Problem 74 - Chemistry The Science in Context

Question

Determine the boiling point of an aqueous solution that is \(2.50 \mathrm{m}\) ethylene glycol \(\left(\mathrm{HOCH}_{2} \mathrm{CH}_{2} \mathrm{OH}\right) ; K_{\mathrm{b}}\) for water is \(0.52^{\circ} \mathrm{C} / \mathrm{m} .\) Assume that the boiling point of pure water is \(100.00^{\circ} \mathrm{C}\)

Step-by-Step Solution

Answer

Answer: The boiling point of a 2.50 m ethylene glycol aqueous solution is 101.3 °C.

Step 1: Determine the boiling point elevation

Using the given molality of the ethylene glycol solution (\(m = 2.50 \, \mathrm{m}\)) and the molal boiling-point-elevation constant for water (\(K_{\mathrm{b}} = 0.52 ^{\circ} \mathrm{C} / \mathrm{m}\)), we can calculate the boiling point elevation (\(\Delta T_{\mathrm{b}}\)) of the solution using the formula: $$ \Delta T_{\mathrm{b}} = m \cdot K_{\mathrm{b}} $$

Step 2: Calculate the boiling point elevation

Now, substituting the given values into the formula, we get: $$ \Delta T_{\mathrm{b}} = (2.50 \, \mathrm{m})\cdot(0.52 ^{\circ} \mathrm{C} / \mathrm{m}) = 1.3 ^{\circ} \mathrm{C} $$ So, the boiling point of the solution is elevated by \(1.3^{\circ} \mathrm{C}\).

Step 3: Find the boiling point of the aqueous solution

Now that we have the boiling point elevation, we can determine the boiling point of the aqueous solution. Add the boiling point elevation to the boiling point of pure water, which is given as \(100.00^{\circ} \mathrm{C}\): $$ \mathrm{Boiling \, Point \, of \, Solution} = \mathrm{Boiling \, Point \, of \, Pure \, Water} + \Delta T_{\mathrm{b}} $$

Step 4: Calculate the boiling point of the aqueous solution

Substitute the boiling point of pure water and the boiling point elevation into the formula: $$ \mathrm{Boiling \, Point \, of \, Solution} = 100.00^{\circ} \mathrm{C} + 1.3^{\circ} \mathrm{C} = 101.3^{\circ} \mathrm{C} $$ The boiling point of the \(2.50 \mathrm{m}\) ethylene glycol aqueous solution is \(101.3^{\circ} \mathrm{C}\).

Key Concepts

Aqueous Solutions

When we talk about aqueous solutions, we refer to mixtures where water acts as the solvent. This means water is the medium in which other substances—like solids, liquids, or gases—are dissolved. In our day-to-day life, many substances form aqueous solutions, such as salty water or sugar dissolved in water.
A key aspect of aqueous solutions is their ability to undergo property changes, such as boiling and freezing, due to the presence of solutes. Solutes interact with the solvent, potentially altering how the solution behaves compared to pure water:

• Boiling Point Elevation: A solution often boils at a higher temperature than pure water because of solute-solvent interactions.
• Freezing Point Depression: Conversely, a solution can freeze at a lower temperature.

Understanding these properties helps explain many everyday phenomena and is crucial in scientific applications.

Molality

Molality is one way to express the concentration of a solution. Unlike molarity, which is based on the volume of solution, molality is based on the amount of solvent. Specifically, molality is defined as the number of moles of solute per kilogram of solvent:\[ \text{molality} (m) = \frac{\text{moles of solute}}{\text{kilograms of solvent}} \]This unit of concentration is particularly useful under conditions where temperature or pressure changes, as molality does not change with temperature or pressure due to its mass-based nature.
For example, when dealing with the problem of boiling point elevation, molality provides an accurate measure because it reflects the amount of solute affecting the solvent's particles directly. Students often encounter this in calculations related to colligative properties.

Boiling Point Elevation Constant

The boiling point elevation constant, often symbolized as \(K_{b}\), is a specific number that represents how much the boiling point of a solvent increases per molal concentration of a non-volatile solute. Each solvent has its own unique \(K_{b}\) due to its specific properties, such as intermolecular forces.
In mathematical terms, the boiling point elevation \((\Delta T_b)\) can be calculated by multiplying the molality of a solution by its \(K_{b}\):\[ \Delta T_b = m \times K_b \]Essentially, this constant helps predict how much higher the boiling point of an aqueous solution will be compared to pure water. For example, in the case of water, the \(K_{b}\) is \(0.52^{\circ} C/m\), indicating that for every molal unit of solute added, the boiling point increases by \(0.52^{\circ}C\).
This concept is crucial for applications where controlling boiling points is necessary, such as in chemical synthesis or food preparation.