Problem 32 - Chemistry: Principles and Reactions

Question

What is the freezing point of maple syrup (66\% sucrose)? Sucrose is \(\mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11}\)

Step-by-Step Solution

Answer

Answer: The freezing point of maple syrup (66% sucrose) is approximately -10.57°C.

Step 1: Determine the Molality of Sucrose in the Maple Syrup Solution

To calculate the molality of sucrose in the solution, we need to find the molal concentration of sucrose. We are given that the maple syrup is 66% sucrose. Let's assume we have a 1 kg (1000 g) solution of maple syrup. Therefore, the mass of sucrose in the solution will be 0.66 * 1000 g = 660 g. Now, calculate the number of moles of sucrose present in the solution. The molar mass of sucrose (C12H22O11) is: (12 * 12) + (22 * 1) + (11 * 16) = 144 + 22 + 176 = 342 g/mol Number of moles of sucrose = (mass of sucrose) / (molar mass of sucrose) = 660 g / 342 g/mol = 1.93 mol Molality (m) = moles of solute / mass of solvent (kg) The mass of the solvent (water) in the solution will be the total mass of maple syrup - mass of sucrose = 1000 g - 660 g = 340 g = 0.34 kg Molality (m) = 1.93 mol / 0.34 kg = 5.68 mol/kg

Step 2: Use the Freezing Point Depression Formula

Now, we will use the freezing point depression formula to determine the freezing point of the solution. ΔTf = Kf * m * i where ΔTf = Freezing point depression, Kf = Cryoscopic constant (1.86 °C·kg/mol, for water), m = Molality (calculated in Step 1), i = Van't Hoff factor (Number of particles formed from one molecule of solute, for sucrose it's 1 because it's a non-electrolyte and doesn't dissociate in water) ΔTf = 1.86 °C·kg/mol * 5.68 mol/kg * 1 = 10.57 °C

Step 3: Determine Freezing Point of the Maple Syrup Solution

To find the freezing point of the maple syrup solution, subtract the freezing point depression (ΔTf) from the freezing point of water (0°C). Freezing point of maple syrup = Freezing point of water - ΔTf = 0°C - 10.57°C = -10.57 °C The freezing point of maple syrup (66% sucrose) is approximately -10.57°C.

Key Concepts

Understanding Molality

Molality is a measure used in chemistry to express the concentration of a solute in a solution. Unlike molarity, which measures the number of moles of solute per liter of solution, molality measures the number of moles of solute per kilogram of solvent. This makes molality particularly useful when dealing with changes in temperature and pressure because it doesn't change with the volume of the solution.

To compute molality (\( m \)), you need to know the number of moles of the solute and the mass of the solvent in kilograms. The formula for calculating molality is: \
\[ m = \frac{\text{moles of solute}}{\text{mass of solvent (kg)}} \]
For instance, if you dissolve 1.93 moles of sucrose in 0.34 kilograms of water, the molality of that solution is approximately 5.68 mol/kg.

Diving Into the Cryoscopic Constant

The cryoscopic constant (Kf) is a proportionality constant that relates the freezing point depression of a solvent to the molality of a solution. It is unique to each solvent and represents the freezing point lowering accomplished by dissolving a mole of solute per kilogram of solvent.

The cryoscopic constant is a key factor in the freezing point depression formula: \
\[ \Delta T_f = K_f \times m \times i \]
where \
\( \Delta T_f \) is the freezing point depression,\( K_f \) is the cryoscopic constant,\( m \) is the molality,and \( i \) is the Van't Hoff factor, which accounts for the number of particles the solute dissociates into. For compounds like sucrose that don't dissociate in water, \( i \) is equal to 1. Water's cryoscopic constant is 1.86 °C·kg/mol.

Properties and Behavior of Sucrose

Sucrose, commonly known as table sugar, has the chemical formula \( \mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11} \). It is a non-electrolyte, meaning it does not dissociate into ions when dissolved in water. Instead, it forms a single molecule in solution which is why, for sucrose, the Van't Hoff factor (\( i \)) is 1.

Sucrose has a molar mass of 342 g/mol and is characterized by sweet taste, solubility in water, and stability under normal conditions. When dissolved, each molecule of sucrose can lower the freezing point of water, a phenomenon explained by freezing point depression. This property allows sucrose to be used in applications like making ice-cream and antifreeze solutions, where lower freezing points are desirable.

In the context of the problem, the properties of sucrose play a significant role in determining the freezing point of the maple syrup solution. Because it is a non-electrolyte, the calculation simplifies, as we don't need to factor in multiple particles affecting the freezing point depression.