If `20 mL` of ethanol (density `=0.7893g//mL)` is mixed with `40mL` water (density `= 0.9971g//mL)` at `25^(@)C`, the final slution has density of `0.9571g//mL`. Calculate the percentage change in total volume of mixing. Also calculate the molality of alcohol in the final solution.

Explanation

To solve the problem step-by-step, we will calculate the percentage change in total volume of mixing and the molality of alcohol in the final solution.

### Step 1: Calculate the mass of ethanol
Given:
- Volume of ethanol (V1) = 20 mL
- Density of ethanol (d1) = 0.7893 g/mL

Using the formula for mass:
\[ \text{Mass of ethanol} = \text{Volume} \times \text{Density} = V1 \times d1 = 20 \, \text{mL} \times 0.7893 \, \text{g/mL} = 15.786 \, \text{g} \]

### Step 2: Calculate the mass of water
Given:
- Volume of water (V2) = 40 mL
- Density of water (d2) = 0.9971 g/mL

Using the formula for mass:
\[ \text{Mass of water} = \text{Volume} \times \text{Density} = V2 \times d2 = 40 \, \text{mL} \times 0.9971 \, \text{g/mL} = 39.884 \, \text{g} \]

### Step 3: Calculate the total mass of the solution
\[ \text{Total mass} = \text{Mass of ethanol} + \text{Mass of water} = 15.786 \, \text{g} + 39.884 \, \text{g} = 55.670 \, \text{g} \]

### Step 4: Calculate the total initial volume of the solution
\[ \text{Total initial volume} = V1 + V2 = 20 \, \text{mL} + 40 \, \text{mL} = 60 \, \text{mL} \]

### Step 5: Calculate the final volume of the solution using density
Given:
- Final density of the solution (d_final) = 0.9571 g/mL

Using the formula for volume:
\[ \text{Final volume} = \frac{\text{Total mass}}{\text{Final density}} = \frac{55.670 \, \text{g}}{0.9571 \, \text{g/mL}} \approx 58.14 \, \text{mL} \]

### Step 6: Calculate the percentage change in volume
\[ \text{Percentage change in volume} = \frac{\text{Initial volume} - \text{Final volume}}{\text{Initial volume}} \times 100 \]
\[ = \frac{60 \, \text{mL} - 58.14 \, \text{mL}}{60 \, \text{mL}} \times 100 \approx 3.1\% \]

### Step 7: Calculate the molality of ethanol in the solution
- Molar mass of ethanol = 46 g/mol

First, calculate the number of moles of ethanol:
\[ \text{Moles of ethanol} = \frac{\text{Mass of ethanol}}{\text{Molar mass of ethanol}} = \frac{15.786 \, \text{g}}{46 \, \text{g/mol}} \approx 0.343 \, \text{mol} \]

Next, calculate the weight of the solvent (water) in kilograms:
\[ \text{Weight of water in kg} = \frac{39.884 \, \text{g}}{1000} = 0.039884 \, \text{kg} \]

Now, calculate the molality:
\[ \text{Molality} = \frac{\text{Moles of solute}}{\text{Weight of solvent in kg}} = \frac{0.343 \, \text{mol}}{0.039884 \, \text{kg}} \approx 8.6 \, \text{mol/kg} \]

### Final Answers:
- Percentage change in volume: **3.1%**
- Molality of ethanol in the final solution: **8.6 mol/kg**