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 use the concept of dilution and the relationship between the concentrations and volumes of the solutions involved.

### Step-by-Step Solution:

1. **Define Variables:**
Let \( x \) be the volume of the 80% solution and \( y \) be the volume of the 20% solution that we need to mix.

2. **Write the Dilution Equation:**
The dilution equation can be expressed as:
\[ (C_1 \cdot V_1) + (C_2 \cdot V_2) = C_f \cdot V_f \]
Where:
- \( C_1 = 80\% \) (concentration of the first solution)
- \( C_2 = 20\% \) (concentration of the second solution)
- \( C_f = 40\% \) (concentration of the final solution)
- \( V_f = x + y \) (total volume of the final solution)

3. **Substituting Values:**
Substitute the values into the equation:
\[ (80\% \cdot x) + (20\% \cdot y) = 40\% \cdot (x + y) \]

4. **Convert Percentages to Decimals:**
Convert the percentages to decimals for easier calculations:
\[ 0.8x + 0.2y = 0.4(x + y) \]

5. **Expand the Equation:**
Expanding the right side gives:
\[ 0.8x + 0.2y = 0.4x + 0.4y \]

6. **Rearranging Terms:**
Rearranging the equation to group like terms:
\[ 0.8x - 0.4x = 0.4y - 0.2y \]
This simplifies to:
\[ 0.4x = 0.2y \]

7. **Finding the Ratio:**
Dividing both sides by \( 0.2 \):
\[ 2x = y \]
Thus, the ratio of \( x \) to \( y \) is:
\[ \frac{x}{y} = \frac{1}{2} \]

8. **Final Ratio:**
Therefore, the final ratio of the volumes of the two solutions is:
\[ 1 : 2 \]