To what volume must 50 mL of 3.50 M `H_2SO_4` be diluted in order to make 2 M `H_(2)SO_(4)` ? [Hint: Use Rule 5]

Answer

To solve the problem of diluting 50 mL of 3.50 M \( H_2SO_4 \) to make a 2 M solution, we can use the dilution formula:

\[ M_1 V_1 = M_2 V_2 \]

where:
- \( M_1 \) = initial molarity (3.50 M)
- \( V_1 \) = initial volume (50 mL)
- \( M_2 \) = final molarity (2 M)
- \( V_2 \) = final volume (unknown)

### Step-by-Step Solution:

1. **Identify the Known Values:**
- \( M_1 = 3.50 \, \text{M} \)
- \( V_1 = 50 \, \text{mL} \)
- \( M_2 = 2 \, \text{M} \)

2. **Set Up the Dilution Equation:**
\[ M_1 V_1 = M_2 V_2 \]

3. **Substitute the Known Values into the Equation:**
\[ 3.50 \, \text{M} \times 50 \, \text{mL} = 2 \, \text{M} \times V_2 \]

4. **Calculate \( M_1 V_1 \):**
\[ 3.50 \times 50 = 175 \, \text{M mL} \]

5. **Rearrange the Equation to Solve for \( V_2 \):**
\[ V_2 = \frac{M_1 V_1}{M_2} \]

6. **Substitute the Values into the Rearranged Equation:**
\[ V_2 = \frac{175 \, \text{M mL}}{2 \, \text{M}} = 87.5 \, \text{mL} \]

7. **Determine the Volume of Water to be Added:**
- The initial volume is 50 mL, and the final volume after dilution is 87.5 mL.
- Volume of water to be added = \( V_2 - V_1 = 87.5 \, \text{mL} - 50 \, \text{mL} = 37.5 \, \text{mL} \)

### Final Answer:
To make a 2 M \( H_2SO_4 \) solution, the 50 mL of 3.50 M \( H_2SO_4 \) must be diluted to a total volume of 87.5 mL, which requires adding 37.5 mL of water.