Problem 41 - Chemistry The Central Science In Si Units

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 \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH},(\mathbf{c})\) the molality of \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\).

Step-by-Step Solution

Answer

Mole fraction: 0.0193, Mass percent: 4.76%, Molality: 1.085 mol/kg.

Step 1: Calculate the Moles of Ethanol

First, find the molar mass of ethanol (\(\mathrm{C}_2\mathrm{H}_5\mathrm{OH}\)). Ethanol has 2 carbon atoms, 6 hydrogen atoms, and 1 oxygen atom. Therefore, its molar mass is calculated as:\[\left(2 \times 12.01\right) + \left(6 \times 1.008\right) + \left(1 \times 16.00\right) = 46.08 \, \text{g/mol}\]. Now, calculate the number of moles of ethanol using the formula: \(\text{moles} = \frac{\text{mass}}{\text{molar mass}}\). So, \[\text{moles of } \mathrm{C}_2\mathrm{H}_5\mathrm{OH} = \frac{50.0 \text{ g}}{46.08 \text{ g/mol}} \approx 1.085 \text{ moles}\].

Step 2: Calculate the Mole Fraction of Ethanol

Find the moles of water (\(\mathrm{H}_2\mathrm{O}\)). The molar mass of water is \(18.015\, \text{g/mol}\). Thus, \[\text{moles of } \mathrm{H}_2\mathrm{O}= \frac{1000 \text{ g}}{18.015 \text{ g/mol}} \approx 55.51 \text{ moles}\]. Now, use the mole fraction formula: \(\chi = \frac{\text{moles of solute}}{\text{moles of solute} + \text{moles of solvent}}\). Substitute the known values: \[\chi_{\mathrm{C}_2\mathrm{H}_5\mathrm{OH}} = \frac{1.085}{1.085 + 55.51} \approx 0.0193\].

Step 3: Calculate the Mass Percent of Ethanol

Mass percent is calculated by the formula \(\text{mass percent} = \left(\frac{\text{mass of solute}}{\text{mass of solution}} \right) \times 100\%\). The mass of the solution is \(50.0 \text{ g (ethanol)} + 1000 \text{ g (water)} = 1050 \text{ g}\). Therefore, the mass percent of ethanol is: \[\frac{50.0}{1050} \times 100\% \approx 4.76\%\].

Step 4: Calculate the Molality of Ethanol

Molality is defined as the moles of solute per kilogram of solvent. Using the moles for ethanol calculated earlier (1.085 moles) and the mass of the solvent (water) in kilograms, which is 1.000 kg, compute: \[\text{molality} = \frac{1.085}{1.000} = 1.085 \text{ mol/kg}\].

Key Concepts

Mole Fraction

The mole fraction is a way to express the concentration of a component in a mixture. It's the ratio of the number of moles of one component to the total number of moles of all components in the solution. In this exercise, the mole fraction of ethanol was calculated by dividing the moles of ethanol by the sum of moles of ethanol and water. This provides a dimensionless number, which indicates the proportion of ethanol molecules among all molecules in the solution.

• Number of moles of ethanol: Calculated using its mass and molar mass.
• Number of moles of water: Found by dividing the mass of water by its molar mass.
• Mole fraction formula: \[ \chi = \frac{\text{moles of solute}}{\text{moles of solute} + \text{moles of solvent}} \]
• Result: 0.0193, indicating ethanol is a minor component relative to water.

Knowing the mole fraction helps in understanding the composition and predicting the behavior of solutions, especially in physical chemistry.

Mass Percent

Mass percent is another way to describe the concentration of a substance in a solution. It tells you what fraction of the total mass of the solution is made up of a particular solute. This is useful for practical tasks like preparing a solution to a specific concentration.

• The formula used is: \[ \text{Mass percent} = \left(\frac{\text{mass of solute}}{\text{mass of solution}} \right) \times 100\% \]
• In this exercise, the mass of the solution is the sum of the mass of ethanol and water.
• The calculated mass percent for ethanol is approximately 4.76%.

Mass percent is commonly used in laboratory settings for creating solutions with precise amounts of solute. It's a straightforward way to communicate how much of a substance is contained in a solution.

Molality

Molality is an important concentration unit in solution chemistry, especially when dealing with temperature-dependent scenarios. Unlike molarity, molality is independent of temperature since it is based on the mass of the solvent, not its volume.

• Calculated using: \[ \text{Molality} = \frac{\text{moles of solute}}{\text{kilograms of solvent}} \]
• For our solution, the number of moles of ethanol was divided by the mass of water in kilograms.
• The result was a molality of 1.085 mol/kg.

Molality is particularly useful when the solution undergoes temperature changes, as the solvent's volume and solute's concentration might vary, but the mass remains constant. Thus, it's often preferred in thermodynamic calculations.