If at a particular temperature, the density of `18MH_(2)SO_(4)` is `1.8g cm^(-3)`, calculate (a) molality, (b) `%` concentrating by weight of solute and solvent (c) mole fraction of water and `H_(2)SO_(4)`.

Explanation

To solve the problem step by step, we will calculate the molality, the percentage concentration by weight of solute and solvent, and the mole fraction of water and H₂SO₄.

### Given Data:
- Density of 18M H₂SO₄ solution = 1.8 g/cm³
- Molarity (M) = 18 M
- Molar mass of H₂SO₄ = 98 g/mol

### Step 1: Calculate the mass of the solution
Since the density of the solution is given, we can calculate the mass of 1000 mL (1 L) of the solution.

\[ \text{Mass of solution} = \text{Density} \times \text{Volume} = 1.8 \, \text{g/cm}^3 \times 1000 \, \text{cm}^3 = 1800 \, \text{g} \]

### Step 2: Calculate the mass of solute (H₂SO₄)
The molarity of the solution tells us that there are 18 moles of H₂SO₄ in 1 L of solution.

\[ \text{Mass of H₂SO₄} = \text{Number of moles} \times \text{Molar mass} = 18 \, \text{mol} \times 98 \, \text{g/mol} = 1764 \, \text{g} \]

### Step 3: Calculate the mass of solvent (water)
The mass of the solvent (water) can be calculated by subtracting the mass of the solute from the mass of the solution.

\[ \text{Mass of solvent} = \text{Mass of solution} - \text{Mass of solute} = 1800 \, \text{g} - 1764 \, \text{g} = 36 \, \text{g} \]

### Step 4: Calculate molality (m)
Molality is defined as the number of moles of solute per kilogram of solvent.

\[ \text{Molality} (m) = \frac{\text{Number of moles of solute}}{\text{Mass of solvent in kg}} = \frac{18 \, \text{mol}}{0.036 \, \text{kg}} = 500 \, \text{mol/kg} \]

### Step 5: Calculate percentage concentration by weight of solute
The percentage by weight of the solute is calculated as follows:

\[ \text{Percentage of solute} = \left( \frac{\text{Mass of solute}}{\text{Mass of solution}} \right) \times 100 = \left( \frac{1764 \, \text{g}}{1800 \, \text{g}} \right) \times 100 \approx 98\% \]

### Step 6: Calculate percentage concentration by weight of solvent
The percentage by weight of the solvent is calculated similarly:

\[ \text{Percentage of solvent} = \left( \frac{\text{Mass of solvent}}{\text{Mass of solution}} \right) \times 100 = \left( \frac{36 \, \text{g}}{1800 \, \text{g}} \right) \times 100 \approx 2\% \]

### Step 7: Calculate mole fraction of H₂SO₄ and water
First, we need to calculate the number of moles of water (H₂O):

\[ \text{Number of moles of H₂O} = \frac{\text{Mass of H₂O}}{\text{Molar mass of H₂O}} = \frac{36 \, \text{g}}{18 \, \text{g/mol}} = 2 \, \text{mol} \]

Now, we can calculate the mole fractions:

\[ \text{Mole fraction of H₂SO₄} = \frac{\text{Number of moles of H₂SO₄}}{\text{Total moles}} = \frac{18}{18 + 2} = \frac{18}{20} = 0.9 \]

\[ \text{Mole fraction of H₂O} = \frac{\text{Number of moles of H₂O}}{\text{Total moles}} = \frac{2}{20} = 0.1 \]

### Summary of Results:
- (a) Molality = 500 mol/kg
- (b) % concentration by weight of solute = 98%, solvent = 2%
- (c) Mole fraction of H₂SO₄ = 0.9, Mole fraction of H₂O = 0.1