Calculate the mass of potassium sulfate required to prepare its 45% solution in 100g of water.

Options

A. 14g
B. 54g
C. 74g
D. 81.8g

Explanation

To solve the problem of calculating the mass of potassium sulfate required to prepare a 45% solution in 100g of water, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Definition of Mass Percentage**:
The mass percentage of a solution is defined as:
\[ \text{Mass \%} = \left( \frac{\text{Mass of solute}}{\text{Mass of solution}} \right) \times 100 \]
In this case, the solute is potassium sulfate (K₂SO₄) and the solvent is water.

2. **Set Up the Equation**:
Let the mass of potassium sulfate required be \( x \) grams. The total mass of the solution will be the mass of the solute (potassium sulfate) plus the mass of the solvent (water). Therefore:
\[ \text{Mass of solution} = x + 100 \text{ g} \]
Given that the mass percentage is 45%, we can set up the equation:
\[ 45 = \left( \frac{x}{x + 100} \right) \times 100 \]

3. **Rearranging the Equation**:
To remove the percentage, we can simplify the equation:
\[ 45 = \frac{x}{x + 100} \times 100 \]
Dividing both sides by 100:
\[ 0.45 = \frac{x}{x + 100} \]

4. **Cross-Multiplying**:
Now, we can cross-multiply to solve for \( x \):
\[ 0.45(x + 100) = x \]
Expanding the left side:
\[ 0.45x + 45 = x \]

5. **Isolating \( x \)**:
Rearranging the equation to isolate \( x \):
\[ 45 = x - 0.45x \]
This simplifies to:
\[ 45 = 0.55x \]

6. **Solving for \( x \)**:
Finally, divide both sides by 0.55 to find \( x \):
\[ x = \frac{45}{0.55} \approx 81.82 \text{ g} \]

### Conclusion:
The mass of potassium sulfate required to prepare a 45% solution in 100g of water is approximately **81.82 grams**.