Problem 41 - Chemistry: The Central Science in SI Units, Global Edition

Question

A solution is made containing \(50.0 \mathrm{~g}\) of ethanol \(\left(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\right)\) in \(1000 \mathrm{~g}\) of \(\mathrm{H}_{2} \mathrm{O} .\) Calculate \((\mathbf{a})\) the mole fraction of \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH},\) (b) the mass percent of \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\), (c) the molality of \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\).

Step-by-Step Solution

Answer

(a) Mole fraction of ethanol: \(0.019\) (b) Mass percent of ethanol: \(4.76\%\) (c) Molality of ethanol: \(1.085 \mathrm{~mol/kg}\)

Step 1: (Step 1: Molar Mass of Ethanol and Water)

(To find the mole fraction, mass percent, and molality of ethanol in the solution, we first need to find the molar mass of ethanol and water. For ethanol, the molar mass will be the sum of the molar masses of 2 carbon atoms, 6 hydrogen atoms, and 1 oxygen atom. For water, the molar mass will be the sum of the molar masses of 2 hydrogen atoms and 1 oxygen atom.)

Step 2: (Step 2: Moles of Ethanol and Water)

(Next, we need to calculate the moles of ethanol and water in the solution. We do this by dividing the mass of each substance by its molar mass.)

Step 3: (Step 3: Mole Fraction of Ethanol)

(We can find the mole fraction of ethanol by dividing the moles of ethanol by the total moles of both ethanol and water.)

Step 4: (Step 4: Mass Percent of Ethanol)

(The mass percent of ethanol can be calculated by dividing the mass of ethanol in the solution by the total mass of the solution and multiplying the result by 100.)

Step 5: (Step 5: Molality of Ethanol)

(The molality of ethanol can be found by dividing the moles of ethanol by the mass of the solvent (water) in kilograms.) Now let's perform the calculations. Step 1: Molar Mass of Ethanol and Water Ethanol (\(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\)): Molar mass = (2 × 12.01) + (6 × 1.01) + (1 × 16.00) = 46.07 g/mol Water (\(\mathrm{H}_{2} \mathrm{O}\)): Molar mass = (2 × 1.01) + (1 × 16.00) = 18.02 g/mol Step 2: Moles of Ethanol and Water Moles of ethanol = Mass of ethanol / Molar mass of ethanol = \(50.0 \mathrm{~g} / 46.07 \mathrm{~g/mol} = 1.085 \mathrm{~mol}\) Moles of water = Mass of water / Molar mass of water = \(1000 \mathrm{~g} / 18.02 \mathrm{~g/mol} = 55.49 \mathrm{~mol}\) Step 3: Mole Fraction of Ethanol Mole fraction of ethanol = Moles of ethanol / Total moles = \(1.085 \mathrm{~mol} / (1.085\mathrm{~mol} + 55.49 \mathrm{~mol}) \approx 0.019\) Step 4: Mass Percent of Ethanol Mass percent of ethanol = (Mass of ethanol / Total mass) × 100 = \((50.0 \mathrm{~g} / (50.0 \mathrm{~g} + 1000 \mathrm{~g})) × 100 = 4.76\%\) Step 5: Molality of Ethanol Molality of ethanol = Moles of ethanol / Mass of water in kg = \(1.085 \mathrm{~mol} / 1.000 \mathrm{~kg} = 1.085 \mathrm{~mol/kg}\) The results are as follows: (a) Mole fraction of ethanol: 0.019 (b) Mass percent of ethanol: 4.76% (c) Molality of ethanol: 1.085 mol/kg

Key Concepts

Mole Fraction

The mole fraction is a way to express the concentration of a component in a solution by comparing it to the total number of moles of all components. It's a dimensionless number and is particularly useful in thermodynamics because it does not change with temperature.
To calculate the mole fraction of ethanol in our solution, we first need the number of moles of ethanol and water. Based on the known masses and molar masses, we find 1.085 moles of ethanol and 55.49 moles of water. The mole fraction formula is:

• Mole fraction of ethanol = \(\frac{\text{moles of ethanol}}{\text{total moles of ethanol and water}}\)

This results in a mole fraction of approximately 0.019 for ethanol. This means that 1.9% of the molecules in the solution are ethanol molecules. Understanding the mole fraction helps in comprehending how different components are distributed in a solution.

Mass Percent

Mass percent gives the concentration of a component in a solution as a percentage of the total mass. It is particularly intuitive because it relates the mass of a solute to the total mass of the solution, making it easy to understand and used widely in fields like chemistry and material science.
To determine the mass percent of ethanol, we divide the mass of ethanol by the total mass of the solution and then multiply by 100 to get a percentage. With ethanol weighing 50.0 grams and water 1000 grams, the formula is:

• Mass percent of ethanol = \(\frac{\text{mass of ethanol}}{\text{total mass of solution}} \times 100\)

This calculation results in a mass percent of 4.76%, indicating that ethanol constitutes 4.76% of the total solution mass. Mass percent is a practical measure for quickly assessing how much of a substance is present, making it an essential calculation in solution preparation.

Molality

Molality is another concentration measure that expresses moles of solute per kilogram of solvent. It is particularly helpful because it remains unchanged regardless of temperature, unlike molarity. It's used in situations where temperature changes might occur, such as in boiling point elevation or freezing point depression.
For our solution, the molality of ethanol is found by dividing the number of moles of ethanol by the mass of the solvent (water) in kilograms. The formula is as follows:

• Molality of ethanol = \(\frac{\text{moles of ethanol}}{\text{mass of water in kg}}\)

With 1.085 moles of ethanol and the water mass being 1 kg, the molality is calculated as 1.085 mol/kg. This tells us how many moles of ethanol are present for every kilogram of water. Molality is advantageous in certain calculations because it does not change with temperature, making it reliable for experiments that involve heating or cooling.