Question

Vinegar is a \(5 \%\) solution (by weight) of acetic acid in water. Determine the mole fraction and molality of acetic acid. What is the concentration of acetic acid in parts per million (ppm)? Explain why it is not possible to calculate the molarity of this solution from the information provided.

Short answer

Mole fraction: 0.0155, Molality: 0.877 mol/kg, ppm: 50,000. Molarity can't be calculated due to lack of volume data.

Step-by-step solution

Determine Composition

First, define the components of the vinegar solution. Vinegar is a mixture of acetic acid (\( \text{C}_2\text{H}_4\text{O}_2 \)) and water. A 5% solution by weight means there are 5 grams of acetic acid in every 100 grams of solution.

Calculate Moles of Acetic Acid

To find the moles of acetic acid, use its molar mass, which is 60.05 g/mol. Calculate moles:\[\text{Moles of acetic acid} = \frac{5 \text{ g}}{60.05 \text{ g/mol}} \approx 0.0833 \text{ mol}\]

Calculate Mass of Water

To find the mass of water in the solution, subtract the mass of acetic acid from the total mass of the solution:\[\text{Mass of water} = 100 \text{ g} - 5 \text{ g} = 95 \text{ g}\]

Calculate Mole Fraction

To calculate the mole fraction of acetic acid, determine the moles of water using its molar mass (18.02 g/mol):\[\text{Moles of water} = \frac{95 \text{ g}}{18.02 \text{ g/mol}} \approx 5.273 \text{ mol}\]Now calculate the mole fraction of acetic acid:\[\text{Mole fraction of acetic acid} = \frac{0.0833 \text{ mol}}{0.0833 \text{ mol} + 5.273 \text{ mol}} \approx 0.0155\]

Calculate Molality

Molality (m) is moles of solute per kilogram of solvent:\[\text{Molality} = \frac{0.0833 \text{ mol}}{0.095 \text{ kg}} \approx 0.877 \text{ mol/kg}\]

Calculate Concentration in ppm

To calculate parts per million (ppm) of acetic acid:\[\text{ppm} = \left(\frac{5 \text{ g}}{100 \text{ g}}\right) \times 10^6 = 50,000 \text{ ppm}\]

Discuss Molarity

Molarity requires the volume of the solution, which is not provided. Since density is unknown, molarity cannot be directly calculated from given information.

Key concepts

Understanding Acetic Acid in Vinegar

Acetic acid, with the chemical formula \( \text{C}_2\text{H}_4\text{O}_2 \), is the main component that gives vinegar its distinctive sour taste. When we refer to a 5% solution by weight, it implies that in 100 grams of vinegar, 5 grams are acetic acid and the remaining 95 grams are water. This acidic compound has a molar mass of approximately 60.05 g/mol. Acetic acid is not only used in culinary applications but also in various industrial and cleaning processes due to its antimicrobial properties. When studying solutions like vinegar, knowing the weight percentage helps us understand the proportion of acetic acid present, which is essential for calculating mole fractions and concentrations such as molality and ppm. With this information, the next steps involve understanding how we calculate various concentrations like molality, and parts per million for a comprehensive analysis of the solution.

Calculating Molality for Acetic Acid

Molality is a way to express the concentration of a solution, defined as moles of solute per kilogram of solvent. In this context, the solute is acetic acid, and the solvent is water. It is calculated using the formula:\[\text{Molality (m)} = \frac{\text{moles of solute}}{\text{kilograms of solvent}} \]In our solution, we found there are roughly 0.0833 moles of acetic acid. Given that the mass of the solvent (water) is 95 grams, or 0.095 kilograms, the molality can be calculated as:\[\text{Molality} = \frac{0.0833 \text{ mol}}{0.095 \text{ kg}} \approx 0.877 \text{ mol/kg} \]Molality is particularly useful in various chemical calculations because it depends only on the mass of the solvent, not its volume. This makes it unaffected by temperature or pressure changes, unlike molarity, which relies on the solution's volume.Understanding the molality helps provide insights into the concentration effects within chemical kinetics and thermodynamics applications.

Understanding Parts Per Million (ppm)

Parts per million (ppm) is a unit of concentration that describes the amount of a substance in a million parts of the mixture. It is a very convenient measure for extremely dilute solutions, where tiny amounts of solute need to be accurately measured. For the given vinegar solution, with a 5% weight of acetic acid, we calculate ppm using the formula:\[\text{ppm} = \left(\frac{\text{grams of solute}}{\text{grams of solution}}\right) \times 10^6 \]Plugging in the values for acetic acid:\[\text{ppm} = \left(\frac{5 \text{ g}}{100 \text{ g}}\right) \times 10^6 = 50,000 \text{ ppm} \]This shows that the acetic acid concentration in vinegar is quite high, about 50,000 parts in every million parts of the solution. Understanding ppm is essential for environmental science and engineering fields where substances present in small doses can have significant effects.By using ppm, we can easily convey the concentration of substances like pollutants in air, beverages, or cleaning agents, which is crucial while assessing their impact or ensuring compliance with safety standards.