Problem 65 - Chemistry: An Atoms-Focused Approach

Question

Cinnamon owes its flavor and odor to cinnamaldehyde (C \(_{9} \mathrm{H}_{8} \mathrm{O}\) ). Determine the freezing point of a solution of 75 mg of cinnamaldehyde dissolved in \(1.00 \overline{\mathrm{g}}\) of benzene \(\left(K_{f}=4.3^{\circ} \mathrm{C} / \mathrm{m} ; \text { normal freezing point }=5.5^{\circ} \mathrm{C}\right)\).

Step-by-Step Solution

Answer

Given the cryoscopic constant (Kf) for benzene is 4.3°C/m. Solution: Step 1: Calculate the number of moles of C9H8O: n(C9H8O) = (0.075 g) / (144.15 g/mol) ≈ 5.20 x 10⁻⁴ mol Step 2: Calculate the molality of the solution: molality ≈ (5.20 x 10⁻⁴ mol) / (1 g) * (1 kg / 1000 g) ≈ 0.000520 mol/kg Step 3: Calculate the freezing point depression: ΔTf ≈ 4.3°C/m * 0.000520 mol/kg ≈ 0.002234°C Step 4: Determine the new freezing point of the solution: f.p(solution) ≈ 5.5°C - 0.002234°C ≈ 5.4978°C The freezing point of the solution is approximately 5.4978°C.

Step 1: Calculate the number of moles of cinnamaldehyde

We are given the mass of cinnamaldehyde (75 mg), so we can calculate the number of moles using the molar mass of cinnamaldehyde, which is \(C_{9} \mathrm{H}_{8} \mathrm{O}\) or (9x12.01 + 8x1.01 + 1x16.00) g/mol: n_{C_{9} H_{8} O} = \frac{mass}{molar\_mass} = \frac{0.075 \,\text{g}}{(9 \cdot 12.01) + (8 \cdot 1.01) + (1 \cdot 16.00)\,\text{g/mol}} = \frac{0.075}{144.15} mol

Step 2: Calculate the molality of the solution

Now that we have the number of moles of cinnamaldehyde, we can calculate the molality, which is defined as the number of moles of solute per kilogram of solvent: molality = \(\frac{n_{C_{9} H_{8} O}}{mass _{benzene}}\) = \(\frac{\frac{0.075}{144.15},\text{mol} }{1.00 \,\text{g}}\cdot\frac{1 \,\text{kg}}{1000\, \text{g}}\)

Step 3: Calculate the freezing point depression

Now that we have the molality, we can use the formula for freezing point depression to find the change in freezing point: \(\Delta T_f = K_f \cdot m = 4.3 ^{\circ} \mathrm{C} / \mathrm{m} \cdot \frac{\frac{0.075}{144.15},\text{mol} }{1.00 \,\text{g}}\cdot\frac{1 \,\text{kg}}{1000\, \text{g}}\)

Step 4: Determine the new freezing point of the solution

Finally, we can subtract the freezing point depression from the normal freezing point of benzene to find the freezing point of the solution: f.p._{solution} = f.p._{benzene} - \(\Delta T_f = 5.5^{\circ} \mathrm{C} - 4.3 ^{\circ} \mathrm{C} / \mathrm{m} \cdot \frac{\frac{0.075}{144.15},\text{mol} }{1.00 \,\text{g}}\cdot\frac{1 \,\text{kg}}{1000\, \text{g}}\) After calculating the values and solving for the freezing point of the solution, we get the final answer.

Key Concepts

Molality Calculation

Molality is a measure of the concentration of a solution. It is defined as the moles of solute per kilogram of solvent. Calculating molality is key in determining how a solute affects the properties of a solution, such as its freezing point. To calculate molality, first determine the number of moles of the solute. This is done using the formula:\[\text{Number of moles} = \frac{\text{mass of solute (g)}}{\text{molar mass of solute (g/mol)}}\]In the exercise, we calculated the moles of cinnamaldehyde, using its given mass and molar mass. With the number of moles known, proceed to find the molality:\[\text{Molality} = \frac{\text{number of moles of solute}}{\text{mass of solvent (kg)}}\]Remember to convert the mass of the solvent from grams to kilograms by dividing by 1000. The molality gives us insight into how concentrated the solution is, playing a crucial role in changes like freezing point depression.

Cinnamaldehyde

Cinnamaldehyde is the organic compound responsible for the distinctive aroma and taste of cinnamon. Its chemical formula is \(C_{9}H_{8}O\), comprising nine carbon atoms, eight hydrogen atoms, and one oxygen atom. This aromatic oily liquid is an aldehyde, which belongs to a group of organic compounds characterized by the presence of a carbonyl center. Cinnamaldehyde is not just a flavoring agent; it has various applications in fragrances, cosmetics, and as an antimicrobial agent. In the context of solution chemistry, cinnamaldehyde's properties, such as its molar mass, become crucial when it is used as a solute. Understanding its structure and behavior helps in calculating how it will interact in solutions, such as determining its effect on the freezing point of the solvent it's mixed with.

Solution Chemistry

Solution chemistry focuses on understanding how substances dissolve, interact, and change phase in solutions. Key to this is the concept of how different solutes affect the properties of solvents. A solution is a homogeneous mixture composed of a solute and a solvent. The solute is the substance dissolved, whereas the solvent is the medium that dissolves the solute. In the provided exercise, benzene serves as the solvent in which cinnamaldehyde is dissolved as the solute. One of the fascinating aspects of solutions is the colligative properties, which depend on the number of solute particles rather than their nature. Freezing point depression is a colligative property observed when the freezing point of a solvent decreases upon adding a solute. This decrease occurs because the solute particles disrupt the formation of the solid phase, requiring a lower temperature to achieve freezing. Understanding these properties is crucial for applications ranging from anti-freezing solutions in vehicles to determining molecular weights of unknown substances in laboratories.