To solve the problem step by step, we will calculate the molarity, normality, molality, and mole fraction of \( H_2SO_4 \) in the solution.
### Given Data:
- Mass of \( H_2SO_4 \) (solute) = 49 g
- Volume of solution = 1 L
- Density of solution = 1.049 g/cm³ (which is equivalent to 1.049 g/mL)
### Step 1: Calculate the mass of the solution
The mass of 1 L of solution can be calculated using its density:
\[
\text{Mass of solution} = \text{Density} \times \text{Volume} = 1.049 \, \text{g/mL} \times 1000 \, \text{mL} = 1049 \, \text{g}
\]
### Step 2: Calculate the mass of the solvent (water)
The mass of the solvent can be calculated by subtracting the mass of the solute from the mass of the solution:
\[
\text{Mass of solvent} = \text{Mass of solution} - \text{Mass of solute} = 1049 \, \text{g} - 49 \, \text{g} = 1000 \, \text{g} = 1 \, \text{kg}
\]
### Step 3: Calculate the molarity of the solution
Molarity (M) is defined as the number of moles of solute per liter of solution. First, we need to calculate the number of moles of \( H_2SO_4 \):
\[
\text{Molar mass of } H_2SO_4 = 2(1) + 32 + 4(16) = 98 \, \text{g/mol}
\]
\[
\text{Moles of } H_2SO_4 = \frac{\text{Mass of } H_2SO_4}{\text{Molar mass of } H_2SO_4} = \frac{49 \, \text{g}}{98 \, \text{g/mol}} = 0.5 \, \text{mol}
\]
Now, we can calculate the molarity:
\[
\text{Molarity} = \frac{\text{Moles of solute}}{\text{Volume of solution in L}} = \frac{0.5 \, \text{mol}}{1 \, \text{L}} = 0.5 \, \text{M}
\]
### Step 4: Calculate the normality of the solution
Normality (N) is related to molarity by the number of equivalents. For \( H_2SO_4 \), which can donate 2 protons (H⁺ ions), the normality is:
\[
\text{Normality} = \text{Molarity} \times n = 0.5 \, \text{M} \times 2 = 1 \, \text{N}
\]
### Step 5: Calculate the molality of the solution
Molality (m) is defined as the number of moles of solute per kilogram of solvent:
\[
\text{Molality} = \frac{\text{Moles of solute}}{\text{Mass of solvent in kg}} = \frac{0.5 \, \text{mol}}{1 \, \text{kg}} = 0.5 \, \text{m}
\]
### Step 6: Calculate the mole fraction of \( H_2SO_4 \)
Mole fraction (\( X \)) is defined as the number of moles of the solute divided by the total number of moles in the solution:
\[
\text{Moles of water} = \frac{\text{Mass of water}}{\text{Molar mass of water}} = \frac{1000 \, \text{g}}{18 \, \text{g/mol}} \approx 55.56 \, \text{mol}
\]
Now, we can calculate the mole fraction:
\[
X_{H_2SO_4} = \frac{\text{Moles of } H_2SO_4}{\text{Moles of } H_2SO_4 + \text{Moles of water}} = \frac{0.5}{0.5 + 55.56} \approx \frac{0.5}{56.06} \approx 0.0089
\]
### Final Results:
- Molarity = \( 0.5 \, \text{M} \)
- Normality = \( 1 \, \text{N} \)
- Molality = \( 0.5 \, \text{m} \)
- Mole Fraction of \( H_2SO_4 \) = \( 0.0089 \)
