Problem 238 - A Complete Resource Book in Chemistry for JEE Main

Question

\(10 \mathrm{~mL}\) of pure ethanol of density \(0.785 \mathrm{~g} / \mathrm{mL}\) was diluted with water to a final volume of \(100 \mathrm{~mL}\). The density of the resulting solution was \(0.9866 \mathrm{~g} / \mathrm{mL}\). The percentage by weight of ethanol is about

Step-by-Step Solution

Answer

The percentage by weight of ethanol is about 7.96%.

Step 1: Determine Ethanol Mass

The mass of ethanol can be calculated by multiplying its volume by its density. Therefore, we have:\[ \text{Mass of ethanol} = 10 \text{ mL} \times 0.785 \text{ g/mL} = 7.85 \text{ g} \]

Step 2: Determine Total Mass of Solution

The total mass of the solution can be found by multiplying the final volume by the density of the solution. That is:\[ \text{Total mass of solution} = 100 \text{ mL} \times 0.9866 \text{ g/mL} = 98.66 \text{ g} \]

Step 3: Calculate Percentage by Weight

The percentage by weight of ethanol in the solution can be determined using the formula:\[ \text{Percentage by weight} = \left(\frac{\text{Mass of ethanol}}{\text{Total mass of solution}}\right) \times 100 \]Substitute the known values:\[ \text{Percentage by weight} = \left(\frac{7.85 \text{ g}}{98.66 \text{ g}}\right) \times 100 \approx 7.96\% \]

Key Concepts

Density Calculation

Density is an important concept in solutions as it helps us to understand the relationship between mass and volume. In simple terms, density is the amount of mass in a given volume and is usually expressed in units such as grams per milliliter (g/mL) or kilograms per cubic meter (kg/m³). Calculating the density of a substance involves dividing the mass by the volume using the formula: \[\text{Density} = \frac{\text{Mass}}{\text{Volume}}\]In the original exercise, the density of pure ethanol was given as \(0.785 \text{ g/mL}\). This means that for every milliliter of ethanol, there is a mass of 0.785 grams. This density information is crucial when determining how much ethanol was initially available before dilution.

• Key Insight: Remember, density can provide information about how much of a substance is contained in a space, but does not change with the amount of the sample as it is an intensive property.

Mass Percent Calculation

The mass percent of a solution is a way to express the concentration of a solute (in our context, ethanol) in a solution. It represents the ratio of the mass of the solute to the total mass of the solution, expressed as a percentage. This is defined by the formula:\[\text{Mass percent} = \left(\frac{\text{Mass of solute}}{\text{Total mass of solution}}\right) \times 100\]From the exercise, we calculated that the mass of ethanol was \(7.85 \text{ g}\) and the total mass of the resulting solution was \(98.66 \text{ g}\). Using these values, the mass percent calculation resulted in approximately \(7.96\%\). This means that out of the total mass of the solution, approximately 7.96% is ethanol.

• Useful Tip: Always ensure both masses are in the same units. This ensures accurate calculations and avoid inconsistencies.

Dilution Process

Dilution is a process used to decrease the concentration of a solute in a solution by adding more solvent. In the original problem, pure ethanol was diluted with water from \(10 \text{ mL}\) to \(100 \text{ mL}\). This means that water was added to increase the total volume of the solution. The amount of water added can be calculated as:\[\text{Volume of water added} = \text{Final volume} - \text{Initial volume of ethanol} = 100 \text{ mL} - 10 \text{ mL} = 90 \text{ mL}\]During dilution, the total amount of solute (ethanol in this case) does not change, only its concentration does. This means that although we have increased the volume, the absolute mass of ethanol remains at \(7.85 \text{ g}\), just like before dilution.

• Remember: The purpose of dilution is to create a less concentrated solution, but the amount of solute does not change in the process. It only affects its concentration in the resulting solution.