Problem 76 - Chemistry: The Central Science in SI Units, Global Edition

Question

What is the freezing point of an aqueous solution that boils at \(105.0^{\circ} \mathrm{C} ?\)

Step-by-Step Solution

Answer

The freezing point of the aqueous solution is approximately \(-18.166^{\circ}\mathrm{C}\).

Step 1: Determine the change in boiling point

First, we need to determine the change in boiling point of the aqueous solution. To do this, we will subtract the boiling point of pure water from the boiling point of the solution. \[ \Delta T_b = T_{solution} - T_{water} \] where \(T_{solution}\) = boiling point of the solution, which is given as 105.0°C, \(T_{water}\) = boiling point of pure water, which is 100°C, \[ \Delta T_b = 105.0 - 100 = 5.0^{\circ}\mathrm{C} \] So, the change in boiling point is 5.0°C.

Step 2: Calculate the molality of the solute

Next, we need to determine the molality of the solute. To do this, we will use the boiling point elevation equation: \[ \Delta T_b = K_b \cdot m \] where \(\Delta T_b\) = change in boiling point, which is 5.0°C, \(K_b\) = molal boiling point elevation constant of water, which is 0.512°C/molal, \(m\) = molality of the solute. Solving for molality: \[ m = \frac{\Delta T_b}{K_b} = \frac{5.0}{0.512} \approx 9.766\, \mathrm{molal} \] So, the molality of the solute in the aqueous solution is approximately 9.766 molal.

Step 3: Calculate the change in freezing point

Now, we need to determine the change in freezing point of the solution. To do this, we will use the freezing point depression equation: \[ \Delta T_f = K_f \cdot m \] where \(\Delta T_f\) = change in freezing point, \(K_f\) = molal freezing-point depression constant of water, which is 1.86°C/molal, \(m\) = molality of the solute, which is approximately 9.766 molal. Calculating \(\Delta T_f\): \[ \Delta T_f = 1.86 \cdot 9.766 \approx 18.166\,^{\circ}\mathrm{C} \] So, the change in freezing point is approximately 18.166°C.

Step 4: Determine the freezing point of the solution

Finally, we need to determine the freezing point of the aqueous solution. To do this, we will subtract the change in freezing point from the freezing point of pure water. \[ T_{solution} = T_{water} - \Delta T_f \] where \(T_{solution}\) = freezing point of the solution, \(T_{water}\) = freezing point of pure water, which is 0°C, \(\Delta T_f\) = change in freezing point, which is approximately 18.166°C. Calculating the freezing point of the solution: \[ T_{solution} = 0 - 18.166 \approx -18.166\,^{\circ}\mathrm{C} \] So, the freezing point of the aqueous solution is approximately -18.166°C.

Key Concepts

Boiling Point Elevation

When a solute dissolves in a solvent like water, it typically causes the boiling point of the solution to increase compared to the boiling point of the pure solvent. This phenomenon is known as boiling point elevation. The key reason behind this is that the solute molecules interfere with the ability of the solvent molecules to escape into the gas phase. As a result, more heat energy is required to make the solution boil.

To quantify this effect, we use the formula:

• \(\Delta T_b = K_b \cdot m\)

where \(\Delta T_b\) is the change in boiling point, \(K_b\) is the ebullioscopic constant specific to the solvent, and \(m\) is the molality of the solution. Understanding this concept is essential for solving problems related to boiling point changes in solutions.

Freezing Point Depression

Just as boiling point elevation raises the temperature needed to boil a solution, freezing point depression lowers the temperature at which a solution freezes. When a solute is added to a solvent, it disrupts the crystalline structure of the pure solvent, thus lowering the freezing point.

This effect can be calculated using the formula:

• \(\Delta T_f = K_f \cdot m\)

where \(\Delta T_f\) denotes the change in freezing point, \(K_f\) is the cryoscopic constant of the solvent, and \(m\) represents the molality of the solution. For water, the constant \(K_f\) is typically 1.86°C/molal. Calculating the freezing point depression helps determine the new freezing point of the solution, which is crucial in various scientific and industrial applications.

Molality

Molality is a measure of the concentration of a solution, specifically expressed as the number of moles of solute per kilogram of solvent. It is represented using the unit 'molal' and is calculated using the formula:

• \(m = \frac{n_{solute}}{m_{solvent}}\)

where \(n_{solute}\) is the number of moles of the solute and \(m_{solvent}\) is the mass of the solvent in kilograms.

One key advantage of using molality over molarity is that molality is not affected by temperature changes. This makes it particularly useful for studying properties like boiling point elevation and freezing point depression, which depend on the number of solute particles in a solvent.

Aqueous Solution

An aqueous solution is a solution in which water acts as the solvent. "Aqueous" signifies the presence of water, the most common and versatile solvent. This type of solution has a myriad of applications in both natural and industrial processes.

In an aqueous solution, the properties of the solute, such as its boiling and freezing points, are altered due to the interactions with water molecules. This alteration leads to phenomena like colligative properties, which include boiling point elevation and freezing point depression.

• Aqueous solutions are crucial in biological processes where water is the solvent for various biochemical reactions.
• Understanding the behavior of solutes in aqueous solutions allows scientists and engineers to manipulate conditions for desired outcomes in various fields.

Learning about aqueous solutions helps in comprehending a wide range of chemical reactions and processes that rely on water as the primary solvent.