182. In a continuous crystallizer, 100 g of a saturated solution of sugar at 85% (w/w) enters the crystallizer and
leaves the crystallizer at 70% (w/w). The weight of input solids converted to crystals (g) in the crystallizer is
1. 30
2. 50
3. 70
4. 85

Introduction:

Crystallization is a process used to purify or separate components in a solution. In industrial processes, such as sugar production, crystallizers are used to convert a portion of the dissolved solids into crystalline form. In a continuous crystallizer, the input and output concentrations of solids can be used to calculate the weight of solids that are converted to crystals.

In this article, we’ll walk through the calculation of how much sugar, initially dissolved in a saturated solution, is converted into crystals in a continuous crystallizer.

Problem Overview:

You are provided with the following information about the continuous crystallizer process:

• Input Solution: 100 g of a saturated sugar solution at 85% (w/w) sugar.

• Output Solution: The solution leaves the crystallizer at 70% (w/w) sugar.

The task is to determine how much of the sugar (in grams) has been converted into crystals.

Key Concepts:

• W/W (Weight/Weight) Concentration: This refers to the mass of the solute (in this case, sugar) in a given mass of solution.

• Saturated Solution: A solution where the maximum amount of solute is dissolved in the solvent at a specific temperature.

We use the principle of mass balance to solve this problem.

Step-by-Step Calculation:

1. Input Concentration of Sugar:

   • Input Mass of Solution = 100 g

   • Sugar Concentration in Input = 85% (w/w)

   The mass of sugar in the input solution is:

   $$\text{Mass of Sugar (Input)}=100\text{g}\times 0.85=85\text{g}$$

2. Output Concentration of Sugar:

   • Sugar Concentration in Output = 70% (w/w)

   Let the mass of the output solution be x grams. The mass of sugar in the output solution is:

   $$\text{Mass of Sugar (Output)}=x\times 0.70$$

3. Mass Balance: The mass of sugar converted into crystals is the difference between the sugar in the input and the sugar in the output:

   $$\text{Sugar Converted to Crystals}=\text{Mass of Sugar (Input)}-\text{Mass of Sugar (Output)}$$

   Substituting the values:

   $$\text{Sugar Converted to Crystals}=85-(x\times 0.70)$$

4. Total Mass of Output Solution: The total mass of the output solution can be calculated based on the fact that 30% of the output solution is water (since 70% is sugar). Thus, the total mass of the output solution is:

   Mass of Output Solution=Mass of Sugar (Input)0.30=100 g−Sugar Converted to Crystals0.30\text{Mass of Output Solution} = \frac{\text{Mass of Sugar (Input)}}{0.30} = \frac{100 \, \text{g} – \text{Sugar Converted to Crystals}}{0.30}Mass of Output Solution=0.30Mass of Sugar (Input)​=0.30100g−Sugar Converted to Crystals​

5. Solve the Equations: After calculating, we find that the amount of sugar converted to crystals is 50 grams.

Conclusion:

Based on the calculation, 50 grams of the input sugar has been converted into crystals.

Answer:

The correct answer is 2. 50 g.