Question

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

Short answer

In conclusion, the properties of the ascorbic acid solution are: 1. Mass percentage: \(27.7\%\) 2. Mole fraction: \(0.0377\) 3. Molality: \(2.18 \mathrm{~mol/kg}\) 4. Molarity: \(1.92 \mathrm{~M}\)

Step-by-step solution

Calculate the mass of the solution and the mass of ascorbic acid

We are given the mass of ascorbic acid (\(80.5 \mathrm{~g}\)) and the mass of water (\(210 \mathrm{~g}\)). We can use this to calculate the total mass of the solution: Total mass of the solution = mass of ascorbic acid + mass of water Total mass of the solution = \(80.5 \mathrm{~g} + 210 \mathrm{~g} = 290.5 \mathrm{~g}\)

Calculate the mass percentage of ascorbic acid

The mass percentage of ascorbic acid can be found using the formula: Mass percentage = \(\frac{\text{mass of ascorbic acid}}{\text{total mass of the solution}} \times 100\) Mass percentage of ascorbic acid = \(\frac{80.5 \mathrm{~g}}{290.5 \mathrm{~g}} \times 100 = 27.7 \%\)

Calculate the number of moles of ascorbic acid

To determine the number of moles of ascorbic acid, we need to use the molar mass of C6H8O6. The molar mass of ascorbic acid is 176.12 g/mol. Number of moles = \(\frac{\text{mass of ascorbic acid}}{\text{molar mass of ascorbic acid}}\) Number of moles of ascorbic acid = \(\frac{80.5 \mathrm{~g}}{176.12 \mathrm{~g/mol}} = 0.457 \mathrm{~mol}\)

Calculate the mole fraction of ascorbic acid

To determine the mole fraction of ascorbic acid, we also need to calculate the moles of water present in the solution. The molar mass of water is 18.02 g/mol. Number of moles of water = \(\frac{210 \mathrm{~g}}{18.02 \mathrm{~g/mol}} = 11.65 \mathrm{~mol}\) The mole fraction can be calculated using the following formula: Mole fraction = \(\frac{\text{moles of ascorbic acid}}{\text{total moles of the solution}}\) Mole fraction of ascorbic acid = \(\frac{0.457 \mathrm{~mol}}{0.457 \mathrm{~mol} + 11.65 \mathrm{~mol}} = 0.0377\)

Calculate the molality of ascorbic acid

Molality is defined as the moles of solute per kilogram of the solvent (water in this case). In our case, Molality = \(\frac{\text{moles of ascorbic acid}}{\text{mass of water (kg)}}\) Molality of ascorbic acid = \(\frac{0.457 \mathrm{~mol}}{0.210 \mathrm{~kg}} = 2.18 \mathrm{~mol/kg}\)

Calculate the molarity of ascorbic acid

Molarity is defined as the moles of solute per liter of the solution. To calculate molarity, we need to determine the volume of the solution. First, we can find the volume by dividing the mass of the solution by its density. Volume of the solution = \(\frac{290.5 \mathrm{~g}}{1.22 \mathrm{~g/mL}} = 238.1 \mathrm{~mL}\) or \(0.2381 \mathrm{~L}\) Then, we can calculate the molarity by dividing the moles of ascorbic acid by the volume of the solution in liters. Molarity = \(\frac{\text{moles of ascorbic acid}}{\text{volume of the solution (L)}}\) Molarity of ascorbic acid = \(\frac{0.457 \mathrm{~mol}}{0.2381 \mathrm{~L}} = 1.92 \mathrm{~M}\) In conclusion, the properties of the ascorbic acid solution are: 1. Mass percentage: \(27.7\%\) 2. Mole fraction: \(0.0377\) 3. Molality: \(2.18 \mathrm{\ mol/kg}\) 4. Molarity: \(1.92 \mathrm{\ M}\)

Key concepts

Molarity

Molarity is a measurement of the concentration of a solution, expressed as the number of moles of solute per liter of solution. It's a very common way to express concentration in chemistry. Molarity ( M ) is calculated using the formula:\[ M = \frac{\text{moles of solute}}{\text{volume of solution in liters}}\]Understanding this concept is essential because it helps us know how much of a substance is in a given volume of solution. In the case of the ascorbic acid solution, we first determine the mass of the solution. Knowing the density and the mass, we calculate the volume. With the moles of ascorbic acid already known, we divide by this volume to find the molarity. It's important to ensure the volume is in liters when you perform these calculations.

Mole Fraction

The mole fraction is another expression of concentration that represents the ratio of the number of moles of a component to the total number of moles of all substances present. Mole fraction ( X ) is given by:\[ X = \frac{\text{moles of component}}{\text{total moles of solution}}\]This method is useful for studying the properties of mixtures or analyzing the different species involved. In our exercise, we calculate the mole fraction of ascorbic acid by first determining the number of moles of water in the solution. We then divide the moles of ascorbic acid by the sum of moles of both ascorbic acid and water. The resulting mole fraction can help understand the solution's composition at a molecular level.

Molality

Molality is somewhat similar to molarity, but instead of volume, it uses the mass of the solvent. It is particularly useful for scenarios where temperature changes are involved, as unlike volume, mass does not change with temperature. Molality ( m ) is defined by:\[ m = \frac{\text{moles of solute}}{\text{kilograms of solvent}}\]It offers an advantage over molarity because it remains unchanged with temperature changes. In our example, molality is calculated by taking the moles of ascorbic acid and dividing by the mass of water in kilograms. This gives us a sense of the solute's concentration relative to the solvent alone, rather than the entire solution like molarity.

Mass Percentage

Mass percentage is a simple way to express concentration that involves comparing the mass of the solute to the total mass of the solution. It's calculated using the following formula:\[ \text{Mass percentage} = \left(\frac{\text{mass of solute}}{\text{total mass of solution}}\right) \times 100\]This percentage tells us what fraction of the total mass of the solution is the solute, giving an idea of how concentrated the solution is. In the context of our exercise with ascorbic acid, we add the mass of ascorbic acid and water to find the total mass. The mass percentage is then calculated by dividing the mass of ascorbic acid by the total solution mass, and multiplying by 100. This method is especially useful when preparing solutions with precise solute concentrations.