Question

The boiling point of ethanol \(\left(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\right)\) is \(78.5^{\circ} \mathrm{C}\). What is the boiling point of a solution of \(6.4 \mathrm{~g}\) of vanillin \((\mathscr{M}=152.14 \mathrm{~g} / \mathrm{mol})\) in \(50.0 \mathrm{~g}\) of ethanol \(\left(K_{\mathrm{b}}\right.\) of ethanol \(\left.=1.22^{\circ} \mathrm{C} / \mathrm{m}\right) ?\)

Short answer

The boiling point of the solution is approximately 79.52°C.

Step-by-step solution

- Find the molality of the solution

Molality (m) is defined as the number of moles of solute per kilogram of solvent. First, calculate the number of moles of vanillin \(\text{Moles of vanillin} = \frac{6.4\ \text{g}}{152.14\ \text{g/mol}} = 0.042\ \text{mol}\). Next, convert the mass of ethanol from grams to kilograms \(\text{Mass of ethanol} = 50.0\ \text{g} = 0.050\ \text{kg}\). Finally, calculate the molality \(m = \frac{0.042\ \text{mol}}{0.050\ \text{kg}} = 0.84\ \text{m}\).

- Calculate the boiling point elevation

Boiling point elevation \(\Delta T_b\)is given by the formula \(\Delta T_b = K_b \times m\), where \(K_b\) is the ebullioscopic constant and \(m\) is the molality. Using the values, \(\Delta T_b = 1.22\ ^{\circ} \text{C/m} \times 0.84\ \text{m} = 1.0248\ ^{\circ} \text{C}\).

- Determine the new boiling point of the solution

To find the new boiling point of the solution, add the boiling point elevation to the original boiling point of ethanol: \(T_{\text{boiling, solution}} = 78.5\ ^{\circ} \text{C} + 1.0248\ ^{\circ} \text{C} = 79.5248\ ^{\circ} \text{C}\).

Key concepts

Molality

Molality is a measure of solute concentration defined as the number of moles of solute per kilogram of solvent. It is different from molarity, which considers the volume of the solution. Molality is represented by the symbol 'm'.
The formula to calculate molality is: \[ m = \frac{\text{moles of solute}}{\text{kilograms of solvent}} \]
In our exercise, we first calculate the number of moles of vanillin by dividing its mass by its molar mass. Then, we convert the mass of ethanol to kilograms. Finally, we use these values to find the molality of the solution.
This step is crucial because molality is used in calculating other properties, such as boiling point elevation.

Solution Concentration

Solution concentration indicates the amount of solute present in a given quantity of solvent or solution. There are different ways to express concentration, including molarity, molality, and mole fraction.
In this context, we use molality because it simplifies calculations involving temperature changes, as it is not affected by temperature and pressure changes, unlike molarity.
Understanding concentration is essential for predicting how a solution will behave under different conditions, and it helps us in making accurate chemical calculations, especially for colligative properties like boiling point elevation or freezing point depression.

Ebullioscopic Constant

The ebullioscopic constant \(K_b\) is a property of the solvent that quantifies how much the boiling point of the solvent will increase when a solute is added. It is specific to each solvent and is usually given in degrees Celsius per molal (\(^{\text{°}}C/m\)).
The formula for boiling point elevation is: \[ \Delta T_b = K_b \times m \]
In our problem, the ebullioscopic constant for ethanol is 1.22 \(^{\text{°}}C/m\). Using this constant, and the previously calculated molality, we find the boiling point elevation. This means the boiling point of the ethanol solution will be higher than that of pure ethanol.

Chemical Calculations

Chemical calculations are essential for solving problems involving the properties and behavior of solutions. They allow us to predict how much a solvent's boiling or freezing point will change when a given amount of solute is added.
In this context, we use various formulas for steps like finding molality, calculating boiling point elevation, and determining the new boiling point. These steps involve basic arithmetic and understanding of concepts like molar mass, mass conversion, and molality.
By carefully following each calculation step, you can determine how the properties of a solution change when a solute, like vanillin, is added to a solvent, like ethanol.