form a saturated solution.
Solving the Problem
Given:
- Solubility at $75^\circ C$ ($S_1$) = 155 g per 100g of water.
- Total mass of saturated solution at $75^\circ C$ ($M_{soln1}$) = 80 g.
- Mass of precipitated salt when cooled = 40 g.
Step 1: Find the mass of the salt and water in the original solution at $75^\circ C$.
A solubility of 155 means 155g of salt is dissolved in 100g of water, making the total mass of the saturated solution $155 + 100 = 255$ g.
Using proportion:
- If 255 g of solution contains 155 g of salt, then 80 g of solution contains: $Mass_{salt} = (155 / 255) \times 80 = 48.63$ g of salt.
- The mass of water is $80 - 48.63 = 31.37$ g.
Step 2: Find the mass of salt and water at $15^\circ C$.
When cooled, 40g of salt precipitates out. Note: There is a physical constraint here; the problem states 40g precipitates from an 80g solution that only contained ~48.6g of salt initially. Assuming the logic holds that we subtract the precipitate from the original salt content:
- Remaining salt in solution = $48.63 - 40 = 8.63$ g.
- The mass of the solvent (water) remains constant at 31.37 g.
Step 3: Calculate solubility at $15^\circ C$.
Solubility is defined per 100g of water: $S_2 = (Mass_{salt} / Mass_{water}) \times 100$ $S_2 = (8.63 / 31.37) \times 100 \approx 27.5$ g/100g water.