Question

One hundred kilograms of a dilute waste acid containing 30.0% sulphuric acid is to be fortified to 50.0% sulphuric acid using concentrated sulphuric acid of strength 96%. How many kilograms of concentrated sulphuric acid are required for this process?

• A. 33.88 kg
• B. 43.48 kg
• C. 68.54 kg
• D. 50.86 kg

Hint:

For rapid calculation, remember the lever rule or Pearson's square:
Ratio of concentrated acid to dilute acid is given by:
\[ \frac{C}{W} = \frac{w_P - w_W}{w_C - w_P} = \frac{0.50 - 0.30}{0.96 - 0.50} = \frac{0.20}{0.46} \]
This gives the direct ratio and saves precious time during competitive examinations.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
This is a steady-state material balance problem with no chemical reaction.
We are mixing a weak/dilute acid stream with a highly concentrated acid stream to achieve a target acid concentration of \( 50.0\% \) sulphuric acid.

Step 2: Key Formula or Approach:
Let \( W \) be the mass of the dilute waste acid, which is \( 100 \text{ kg} \).
Let \( C \) be the mass of the concentrated sulphuric acid added.
Let \( P \) be the mass of the fortified product acid.
We write two mass conservation equations:
1. Total Mass Balance:
\[ W + C = P \]
2. Sulphuric Acid Component Balance:
\[ w_W \cdot W + w_C \cdot C = w_P \cdot P \]
where \( w_W \), \( w_C \), and \( w_P \) are the weight fractions of sulphuric acid in each stream respectively.

Step 3: Detailed Explanation:

• Identify the given variables:
\( W = 100 \text{ kg} \)
\( w_W = 30.0\% = 0.30 \)
\( w_C = 96\% = 0.96 \)
\( w_P = 50.0\% = 0.50 \)

• Express the total mass balance:
\[ P = 100 + C \]

• Substitute \( P \) into the sulphuric acid balance:
\[ 0.30 \cdot (100) + 0.96 \cdot C = 0.50 \cdot (100 + C) \]

• Perform the algebraic expansions:
\[ 30 + 0.96 \cdot C = 50 + 0.50 \cdot C \]

• Rearrange the terms to solve for \( C \):
\[ 0.96 \cdot C - 0.50 \cdot C = 50 - 30 \]
\[ 0.46 \cdot C = 20 \]
\[ C = \frac{20}{0.46} \approx 43.478 \text{ kg} \]


Step 4: Final Answer:
The mass of concentrated sulphuric acid needed is approximately 43.48 kg.