Question

Ascorbic acid (vitamin \(\left.\mathrm{C}, \mathrm{C}_{6} \mathrm{H}_{8} \mathrm{O}_{6}\right)\) 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, (b) the mole fraction, (c) the molality, (d) the molarity of ascorbic acid in this solution.

Short answer

The mass percentage, mole fraction, molality, and molarity of ascorbic acid in this solution are: (a) Mass percentage = \(27.71\%\) (b) Mole fraction = \(0.0377\) (c) Molality = \(2.18 \: \text{mol/kg}\) (d) Molarity = \(1.92 \: \text{mol/L}\)

Step-by-step solution

Calculate the molar mass of ascorbic acid

To calculate the molar mass of ascorbic acid (C6H8O6), we need to consider the molar mass of each element present in ascorbic acid and add them all together: Molar mass of C = 12.01 g/mol Molar mass of H = 1.01 g/mol Molar mass of O = 16.00 g/mol Molar mass of ascorbic acid = 6 * (12.01) + 8 * (1.01) + 6 * (16.00) = 72.06 + 8.08 + 96.00 = 176.14 g/mol

Calculate the mass percentage

Mass percentage is calculated by dividing the mass of the solute (ascorbic acid) by the total mass of the solution (ascorbic acid and water), then multiplied by 100 to express it as a percentage: Mass percentage = (mass of ascorbic acid / total mass) * 100 Mass percentage = (80.5 g / (80.5 g + 210 g)) * 100 = (80.5 g / 290.5 g) * 100 = 27.71%

Calculate the mole fraction

To determine the mole fraction, we need to find the moles of each component in the solution: - Moles of ascorbic acid = mass / molar mass = 80.5 g / 176.14 g/mol = 0.457 moles - Moles of water = 210 g / 18.02 g/mol (molar mass of water) = 11.65 moles Mole fraction of ascorbic acid = moles of ascorbic acid / total moles Mole fraction = 0.457 moles / (0.457 moles + 11.65 moles) = 0.457 moles / 12.107 moles = 0.0377

Calculate the molality

Molality is the number of moles of solute (ascorbic acid) divided by the mass of the solvent (water) in kilograms: Molality = (moles of ascorbic acid) / (mass of water in kg) Molality = (0.457 moles) / (210 g / 1000) = 0.457 moles / 0.210 kg = 2.18 mol/kg

Calculate the molarity

First, we need to find the volume of the solution. We are given the density (1.22 g/mL) and the total mass (290.5 g), so we can calculate the volume: Volume = mass /density = 290.5 g / (1.22 g/mL) = 238.1 mL Now we can calculate the molarity. Molarity is the number of moles of solute (ascorbic acid) divided by the volume of the solution in liters: Molarity = (moles of ascorbic acid) / (volume of solution in L) Molarity = 0.457 moles / (238.1 mL / 1000) = 0.457 moles / 0.2381 L = 1.92 mol/L In summary, the mass percentage, mole fraction, molality, and molarity of ascorbic acid in this solution are: (a) Mass percentage = 27.71% (b) Mole fraction = 0.0377 (c) Molality = 2.18 mol/kg (d) Molarity = 1.92 mol/L

Key concepts

Molar Mass

Before diving into solution concentration calculations, it’s essential to understand the concept of molar mass. Molar mass is the mass of one mole of a given substance.
To compute the molar mass, you add up the atomic masses of all the atoms in a molecule. Using vitamin C (ascorbic acid, \( \text{C}_6\text{H}_8\text{O}_6 \)) as an example, let's break it down:

• Carbon (C) has an atomic mass of 12.01 g/mol. Since there are six carbons, the total comes to \( 6 \times 12.01 \ g/\text{mol} = 72.06 \ g/\text{mol} \).
• Hydrogen (H) has an atomic mass of 1.01 g/mol, and there are eight hydrogens, so \( 8 \times 1.01 \ g/\text{mol} = 8.08 \ g/\text{mol} \).
• Oxygen (O), at 16.00 g/mol and with six atoms, totals \( 6 \times 16.00 \ g/\text{mol} = 96.00 \ g/\text{mol} \).

Consequently, the molar mass of ascorbic acid adds up to 176.14 g/mol. Understanding this allows you to transition into other types of calculations.

Mass Percentage

Mass percentage is a way to express the concentration of a component in a solution. It tells you how much of the total mass comes from the solute.
You calculate it using the formula: \[ \text{Mass Percentage} = \left( \frac{\text{mass of solute}}{\text{total mass of solution}} \right) \times 100 \] For the problem at hand, the mass of ascorbic acid was 80.5 g, and the mass of water was 210 g, giving a total solution mass of 290.5 g. The calculation:\[ \left( \frac{80.5 g}{290.5 g} \right) \times 100 = 27.71\% \] This tells us that 27.71% of the total solution mass is ascorbic acid.

Mole Fraction

The mole fraction is another concentration measure, indicating the ratio of the moles of one component to the total moles in the solution. It's especially useful when dealing with mixtures of gases.
For vitamin C, the moles are found by dividing its mass by its molar mass (from earlier): \[ \text{Moles of ascorbic acid} = \frac{80.5 g}{176.14 g/mol} = 0.457 \text{ moles} \]
Water has a molar mass of 18.02 g/mol, which means:\[ \text{Moles of water} = \frac{210 g}{18.02 g/mol} = 11.65 \text{ moles} \] Thus, the mole fraction of ascorbic acid is:\[ \frac{0.457}{0.457 + 11.65} = 0.0377 \] This means that 3.77% of the total moles in the solution are from ascorbic acid.

Molality

Molality ( m ) differs from molarity since it depends entirely on the mass of the solvent, not the total solution volume. It's calculated as the moles of solute per kilograms of solvent.
For our case with vitamin C:\[ \text{Molality} = \frac{\text{moles of solute}}{\text{mass of solvent in kg}} \] Given 0.457 moles of ascorbic acid and 210 g of water (0.210 kg), the computation is:\[ \frac{0.457}{0.210} = 2.18 \text{ mol/kg} \] This illustrates that for every kilogram of water, there are 2.18 moles of vitamin C dissolved.

Molarity

Molarity ( M ) describes how many moles of solute are present per liter of solution. It's a commonly used measurement, especially in chemistry labs.
To calculate molarity, the volume of the solution must be considered. Given the problem, the mass is 290.5 g and the density is 1.22 g/mL, so: \[ \text{Volume} = \frac{290.5 g}{1.22 g/mL} = 238.1 \text{ mL} = 0.2381 \text{ L} \] Thus, molarity is:\[ \frac{0.457 \text{ moles}}{0.2381 \text{ L}} = 1.92 \text{ mol/L} \] This value indicates that there are 1.92 moles of ascorbic acid in every liter of this solution, making it a reasonably concentrated solution.