Question

Carbon tetrachloride \(\left(\mathrm{CCl}_{4}\right)\) boils at \(76.8^{\circ} \mathrm{C}\) and has a density of \(1.59 \mathrm{~g} / \mathrm{mL}\) (a) A solution prepared by dissolving \(0.287 \mathrm{~mol}\) of a nonelectrolyte in \(255 \mathrm{~mL}\) of \(\mathrm{CCl}_{4}\) boils at \(80.3^{\circ} \mathrm{C}\). What is the boiling point constant \(\left(k_{\mathrm{b}}\right)\) for \(\mathrm{CCl}_{4}\) ? (b) Another solution is prepared by dissolving \(37.1 \mathrm{~g}\) of an electrolyte \((\mathrm{MM}=167 \mathrm{~g} / \mathrm{mol})\) in \(244 \mathrm{~mL}\) of \(\mathrm{CCl}_{4}\). The resulting solution boils at \(85.2^{\circ} \mathrm{C}\). What is \(i\) for the electrolyte?

Short answer

Question: Calculate the boiling point constant (kb) for carbon tetrachloride (CCl4) and the van't Hoff factor (i) for an electrolyte. Answer: The boiling point constant (kb) for carbon tetrachloride (CCl4) is 4.94 °C/mol/kg, and the van't Hoff factor (i) for the electrolyte is approximately 2.98.

Step-by-step solution

- Calculate the boiling point elevation

First, we need to calculate the boiling point elevation, \(\Delta T_b\), which is the difference between the boiling point of the solution and the boiling point of the pure solvent. For part (a), the boiling point of the solution is \(80.3^{\circ}\mathrm{C}\), and the boiling point of the pure solvent is \(76.8^{\circ}\mathrm{C}\): \(\Delta T_b = 80.3^{\circ}\mathrm{C} - 76.8^{\circ}\mathrm{C} = 3.5^{\circ}\mathrm{C}\)

- Calculate molality

Next, we need to calculate the molality of the solution, which is the number of moles of solute per kilogram of solvent. In this case, we have \(0.287 \ \text{mol}\) of solute dissolved in \(255 \ \text{mL}\) of \(\mathrm{CCl}_{4}\). We can use the density information to convert mL to grams and then divide by 1000 to convert grams to kilograms: \(255 \ \text{mL} \cdot \frac{1.59 \ \text{g}}{\text{mL}} = 405.45 \ \text{g}\) \(405.45 \ \text{g} \cdot \frac{1 \ \text{kg}}{1000 \ \text{g}} = 0.40545 \ \text{kg}\) Now we can calculate the molality: \(m = \frac{\text{moles of solute}}{\text{kg of solvent}} = \frac{0.287 \ \text{mol}}{0.40545 \ \text{kg}} = 0.708 \ \text{mol/kg}\)

- Calculate boiling point constant (kb) for CCl4

Since we know \(\Delta T_b = 3.5^{\circ}\mathrm{C}\), \(i = 1\), and \(m = 0.708 \ \text{mol/kg}\), we can use the formula \(\Delta T_b = i \cdot k_b \cdot m\) to calculate \(k_b\): \(3.5^{\circ}\mathrm{C} = 1 \cdot k_b \cdot 0.708 \ \text{mol/kg}\) \(k_b = \frac{3.5^{\circ}\mathrm{C}}{0.708 \ \text{mol/kg}} = 4.94 \ \frac{^{\circ}\mathrm{C}}{\text{mol/kg}}\) Now we have the boiling point constant for carbon tetrachloride: \(k_{\mathrm{b}} = 4.94 \ \frac{^{\circ}\mathrm{C}}{\text{mol/kg}}\).

- Calculate the molality for part (b)

For part (b), we have \(37.1 \ \text{g}\) of an electrolyte with a molar mass of \(167 \ \text{g/mol}\) dissolved in \(244 \ \text{mL}\) of \(\mathrm{CCl}_{4}\). First, calculate the moles of solute: \(\frac{37.1 \ \text{g}}{167 \ \text{g/mol}} = 0.222 \ \text{mol}\) Now, calculate the mass of solvent in kilograms: \(244 \ \text{mL} \cdot \frac{1.59 \ \text{g}}{\text{mL}} = 387.96 \ \text{g}\) \(387.96 \ \text{g} \cdot \frac{1 \ \text{kg}}{1000 \ \text{g}} = 0.38796 \ \text{kg}\) Calculate the molality for part (b): \(m = \frac{0.222 \ \text{mol}}{0.38796 \ \text{kg}} = 0.572 \ \text{mol/kg}\)

- Calculate the van't Hoff factor (i) for the electrolyte

Now we can use the boiling point elevation formula again to calculate \(i\) for the electrolyte in part (b). This time, we know that the boiling point of the solution is \(85.2^{\circ}\mathrm{C}\): \(\Delta T_b = 85.2^{\circ}\mathrm{C} - 76.8^{\circ}\mathrm{C} = 8.4^{\circ}\mathrm{C}\) Use the formula \(\Delta T_b = i \cdot k_b \cdot m\): \(8.4^{\circ}\mathrm{C} = i \cdot 4.94 \ \frac{^{\circ}\mathrm{C}}{\text{mol/kg}} \cdot 0.572 \ \text{mol/kg}\) To solve for \(i\), divide by \(k_b \cdot m\): \(i = \frac{8.4^{\circ}\mathrm{C}}{(4.94 \ \frac{^{\circ}\mathrm{C}}{\text{mol/kg}})(0.572 \ \text{mol/kg})} = 2.98\) The van't Hoff factor for the electrolyte is approximately \(i = 2.98\).

Key concepts

Colligative Properties

Colligative properties are unique characteristics of solutions that depend only on the number of solute particles, not on their identity. These properties include boiling point elevation, freezing point depression, vapor pressure lowering, and osmotic pressure. In the context of the exercise, we're focusing on boiling point elevation, which describes how the boiling point of a solvent increases when a non-volatile solute is dissolved in it. The extent of the boiling point change is proportional to the number of solute particles present in the solution.

Understanding colligative properties is essential for applications ranging from cooking, such as determining the sugar concentration in candy making, to industrial processes like anti-freeze production.

Van't Hoff Factor

The van't Hoff factor, denoted as 'i', is a crucial concept when discussing colligative properties. This factor represents the number of particles a compound dissociates into when it dissolves in a solvent. For non-electrolyte solutions, such as sugar in water, the van't Hoff factor is typically 1, as they do not dissociate into ions. However, for electrolyte solutions like salts, 'i' becomes crucial. These substances dissociate into their constituent ions, so the van't Hoff factor will be equal to the total number of ions produced.

For example, if you dissolve NaCl in water, it dissociates into two ions (Na+ and Cl-), which means i would ideally be 2. But, in reality, 'i' can be less than 2 due to ion pairing or other interactions. In the provided exercise, 'i' helps us understand the greater boiling point elevation observed for electrolyte solutions compared to non-electrolytes.

Molality

Molality is a measurement of concentration and is defined as the number of moles of solute per kilogram of solvent, not to be confused with molarity, which is moles of solute per liter of solution. The use of molality is especially advantageous in the study of colligative properties because it doesn't change with temperature as volume does. This makes it a more precise metric in conditions involving temperature changes.

In the textbook exercise, we've used molality to determine the boiling point elevation constant for the solvent (carbon tetrachloride) and to understand the impact of different solutes on the boiling point. It's essential to convert the mass of the solvent to kilograms to maintain the proper units for using molality in calculations.

Non-Electrolyte Solutions

Non-electrolyte solutions contain solutes that, when dissolved, don't dissociate into ions. Common examples include sugar, ethanol, and urea. Non-electrolytes have van't Hoff factors that are typically equal to 1 since they don't produce additional particles that contribute to colligative properties. This means they have a relatively straightforward influence on properties like boiling point elevation; the effect is directly proportional to the molality. In our exercise, the non-electrolyte solute in carbon tetrachloride demonstrated a predictable change in boiling point elevation.

Electrolyte Solutions

In sharp contrast, electrolyte solutions contain solutes that when dissolved in a solvent, dissociate into ions—positively and negatively charged particles. These ions increase the solution's ability to conduct electricity, hence 'electrolyte'. Electrolyte solutions have van't Hoff factors greater than 1, depending on the degree of dissociation. The impact of electrolytes on colligative properties is more pronounced because the dissolved substance separates into more particles. This can lead to a significant increase in properties like boiling point elevation, as depicted in the second part of the textbook exercise where the electrolyte solution had a markedly higher boiling point due to the dissociation of more particles, evidenced by the van't Hoff factor calculated to be 2.98.