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 solution is 23.91%, the molality is 1.64 m, the mole fraction is 0.0286, and the normality is 4.11 N.

Step-by-step solution

Calculate the molar mass of citric acid

First, we need to find the molar mass of citric acid: H3C6H5O7. Using the periodic table, we find the molar mass of each element and multiply it by the corresponding subscript: Molar mass of citric acid = (3 × 1.01 g/mol) + (6 × 12.01 g/mol) + (5 × 1.01 g/mol) + (7 × 16.00 g/mol) = 192.12 g/mol

Calculate the mass of citric acid and the mass of the solution

We are given the molarity and density of the solution. We can use these values to find the mass of citric acid and the mass of the solution in 1 L: Mass of citric acid = Molarity × Molar mass × Volume of the solution Mass of citric acid = 1.37 mol/L × 192.12 g/mol × 1 L = 263.00 g Mass of the solution = Density × Volume of the solution Mass of the solution = 1.10 g/cm³ × 1000 cm³ = 1100 g

Calculate the mass percent of citric acid

Now, we can use the mass of citric acid and the mass of the solution to calculate the mass percent of citric acid: Mass percent of citric acid = (mass of citric acid ÷ total mass of the solution) × 100 Mass percent of citric acid = (263.00 g ÷ 1100 g) × 100 = 23.91%

Calculate the molality of citric acid

To find the molality of citric acid, we need to know the moles of citric acid and the mass of the solvent in kg. Moles of citric acid = 1.37 mol/L × 1 L = 1.37 mol Mass of solvent (water) = mass of the solution - mass of citric acid = 1100 g - 263.00 g = 837.00 g = 0.837 kg Now, we can use these values to find the molality: Molality of citric acid = moles of citric acid ÷ mass of solvent (in kg) Molality of citric acid = 1.37 mol ÷ 0.837 kg = 1.64 m

Calculate the mole fraction of citric acid

To find the mole fraction, we need to know the total moles of solute (citric acid) and solvent (water): Moles of water = mass of water ÷ molar mass of water = 837.00 g ÷ 18.02 g/mol = 46.45 mol Now, we can find the mole fraction of citric acid: Mole fraction of citric acid = moles of citric acid ÷ total moles of solute and solvent Mole fraction of citric acid = 1.37 mol ÷ (1.37 mol + 46.45 mol) = 0.0286

Calculate the normality of citric acid

Citric acid has three acidic protons (3 equivalents per mole). To find the normality, we can use the molarity and the number of acidic protons: Normality of citric acid = Molarity × number of acidic protons Normality of citric acid = 1.37 mol/L × 3 = 4.11 N In conclusion, the mass percent of citric acid is 23.91%, the molality is 1.64 m, the mole fraction is 0.0286, and the normality is 4.11 N.

Key concepts

Molarity

Molarity is a measure of concentration indicating the number of moles of solute present in one liter of solution. It is expressed in moles per liter (mol/L). Think of molarity as how crowded the solute molecules are within a solution.

To calculate molarity, you need two things: the number of moles of the solute and the volume of the solution. The formula is:

• Molarity (M) = \( \frac{\text{moles of solute}}{\text{liters of solution}} \)

For instance, in the original exercise, the molarity of the citric acid solution was given as 1.37 M. This means that there are 1.37 moles of citric acid in every liter of the solution.

Molality

Molality, another way to express concentration, differs from molarity by utilizing the mass of the solvent rather than the volume of the solution. This is particularly useful as it remains unaffected by temperature changes. Molality is expressed in moles of solute per kilogram of solvent (mol/kg).

If you're working on a problem where temperature might affect the density of the solution, molality is a reliable choice!

• Molality (m) = \( \frac{\text{moles of solute}}{\text{kilograms of solvent}} \)

In our given problem, the citric acid had a molality of 1.64 m, calculated by dividing the moles of citric acid by the mass in kilograms of the solvent (water).

Mole Fraction

Mole fraction provides a way to express the concentration of a component in a mixture independent of the mass or volume. It is the ratio of the moles of one component to the total moles of all components in the mixture. Because it is a fraction, its value lies between 0 and 1.

Mole fraction, being unitless, is useful in thermodynamics. Knowing this can help calculate properties like vapor pressure more accurately.

• Mole Fraction = \( \frac{\text{moles of component}}{\text{total moles in solution}} \)

Using the example, the mole fraction of citric acid is 0.0286, meaning approximately 2.86% of the total moles in the solution are citric acid.

Normality

Normality is an expression of concentration that emphasizes the reactive capacity of a solution's solute. It is particularly useful in acid-base and redox reactions. Normality is expressed as equivalents per liter (eq/L).

The calculation of normality takes into account the number of reactive units, like acidic protons or electrons, a solute can supply or consume during a reaction.

• Normality (N) = Molarity × Number of Reactive Units

In the problem, citric acid with 3 acidic protons has a normality of 4.11 N, obtained by multiplying its molarity by the number of acidic protons.

Mass Percent

Mass percent, or weight percent, represents the concentration of an element in a mixture or a component in a composite. It is defined as the ratio of the mass of one component to the total mass of the mixture, multiplied by 100.

This measure gives a quick picture of how much of a substance is present relative to the whole mixture. It is often used in industry and laboratories to create specific composition mixtures.

• Mass Percent = \( \frac{\text{mass of component}}{\text{total mass of solution}} \times 100 \)

For example, the mass percent of citric acid in the solution is 23.91%, indicating that this portion of the total mass is due to citric acid.