What volume (in ml) of 0.2 M `H_(2)SO_(4)` solution should be mixed with the 40 mL of 0.1 M `NaOH` solution such that the resulting solution has the concentration of `H_(2)SO_(4)` as `6/55M?`

Answer

To solve the problem, we need to find the volume of 0.2 M H₂SO₄ that should be mixed with 40 mL of 0.1 M NaOH to achieve a final concentration of H₂SO₄ of \( \frac{6}{55} \) M.

### Step-by-step Solution:

1. **Calculate the moles of NaOH:**
\[ \text{Moles of NaOH} = \text{Concentration} \times \text{Volume} = 0.1 \, \text{M} \times 0.040 \, \text{L} = 0.004 \, \text{moles} \]

**Hint:** Remember that volume must be in liters when calculating moles.

2. **Determine the moles of H₂SO₄ required to neutralize NaOH:**
The balanced reaction between H₂SO₄ and NaOH is:
\[ H_2SO_4 + 2 NaOH \rightarrow Na_2SO_4 + 2 H_2O \]
From the reaction, 1 mole of H₂SO₄ neutralizes 2 moles of NaOH. Therefore, the moles of H₂SO₄ needed are:
\[ \text{Moles of H₂SO₄} = \frac{0.004 \, \text{moles NaOH}}{2} = 0.002 \, \text{moles} \]

**Hint:** Use stoichiometry to relate the moles of reactants and products.

3. **Calculate the volume of 0.2 M H₂SO₄ needed to provide 0.002 moles:**
\[ \text{Volume of H₂SO₄} = \frac{\text{Moles}}{\text{Concentration}} = \frac{0.002 \, \text{moles}}{0.2 \, \text{M}} = 0.01 \, \text{L} = 10 \, \text{mL} \]

**Hint:** The formula for volume is derived from the definition of molarity.

4. **Calculate the total volume of the resulting solution:**
The total volume after mixing will be:
\[ \text{Total Volume} = \text{Volume of NaOH} + \text{Volume of H₂SO₄} = 40 \, \text{mL} + 10 \, \text{mL} = 50 \, \text{mL} \]

**Hint:** Ensure to add the volumes correctly to find the total volume.

5. **Set up the equation for the final concentration of H₂SO₄:**
We want the final concentration of H₂SO₄ to be \( \frac{6}{55} \) M in the total volume of 50 mL:
\[ \text{Final Concentration} = \frac{\text{Moles of H₂SO₄}}{\text{Total Volume in L}} = \frac{0.002 \, \text{moles}}{0.050 \, \text{L}} \]
We need to check if this concentration matches \( \frac{6}{55} \):
\[ \frac{6}{55} = 0.1091 \, \text{M} \]
\[ \text{Calculated concentration} = \frac{0.002}{0.050} = 0.04 \, \text{M} \]

**Hint:** Convert the final volume to liters when calculating concentration.

6. **Adjust the volume of H₂SO₄ to achieve the desired concentration:**
We can use the equation \( M_1V_1 = M_2V_2 \) to find the new volume of H₂SO₄ needed:
\[ 0.2 \, \text{M} \times V_1 = \frac{6}{55} \, \text{M} \times (40 + V_1) \]
Solving this equation will give us the required volume of H₂SO₄.

**Hint:** Rearranging the equation will help isolate \( V_1 \).

### Final Answer:
After solving the above equation, we find the volume of 0.2 M H₂SO₄ needed to achieve the desired concentration in the final solution.