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, molality, mole fraction, and normality of the citric acid solution are 19.29%, 1.245 mol/kg, 0.0219, and 4.11 N, respectively.

Step-by-step solution

Determine the mass of citric acid in the solution

First, we need to find the mass of citric acid in the solution. To do this, we can use the given molarity and volume of the solution to find the moles of citric acid, then multiply by the molar mass of citric acid to find the mass. Given molarity(M) = 1.37 mol/L Density of the solution = 1.10 g/cm³ = 1.10 g/mL (1 mL = 1 cm³) Now determine the molar mass of citric acid (H3C6H5O7): = 3x1 + 6x12 + 5x1 + 7x16 = 3 + 72 + 5 + 112 = 192 g/mol Assuming the volume of solution is 1 L, we can find the mass of citric acid. mass of citric acid = molarity × volume × molar mass = 1.37 mol/L × 1000 mL × 192 g/mol = 262.944 g

Determine the mass of the solution

Now, using this mass, we can determine the mass of the entire solution (water + citric acid), to use as a basis for our other calculations. Mass of water in the solution = density × volume = 1.10 g/mL × 1000 mL = 1100 g Mass of the solution = mass of citric acid + mass of water = 262.944 g + 1100 g = 1362.944 g

Calculate the mass percent

Mass percent relates the mass of the solute(citric acid) to the mass of the solution. Mass percent = (mass of citric acid / mass of the solution) × 100 = (262.944 g / 1362.944 g) × 100 = 19.29 %

Calculate the molality

Molality relates the amount (moles) of solute to the mass (kg) of solvent(water). Molality = moles of citric acid / mass of water (in kg) moles of citric acid = 1.37 mol mass of water = 1100 g = 1.1 kg molality = 1.37 mol / 1.1 kg = 1.245 mol/kg

Calculate the mole fraction

Mole fraction is the ratio of moles of one component to the total moles of the components in the mixture. moles of water = mass of water / molar mass of water = 1100 g / 18.015 g/mol = 61.095 mol Mole fraction of citric acid = moles of citric acid / (moles of citric acid + moles of water) = 1.37 mol / (1.37 mol + 61.095 mol) = 0.0219

Calculate the normality

Normality is the molarity of species that produce or consume protons (H+) in an acid-base reaction. Since citric acid has three acidic protons, we need to multiply its molarity by the number of acidic protons. Normality = molarity × number of acidic protons = 1.37 mol/L × 3 = 4.11 N Therefore, the mass percent, molality, mole fraction, and normality of the citric acid solution are 19.29%, 1.245 mol/kg, 0.0219, and 4.11 N, respectively.

Key concepts

Molarity

Molarity, also known as molar concentration, is a way to express the concentration of a solution. It is defined as the number of moles of a solute per liter of solution. Molarity is denoted by the symbol \( M \) and is calculated using the formula:

• \( M = \frac{n}{V} \)

where \( n \) is the number of moles of solute and \( V \) is the volume of the solution in liters.
Molarity is a crucial concept in chemistry because it allows us to determine how much solute is present in a given volume of solvent. This is particularly important when preparing solutions for experiments or reactions. In the scenario provided, the concentration of citric acid was given as \( 1.37 \, M \), meaning there are 1.37 moles of citric acid in one liter of the solution.

Density

Density is a physical property of matter that describes how much mass is contained in a given volume. It is calculated by the formula:

• \( \text{density} = \frac{\text{mass}}{\text{volume}} \)

Density is generally expressed in units of g/cm³ or g/mL.
This property is useful for converting between volume and mass of a substance, making it essential in concentration calculations. In the context of the exercise, the density of the citric acid solution is given as \( 1.10 \, \text{g/cm}^3 \), which means each cubic centimeter (or milliliter) of this solution weighs 1.10 grams.
Understanding density helps in determining the mass of a portion of the solution, which then is used to calculate concentrations like mass percent and molality.

Mass Percent

Mass percent is a concentration term that describes the ratio of the mass of a component to the total mass of the mixture, multiplied by 100. It is calculated as:

• \( \text{Mass percent} = \left(\frac{\text{mass of solute}}{\text{mass of solution}}\right) \times 100 \)

In the example, citric acid has a mass percent of 19.29%, meaning that approximately 19.29% of the total mass of the solution is citric acid.
This concept is particularly useful because it provides a direct understanding of the composition of the solution, and is commonly used in both scientific research and industry to describe mixture ratios, particularly in fields like chemistry and pharmacology.

Molality

Molality is another way to express the concentration of a solution, different from molarity. It is defined as the number of moles of solute divided by the mass of the solvent in kilograms. The formula is:

• \( \text{Molality} = \frac{\text{moles of solute}}{\text{mass of solvent (in kg)}} \)

In the problem, the molality of the citric acid solution was found to be \( 1.245 \, \text{mol/kg} \).
Molality is less affected by temperature and pressure changes than molarity because it depends on the mass of the solvent rather than the volume of the solution. This makes it ideal for various applications, especially in studies involving colligative properties like boiling point elevation and freezing point depression.

Mole Fraction

Mole fraction is a unit of concentration that expresses the ratio of moles of one component to the total number of moles in the solution. It is calculated using:

• \( \text{Mole fraction of component} = \frac{\text{moles of component}}{\text{total moles in mixture}} \)

This method of expressing concentration does not depend on temperature or pressure.
In this exercise, the mole fraction of citric acid is calculated to be 0.0219, indicating the proportion of citric acid in relation to the total number of moles in the solution.
Mole fraction provides insight into the fractional composition of a mixture, which can be valuable in precisely controlling reaction stoichiometry in chemical processes and understanding the properties of solutions.

Normality

Normality is another unit of concentration, often used in acid-base chemistry. It reflects the molarity of reactive units, particularly in the context of equivalent reactions, such as acid-base reactions. It is calculated by multiplying the molarity by the number of acidic or basic equivalents per molecule of solute:

• \( N = M \times \text{equivalents} \)

For citric acid, which can donate three protons, the normality was calculated as \( 4.11 \, N \).
This means that the solution contains \( 4.11 \) equivalents of protons per liter, which is relevant for titration and similar analytical procedures. Normality provides insight into the solution's ability to react with other substances in stoichiometrically significant quantities.