Problem 78 - Chemistry: The Central Science in SI Units, Global Edition

Question

Glycerol, \(\mathrm{C}_{3} \mathrm{H}_{8} \mathrm{O}_{3},\) is a substance used extensively in the manufacture of cosmetic s, foodstuffs, antifreeze, and plastics. Glycerol is a water-soluble liquid with a density of \(1.2656 \mathrm{~g} / \mathrm{mL}\) at \(15^{\circ} \mathrm{C}\). Calculate the molarity of a solution of glycerol made by dissolving \(50.000 \mathrm{~mL}\) glycerol at \(15^{\circ} \mathrm{C}\) in enough water to make \(250.00 \mathrm{~mL}\) of solution.

Step-by-Step Solution

Answer

The molarity of the glycerol solution is 2.7464 M.

Step 1: Calculate the mass of glycerol

Using the given density (1.2656 g/mL) and volume (50.000 mL), we can calculate the mass of glycerol using the formula: mass = density × volume Mass of glycerol = (1.2656 g/mL) × (50.000 mL) Mass of glycerol = 63.28 g

Step 2: Calculate the moles of glycerol

We'll now convert the mass of glycerol into moles using the molar mass of glycerol, which is given by the sum of the atomic masses of all atoms in the molecule: Molar mass of glycerol = (3 × M_C) + (8 × M_H) + (3 × M_O) where \(M_C = 12.01 \, g/mol\), \(M_H = 1.008 \, g/mol\), and \(M_O = 16.00 \, g/mol\). Molar mass of glycerol = (3 × 12.01 g/mol) + (8 × 1.008 g/mol) + (3 × 16.00 g/mol) = 92.093 g/mol Now, we can calculate the moles of glycerol using the formula: moles = mass / molar mass Moles of glycerol = (63.28 g) / (92.093 g/mol) = 0.6866 mol

Step 3: Calculate the molarity of the solution

Now we have the moles of glycerol in the solution, and we need to find the molarity. First, we need to convert the solution volume from mL to L: volume = 250.00 mL = 0.25000 L Now use the formula to calculate the molarity: Molarity = moles / volume Molarity of the glycerol solution = (0.6866 mol) / (0.25000 L) = 2.7464 M Thus, the molarity of the glycerol solution is 2.7464 M.

Key Concepts

Glycerol

Glycerol is a versatile compound known for its sweet taste and syrupy consistency. It is used in various industries and has many practical applications. Its chemical formula is \(\mathrm{C}_{3} \mathrm{H}_{8} \mathrm{O}_{3}\), which indicates that it contains three carbon atoms, eight hydrogen atoms, and three oxygen atoms. Glycerol is known for:

• Being a triple hydroxyl compound, which means it has three hydroxyl \((-\mathrm{OH})\) groups. This structure makes it highly soluble in water and alcohols.
• It plays a crucial role in the cosmetic industry, as it helps to retain moisture, leaving products softer and smoother.
• It's utilized in the food industry as a sweetening agent and a preservative due to its antifungal properties.

Glycerol's viscosity and lubricating qualities make it important in the production of antifreeze and certain types of plastics as well.

Density

Density is an important concept in chemistry that explains the relationship between mass and volume of a substance. It is defined as the mass per unit volume, often expressed in grams per milliliter (g/mL) or kilograms per liter (kg/L).The density formula can be described as:\[\text{Density} = \frac{\text{Mass}}{\text{Volume}}\]In the context of glycerol, with a density of \(1.2656 \text{ g/mL}\) at \(15^{\circ} \text{C}\), it indicates that each milliliter of glycerol weighs 1.2656 grams.Key points on density:

• Density can be used to identify a substance and is a crucial property for the formulation of mixtures and solutions.
• In solution chemistry, knowing the density of a component helps calculate other quantities, like mass or molarity, as seen in our exercise.
• Temperature can affect density, with most substances becoming less dense as temperature increases.

Molar Mass

Molar mass is the mass of one mole of a substance, usually expressed in grams per mole (g/mol). It is calculated by summing the atomic masses of the constituent elements in a molecule. For glycerol, the molar mass is computed as follows:\[\text{Molar mass of glycerol} = (3 \times 12.01 \text{ g/mol}) + (8 \times 1.008 \text{ g/mol}) + (3 \times 16.00 \text{ g/mol}) = 92.093 \text{ g/mol}\]Understanding molar mass is crucial because:

• It allows conversion between moles and grams, bridging the gap between atomic/molecular scale and laboratory scale measurements.
• It is essential for stoichiometric calculations, enabling chemists to stick to the proportions of reactants and products in a chemical reaction.
• When determining solution concentrations like molarity, molar mass is used to convert a measured mass into an amount of substance (in moles).

Solution Chemistry

Solution chemistry revolves around understanding how substances dissolve, interact, and react in liquid solutions. A solution consists of a solute – the substance being dissolved – and a solvent – the medium in which the solute is dissolved. Important concepts in solution chemistry include:

• Molarity: Defined as the number of moles of solute per liter of solution (mol/L). It is a way to express concentration and is key to calculating the quantities needed for reactions.
• Saturation: Refers to the maximum amount of solute that can dissolve in a solvent at a given temperature. Beyond this point, any added solute will not dissolve.
• Dilution: The process of reducing the concentration of a solute in a solution, usually by adding more solvent. This is often performed to achieve desired concentration levels.

Solutions are pivotal in scientific processes, medication preparation, food applications, and numerous other fields. Understanding these fundamental concepts enables the practical application and manipulation of chemical reactions in a controlled and predictable manner.