Problem 48 - Chemistry The Central Science In Si Units

Question

Ascorbic acid (vitamin C, \(\mathrm{C}_{6} \mathrm{H}_{8} \mathrm{O}_{6}\) ) is a water-soluble vitamin. A solution containing \(80.5 \mathrm{~g}\) of ascorbic acid dissolved in \(210 \mathrm{~g}\) of water has a density of \(1.22 \mathrm{~g} / \mathrm{mL}\) at \(55^{\circ} \mathrm{C}\). Calculate (a) the mass percentage, \((\mathbf{b})\) the mole fraction, \((\mathbf{c})\) the molality, \((\mathbf{d})\) the molarity of ascorbic acid in this solution.

Step-by-Step Solution

Answer

a) 27.7%, b) 0.0378, c) 2.18 mol/kg, d) 1.92 M.

Step 1: Calculate the Mass Percentage

The mass percentage is given by the formula:\[\text{Mass Percentage} = \left( \frac{\text{Mass of Solute}}{\text{Total Mass of Solution}} \right) \times 100\%\]Substitute the given values:\[= \left( \frac{80.5 \text{ g}}{80.5 \text{ g} + 210 \text{ g}} \right) \times 100\%\]\[= \left( \frac{80.5}{290.5} \right) \times 100\% \approx 27.7\% \]

Step 2: Calculate the Mole Fraction

First, calculate the moles of each component. The molar mass of ascorbic acid (\(\mathrm{C}_{6} \mathrm{H}_{8} \mathrm{O}_{6}\)) is approximately 176.12 g/mol.Moles of ascorbic acid:\[= \frac{80.5 \text{ g}}{176.12 \text{ g/mol}} \approx 0.457 \text{ mol}\]The molar mass of water \((\mathrm{H}_{2}\mathrm{O})\) is 18.02 g/mol, so moles of water:\[= \frac{210 \text{ g}}{18.02 \text{ g/mol}} \approx 11.66 \text{ mol}\]Now calculate the mole fraction of ascorbic acid:\[\text{Mole Fraction of Ascorbic Acid} = \frac{0.457}{0.457 + 11.66} \approx 0.0378\]

Step 3: Calculate the Molality

Molality is calculated using the formula:\[\text{Molality (m)} = \frac{\text{Moles of solute}}{\text{Mass of solvent in kg}}\]Substitute the given values:\[= \frac{0.457}{0.210} \approx 2.18 \text{ mol/kg}\]

Step 4: Calculate the Molarity

First, find the volume of the solution using its density. The total mass of the solution is 290.5 g, and its density is 1.22 g/mL:\[\text{Volume of solution} = \frac{290.5 \text{ g}}{1.22 \text{ g/mL}} = 238.11 \text{ mL} = 0.23811 \text{ L}\]Molarity is calculated using:\[\text{Molarity (M)} = \frac{\text{Moles of solute}}{\text{Volume of solution in L}}\]Substitute the values:\[= \frac{0.457}{0.23811} \approx 1.92 \text{ M}\]

Key Concepts

Mass Percentage

When thinking about mass percentage, it helps to picture it as the slice of the total pie represented by the ingredient you're interested in. In this case, we want to know how much of the total solution is made up by ascorbic acid.
We'll use the formula:

• Mass Percentage = \( \left( \frac{\text{Mass of Solute}}{\text{Total Mass of Solution}} \right) \times 100\% \)

You simply take the mass of ascorbic acid (our solute) over the total mass of the solution (ascorbic acid plus water), then multiply by 100 to get a percentage. This gives you the percentage of the solution that is ascorbic acid.
Here, it turns out to be about 27.7%, meaning out of 100 parts of the solution, roughly 27.7 parts are ascorbic acid. It's a useful way to express concentration when you're dealing with solutions.

Mole Fraction

Mole fraction is a way to express concentration by comparing the number of moles of one component to the total number of moles in the solution. It's all about the ratio of particles rather than mass.

• Mole Fraction = \( \frac{\text{Moles of Solute}}{\text{Total Moles in Solution}} \)

For this exercise, we calculate the moles of both ascorbic acid and water separately, since they are both contributing to the total number of moles in the solution. We found that ascorbic acid accounts for about 0.0378 of the mole fraction.
This means that approximately 3.78% of all the particles in this solution are ascorbic acid molecules. Unlike mass percentage, mole fraction doesn't deal with weights, it focuses purely on the count of molecules.

Molality

Molality provides another angle on concentration, centered around the number of moles of solute per kilogram of solvent.
It is distinct from molarity, as molality depends on the mass of the solvent, and not the total solution volume. Here’s the formula:

• Molality (m) = \( \frac{\text{Moles of solute}}{\text{Mass of solvent in kg}} \)

This measure comes in handy when temperature and pressure fluctuations occur, because unlike volume, mass remains stable under such changes. For this solution, the molality is about 2.18 mol/kg. This tells us that there are 2.18 moles of ascorbic acid for every kilogram of water acting as the solvent.
This measure is crucial in situations where you need precision regardless of environmental conditions.

Molarity

Among the more common ways to express solution concentration, molarity focuses on the number of moles of solute within a specific volume of solution. It's used in many chemical applications and is relatively easy to calculate.
The formula is:

• Molarity (M) = \( \frac{\text{Moles of solute}}{\text{Volume of solution in L}} \)

For our ascorbic acid solution, once we calculated the volume from the given total mass and density, we found the molarity to be about 1.92 M. This means there are 1.92 moles of ascorbic acid in every liter of solution.
Molarity is commonly used in laboratory settings due to its direct relationship with reaction stoichiometry, via volume-based dosing.