Find the molality fo `H_(2)SO_(4)` solution whose specific gravity is `1.98 g mL^(-1)` and 98% (Weight/volume) `H_(2)SO_(4)`.

Explanation

To find the molality of the `H₂SO₄` solution with a specific gravity of `1.98 g/mL` and `98% (weight/volume)` concentration, we can follow these steps:

### Step 1: Understand the given information
- Specific gravity of the solution = `1.98 g/mL`
- Concentration of `H₂SO₄` = `98% (weight/volume)`

### Step 2: Assume a volume of the solution
For ease of calculation, we can assume a volume of `100 mL` of the solution. This means:
- Volume of solution = `100 mL`

### Step 3: Calculate the weight of `H₂SO₄`
Since the solution is `98% (weight/volume)`, it means that in `100 mL` of the solution, there are `98 g` of `H₂SO₄`.

### Step 4: Calculate the moles of `H₂SO₄`
To find the number of moles of `H₂SO₄`, we use the formula:
\[ \text{Moles of } H₂SO₄ = \frac{\text{Weight of } H₂SO₄}{\text{Molar mass of } H₂SO₄} \]
The molar mass of `H₂SO₄` is `98 g/mol`. Therefore:
\[ \text{Moles of } H₂SO₄ = \frac{98 g}{98 g/mol} = 1 \text{ mole} \]

### Step 5: Calculate the weight of the solution
Using the specific gravity, we can find the weight of the solution:
\[ \text{Weight of solution} = \text{Specific gravity} \times \text{Volume of solution} \]
\[ \text{Weight of solution} = 1.98 \, \text{g/mL} \times 100 \, \text{mL} = 198 \, \text{g} \]

### Step 6: Calculate the weight of the solvent (water)
To find the weight of the solvent (water), we subtract the weight of `H₂SO₄` from the total weight of the solution:
\[ \text{Weight of water} = \text{Weight of solution} - \text{Weight of } H₂SO₄ \]
\[ \text{Weight of water} = 198 \, \text{g} - 98 \, \text{g} = 100 \, \text{g} \]

### Step 7: Convert the weight of water to kg
Since molality is defined as moles of solute per kilogram of solvent, we convert the weight of water from grams to kilograms:
\[ \text{Weight of water in kg} = \frac{100 \, \text{g}}{1000} = 0.1 \, \text{kg} \]

### Step 8: Calculate the molality
Now we can calculate the molality using the formula:
\[ \text{Molality} = \frac{\text{Moles of solute}}{\text{Weight of solvent in kg}} \]
\[ \text{Molality} = \frac{1 \text{ mole}}{0.1 \text{ kg}} = 10 \, \text{mol/kg} \]

### Final Answer
The molality of the `H₂SO₄` solution is `10 mol/kg`.

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