`30 mL` of `CH_(3)OH (d = 0.8 g//cm^(3))` is mixed with `60 mL` of `C_(2)H_(5)OH(d = 0.92 g//cm^(2))` at `25^(@)C` to form a solution of density `0.88 g//cm^(3)`. Select the correct option(s) :

Options

A. Molarity and molality of resulting solution are 6.33 and 13.59 respectively
B. The mole fraction of solute and molality are 0.385 and 13.59 respectively
C. Molarity and % change in volume are 0.615 and zero respectively
D. Mole fraction of solvent and molality are 0.615 and 13.59 respectively

Answer

To solve the problem, we need to calculate the mass of methanol (CH₃OH) and ethanol (C₂H₅OH) from their respective densities and volumes, then determine the molarity, molality, and mole fractions of the components in the solution. Here’s a step-by-step breakdown of the solution:

### Step 1: Calculate the mass of methanol (CH₃OH)
Given:
- Volume of methanol (V₁) = 30 mL
- Density of methanol (d₁) = 0.8 g/cm³

Using the formula:
\[ \text{Mass} = \text{Density} \times \text{Volume} \]

Convert volume to cm³ (1 mL = 1 cm³):
\[ \text{Mass of CH₃OH} = 0.8 \, \text{g/cm³} \times 30 \, \text{cm³} = 24 \, \text{g} \]

### Step 2: Calculate the mass of ethanol (C₂H₅OH)
Given:
- Volume of ethanol (V₂) = 60 mL
- Density of ethanol (d₂) = 0.92 g/cm³

Using the same formula:
\[ \text{Mass of C₂H₅OH} = 0.92 \, \text{g/cm³} \times 60 \, \text{cm³} = 55.2 \, \text{g} \]

### Step 3: Calculate the total mass of the solution
Total mass of the solution is the sum of the masses of methanol and ethanol:
\[ \text{Mass of solution} = \text{Mass of CH₃OH} + \text{Mass of C₂H₅OH} = 24 \, \text{g} + 55.2 \, \text{g} = 79.2 \, \text{g} \]

### Step 4: Calculate the total volume of the solution
Total volume of the solution is the sum of the volumes of methanol and ethanol:
\[ \text{Total volume} = V₁ + V₂ = 30 \, \text{mL} + 60 \, \text{mL} = 90 \, \text{mL} \]

### Step 5: Calculate the molarity of the solution
First, calculate the number of moles of methanol:
- Molar mass of CH₃OH = 32 g/mol
\[ \text{Moles of CH₃OH} = \frac{\text{Mass}}{\text{Molar mass}} = \frac{24 \, \text{g}}{32 \, \text{g/mol}} = 0.75 \, \text{mol} \]

Now, convert the total volume to liters:
\[ \text{Total volume in liters} = 90 \, \text{mL} = 0.09 \, \text{L} \]

Now calculate molarity (M):
\[ \text{Molarity (M)} = \frac{\text{Moles of solute}}{\text{Volume of solution in L}} = \frac{0.75 \, \text{mol}}{0.09 \, \text{L}} \approx 8.33 \, \text{M} \]

### Step 6: Calculate the molality of the solution
Using the mass of ethanol (solvent) in kg:
\[ \text{Mass of solvent in kg} = \frac{55.2 \, \text{g}}{1000} = 0.0552 \, \text{kg} \]

Now calculate molality (m):
\[ \text{Molality (m)} = \frac{\text{Moles of solute}}{\text{Mass of solvent in kg}} = \frac{0.75 \, \text{mol}}{0.0552 \, \text{kg}} \approx 13.59 \, \text{mol/kg} \]

### Step 7: Calculate the mole fractions
1. **Mole fraction of solute (methanol)**:
\[ \text{Mole fraction of CH₃OH} = \frac{\text{Moles of CH₃OH}}{\text{Moles of CH₃OH} + \text{Moles of C₂H₅OH}} \]

Calculate moles of ethanol:
\[ \text{Moles of C₂H₅OH} = \frac{55.2 \, \text{g}}{46 \, \text{g/mol}} \approx 1.20 \, \text{mol} \]

Now calculate the mole fraction:
\[ \text{Mole fraction of CH₃OH} = \frac{0.75}{0.75 + 1.20} = \frac{0.75}{1.95} \approx 0.385 \]

2. **Mole fraction of solvent (ethanol)**:
\[ \text{Mole fraction of C₂H₅OH} = 1 - \text{Mole fraction of CH₃OH} \approx 1 - 0.385 \approx 0.615 \]

### Summary of Results
- Molarity of the solution = 8.33 M
- Molality of the solution = 13.59 mol/kg
- Mole fraction of methanol = 0.385
- Mole fraction of ethanol = 0.615