In what ratio a 80% (wt./vol) solution of `H_2SO_4` be mixed to 20% (wt./vol) of `H_2SO_4` to produce 40% (wt/vol) `H_2SO_4` solution

Options

A. `2:1`
B. `1:2`
C. `1:3`
D. `3:1`

Answer

To solve the problem of mixing an 80% (wt/vol) solution of H₂SO₄ with a 20% (wt/vol) solution to produce a 40% (wt/vol) solution, we can follow these steps:

### Step-by-Step Solution:

1. **Define Variables:**
Let \( V_1 \) be the volume of the 80% H₂SO₄ solution and \( V_2 \) be the volume of the 20% H₂SO₄ solution.

2. **Set Up the Equation:**
The total amount of H₂SO₄ in the mixture can be expressed as:
\[ 0.80 V_1 + 0.20 V_2 = 0.40 (V_1 + V_2) \]
Here, \( 0.80 V_1 \) represents the amount of H₂SO₄ from the 80% solution, \( 0.20 V_2 \) from the 20% solution, and \( 0.40 (V_1 + V_2) \) is the total H₂SO₄ in the final 40% solution.

3. **Simplify the Equation:**
Distributing the right side:
\[ 0.80 V_1 + 0.20 V_2 = 0.40 V_1 + 0.40 V_2 \]

4. **Rearrange the Equation:**
Move all terms involving \( V_1 \) to one side and all terms involving \( V_2 \) to the other side:
\[ 0.80 V_1 - 0.40 V_1 = 0.40 V_2 - 0.20 V_2 \]
This simplifies to:
\[ 0.40 V_1 = 0.20 V_2 \]

5. **Express the Ratio:**
Dividing both sides by \( V_2 \) and then by 0.20 gives:
\[ \frac{V_1}{V_2} = \frac{0.20}{0.40} = \frac{1}{2} \]

6. **Final Ratio:**
The required ratio of the volumes of the two solutions is:
\[ V_1 : V_2 = 1 : 2 \]

### Conclusion:
The ratio in which the 80% and 20% solutions should be mixed to produce a 40% solution is **1:2**.