Question

A \(1.37 M\) solution of citric acid \(\left(\mathrm{H}_{3} \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{O}_{7}\right)\) in water has a density of \(1.10 \mathrm{~g} / \mathrm{cm}^{3} .\) Calculate the mass percent, molality, mole fraction, and normality of the citric acid. Citric acid has three acidic protons.

Short answer

The mass percent of citric acid in the given solution is 23.93%. The molality is 1.636 mol/kg, and the mole fraction is 0.0286. The normality of the citric acid is 4.11 N.

Step-by-step solution

Calculate the molar mass of citric acid

First, we have to find out the molar mass of citric acid: Citric acid (H₃C₆H₅O₇): - 3 Hydrogen atoms: 3 * 1.008 g/mol = 3.024 g/mol - 6 Carbon atoms: 6 * 12.01 g/mol = 72.06 g/mol - 5 Hydrogen atoms: 5 * 1.008 g/mol = 5.04 g/mol - 7 Oxygen atoms: 7 * 16.00 g/mol = 112.00 g/mol Total molar mass = 3.024 + 72.06 + 5.04 + 112.00 = 192.124 g/mol

Calculate mass of solute and solvent

As we know, the solution has a concentration of 1.37 M, which means 1.37 moles of citric acid is found in 1 liter of the solution. We can find the mass of citric acid in 1 L of the solution by multiplying the moles with the molar mass: Mass of citric acid = moles × molar mass = 1.37 mol × 192.124 g/mol = 263.210 g Now, we can find the mass of 1 L solution using the given density: Mass of 1 L solution = mass/volume = 1.10 g/cm³ × 1000 cm³ = 1100 g The mass of the solvent (water) can be calculated by subtracting the mass of citric acid from the mass of the solution: Mass of water = mass of solution - mass of citric acid = 1100 g - 263.210 g = 836.790 g

Calculate mass percent

Mass percent is given by the formula: (mass of solute / mass of solution) × 100 Mass percent of citric acid = (263.210 g / 1100 g) × 100 = 23.93%

Calculate molality

Molality is defined as the number of moles of solute per kilogram of solvent: Molality = moles of citric acid / mass of water (in kg) = 1.37 mol / 0.83679 kg = 1.636 mol/kg

Calculate mole fraction

Mole fraction is the ratio of the moles of solute to the total moles present in the solution: Moles of water = mass of water / molar mass of water = 836.790 g / 18.015 g/mol = 46.468 mol Mole fraction of citric acid = moles of citric acid / (moles of citric acid + moles of water) = 1.37 mol / (1.37 mol + 46.468 mol) = 0.0286

Calculate normality

Normality is defined as the number of equivalents of solute per liter of solution. Since citric acid has three acidic protons, the number of equivalents is three times the number of moles of citric acid: Normality = equivalents of citric acid × molarity = 3 × 1.37 M = 4.11 N Now we have all the required values: - Mass percent of citric acid: 23.93% - Molality of citric acid: 1.636 mol/kg - Mole fraction of citric acid: 0.0286 - Normality of citric acid: 4.11 N

Key concepts

Molar Mass

The molar mass of a compound is the total mass in grams of one mole of that substance. It's a vital starting point for many stoichiometric calculations in chemistry. For instance, the molar mass of citric acid (H₃C₆H₅O₇) is calculated by summing the masses of each individual atom within the molecule. When doing these calculations, it's important to use the most accurate atomic masses and ensure all elements are accounted for in the molecule.

Mass Percent

The mass percent, also known as mass fraction, is a measurement of the concentration of a component in a mixture, expressed as the mass of the solute divided by the total mass of the solution, then multiplied by 100 to get a percentage. For students learning to calculate mass percent, it's important to distinguish between solute and solvent and correctly measure their masses. This calculation is very practical since it doesn't require knowledge of the volume of the solution, only the mass.

Molality

Molality, marked as mol/kg, is a way to express the concentration of a solution by indicating the number of moles of solute per kilogram of solvent. One of the unique properties of molality is that it's independent of temperature, making it particularly useful in scenarios where the temperature might fluctuate, as the volume can change with temperature but the mass remains constant. Students need to pay attention to the units of mass used in this calculation, converting grams to kilograms where necessary.

Mole Fraction

The mole fraction is another way to express the concentration of a component within a mixture. It is defined as the ratio of the number of moles of a particular component to the total number of moles of all components in the mixture. When calculating the mole fraction, it can be helpful to use a formula and understand that it's a part of unity—the sum of the mole fractions in a mixture equals one. Mole fractions are useful in gas laws and determining the partial pressures in a mixture of gases.

Normality

Normality is a measure of concentration that is often used in titrations and is defined as the number of equivalents per liter of solution. It can be especially confusing as the concept of equivalents relates to the reactive capacity of the solute. For citric acid, which has three acidic protons, one mole of citric acid is equivalent to three moles of H⁺ ions. Thus, when calculating normality, being aware of the number of reactive species the solute can produce is crucial.