Battery acid is 4.27 M `H_(2)SO_(4)` (aq) and has density of `1.25 mL^(-1)` . What is the molality of `H_(2)SO_(4)` in the solution ?

Explanation

To find the molality of the `H₂SO₄` solution, we will follow these steps:

### Step 1: Write down the given data
- Molarity (M) of `H₂SO₄` = 4.27 M
- Density of the solution = 1.25 g/mL

### Step 2: Calculate the mass of the solution
Since molarity is defined as moles of solute per liter of solution, we can calculate the mass of the solution using its density.

1. **Convert the volume of the solution to liters**:
- Volume = 1 L = 1000 mL

2. **Use the density to find the mass of the solution**:
\[ \text{Mass of solution} = \text{Density} \times \text{Volume} = 1.25 \, \text{g/mL} \times 1000 \, \text{mL} = 1250 \, \text{g} \]

### Step 3: Calculate the mass of `H₂SO₄`
1. **Calculate the number of moles of `H₂SO₄`**:
\[ \text{Moles of } H₂SO₄ = 4.27 \, \text{moles} \]

2. **Calculate the molar mass of `H₂SO₄`**:
- Molar mass of `H₂SO₄` = 2(1.01 g/mol) + 32.07 g/mol + 4(16.00 g/mol) = 98.09 g/mol

3. **Calculate the mass of `H₂SO₄`**:
\[ \text{Mass of } H₂SO₄ = \text{Moles} \times \text{Molar mass} = 4.27 \, \text{moles} \times 98.09 \, \text{g/mol} = 418.46 \, \text{g} \]

### Step 4: Calculate the mass of the solvent (water)
1. **Subtract the mass of `H₂SO₄` from the mass of the solution**:
\[ \text{Mass of water} = \text{Mass of solution} - \text{Mass of } H₂SO₄ = 1250 \, \text{g} - 418.46 \, \text{g} = 831.54 \, \text{g} \]

### Step 5: Convert the mass of water to kilograms
\[ \text{Mass of water in kg} = \frac{831.54 \, \text{g}}{1000} = 0.83154 \, \text{kg} \]

### Step 6: Calculate the molality
1. **Use the formula for molality (m)**:
\[ \text{Molality (m)} = \frac{\text{Moles of solute}}{\text{Mass of solvent in kg}} = \frac{4.27 \, \text{moles}}{0.83154 \, \text{kg}} \approx 5.135 \, \text{mol/kg} \]

### Final Answer:
The molality of `H₂SO₄` in the solution is approximately **5.135 mol/kg**.

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