The concentration of `H_(2)SO_(4)` in a botal labelled ''conc. Sulphuric acid'' is 18 M. The solution has a density of `"1.84 g cm"^(-3)`. What is the mole fraction and weight percentage of `H_(2)SO_(4)` in this solution?

Explanation

To solve the problem of finding the mole fraction and weight percentage of `H₂SO₄` in a concentrated solution, we can follow these steps:

### Step 1: Calculate the number of moles of `H₂SO₄`
Given that the molarity (M) of the solution is 18 M, this means there are 18 moles of `H₂SO₄` in 1 liter (1000 mL) of solution.

**Moles of `H₂SO₄` = 18 moles**

### Step 2: Calculate the mass of `H₂SO₄`
To find the mass of `H₂SO₄`, we use the formula:
\[ \text{Mass} = \text{Number of moles} \times \text{Molar mass} \]

The molar mass of `H₂SO₄` is approximately 98 g/mol. Therefore:
\[ \text{Mass of } H₂SO₄ = 18 \, \text{moles} \times 98 \, \text{g/mol} = 1764 \, \text{grams} \]

### Step 3: Calculate the mass of the solution
The density of the solution is given as 1.84 g/cm³. Since we have 1 liter of solution, we can convert this volume to cm³:
\[ 1 \, \text{L} = 1000 \, \text{cm}^3 \]

Now, we can calculate the mass of the solution using the density formula:
\[ \text{Mass} = \text{Density} \times \text{Volume} \]
\[ \text{Mass of solution} = 1.84 \, \text{g/cm}^3 \times 1000 \, \text{cm}^3 = 1840 \, \text{grams} \]

### Step 4: Calculate the mass of the solvent
The mass of the solvent can be calculated by subtracting the mass of the solute from the mass of the solution:
\[ \text{Mass of solvent} = \text{Mass of solution} - \text{Mass of } H₂SO₄ \]
\[ \text{Mass of solvent} = 1840 \, \text{g} - 1764 \, \text{g} = 76 \, \text{grams} \]

### Step 5: Calculate the number of moles of solvent
To find the number of moles of the solvent (water), we use the molar mass of water, which is approximately 18 g/mol:
\[ \text{Number of moles of solvent} = \frac{\text{Mass of solvent}}{\text{Molar mass of water}} \]
\[ \text{Number of moles of solvent} = \frac{76 \, \text{g}}{18 \, \text{g/mol}} \approx 4.22 \, \text{moles} \]

### Step 6: Calculate the mole fraction of `H₂SO₄`
The mole fraction (X) of `H₂SO₄` is given by the formula:
\[ X_{H₂SO₄} = \frac{\text{Number of moles of } H₂SO₄}{\text{Number of moles of } H₂SO₄ + \text{Number of moles of solvent}} \]
\[ X_{H₂SO₄} = \frac{18}{18 + 4.22} \approx \frac{18}{22.22} \approx 0.81 \]

### Step 7: Calculate the weight percentage of `H₂SO₄`
The weight percentage of `H₂SO₄` is calculated using the formula:
\[ \text{Weight percentage} = \left( \frac{\text{Weight of } H₂SO₄}{\text{Weight of solution}} \right) \times 100 \]
\[ \text{Weight percentage} = \left( \frac{1764 \, \text{g}}{1840 \, \text{g}} \right) \times 100 \approx 95.87\% \]

### Final Results
- **Mole Fraction of `H₂SO₄`**: 0.81
- **Weight Percentage of `H₂SO₄`**: 95.87%