Question

What is the boiling point of a solution of \(1.00 \mathrm{~g}\) of naphthalene dissolved in \(94.0 \mathrm{~g}\) of toluene? The normal boiling point of toluene is \(110.75^{\circ} \mathrm{C}\) and \(K_{\mathrm{x}, \mathrm{b}}=32.0 \mathrm{~K}\) for toluene. (a) \(110.95^{\circ} \mathrm{C}\) (b) \(111.0^{\circ} \mathrm{C}\) (c) \(113.41^{\circ} \mathrm{C}\) (d) \(110.75^{\circ} \mathrm{C}\)

Short answer

The boiling point of the solution is (a) 110.95°C.

Step-by-step solution

Determine the molality of the solution

Find the number of moles of naphthalene by using its molecular weight (128.17 g/mol). Molality (m) is moles of solute per kilogram of solvent. Calculate molality using the formula: m = (moles of naphthalene) / (mass of toluene in kg).

Calculate the boiling point elevation

Use the boiling point elevation formula: \(\Delta T_b = i \cdot K_b \cdot m\), where \(i\) is the van't Hoff factor (1 for naphthalene, as it is a non-electrolyte), \(K_b\) is the ebullioscopic constant of toluene (32.0 K kg/mol), and \(m\) is the molality of the solution. Calculate \(\Delta T_b\).

Determine the new boiling point

Add the boiling point elevation (\(\Delta T_b\)) to the normal boiling point of toluene (110.75°C) to find the boiling point of the solution: Boiling point of solution = Normal boiling point of toluene + \(\Delta T_b\).

Key concepts

Colligative Properties

Colligative properties are unique characteristics of solutions that depend solely on the number of solute particles, rather than their identity. These properties arise because solute particles disrupt the solvent's structure, which affects its overall physical behavior.

For instance, when a non-volatile solute dissolves in a solvent, several phenomena can be observed, which include boiling point elevation, freezing point depression, vapor pressure lowering, and osmotic pressure increase. The impact on the boiling point, as in the exercise we are considering, is a classic example of a colligative property. It's important to note that these properties are reliant on the quantity of solute particles, which means that even compounds that seem chemically different can produce the same effects on a solvent if they are present in the same molar amount.

Understanding colligative properties can be critical for applications ranging from cooking, such as adding salt to water to raise the boiling point, to industrial processes, where control over the boiling and freezing points of solutions is necessary for efficient operations.

Molality

Molality (m) is a measure of the concentration of a solution, defined as the number of moles of solute per kilogram of solvent. Unlike molarity, which depends on the total volume of the solution, molality is dependent only on the mass of the solvent, making it unaffected by temperature changes.

This measure is particularly useful in situations involving temperature, as it remains constant even if the solution expands or contracts. To calculate molality, one would divide the moles of solute by the mass of the solvent in kilograms. In our naphthalene in toluene example, molality is essential for determining how much the boiling point of toluene will increase due to the dissolved naphthalene. Using molality allows for a more accurate prediction in scenarios involving temperature shifts, making it ideal for studying the boiling point elevation of solutions.

Ebullioscopic Constant

The ebullioscopic constant, symbolized as \(K_b\), is a property of the solvent being used and is vital in the calculations for boiling point elevation. It quantifies the extent to which the boiling point of a given solvent will increase per molal concentration of a non-volatile solute.

In the given exercise, the ebullioscopic constant for toluene is \(32.0\ K\cdot kg/mol\). This value indicates that for every mole of a non-volatile solute dissolved per kilogram of toluene, the boiling point of toluene will rise by 32 degrees Kelvin. This constant provides a straightforward way to calculate the change in boiling point brought about by the solute, making it crucial for the correct application of the formula \(\Delta T_b = i \cdot K_b \cdot m\) to determine the boiling point elevation in practical situations.

Van't Hoff Factor

The van't Hoff factor, denoted by \(i\), describes the number of particles into which a compound dissociates in solution. For non-electrolytes, substances that do not break into ions in solution, the van't Hoff factor is typically 1, as these compounds dissolve into individual molecules without further separation. This relates directly to colligative properties, where the effect on the boiling point is proportional to the number of particles present in the solution.

For electrolytes, the van't Hoff factor can be greater than 1 because these compounds dissociate into ions. However, in the case of our naphthalene example, naphthalene is a non-electrolyte, and thus the van't Hoff factor is 1. This implies that each molecule of naphthalene dissolved in toluene contributes equally to the boiling point elevation without any additional particles from dissociation. Understanding the van't Hoff factor is crucial for accurately predicting the changes in a solution's boiling point and other colligative properties.