Question

Which aqueous solution has a freezing point closer to its predicted value, \(0.01 \mathrm{~m}\) NaBr or \(0.01 \mathrm{~m} \mathrm{MgCl}_{2} ?\) Explain.

Short answer

0.01 m NaBr has a freezing point closer to its predicted value since its van't Hoff factor is smaller (i = 2) compared to MgCl\text{2} (i = 3).

Step-by-step solution

Understanding Freezing Point Depression

Freezing point depression describes how the addition of solute to a solvent lowers the freezing point of the solvent. The formula for freezing point depression is \(\text{Δ}T_f = i \times K_f \times m \), where \(\text{Δ}T_f\) is the change in freezing point, \(\text{K}_f\) is the freezing point depression constant, \(\text{m}\) is the molality of the solution, and \(\text{i}\) is the van't Hoff factor which represents the number of particles into which a solute dissociates in solution.

Determine the Van't Hoff Factor (i)

For \(0.01 \ \text{m}\) NaBr, NaBr dissociates into \(\text{Na}^+\) and \(\text{Br}^-\), so \(\text{i} = 2\). For \(0.01 \ \text{m}\) MgCl\text{2}, MgCl\text{2} dissociates into \(\text{Mg}^{2+}\) and 2 \(\text{Cl}^-\), so \(\text{i} = 3\).

Freezing Point Depression Calculation

Using \( \text{Δ}T_f = i \times K_f \times m \), we see that since the same solvent (water) and the same molality (0.01 \text{m}) are used, \(\text{K}_f\) and \(\text{m}\) remain constant. So, \(\text{Δ}T_f\) depends only on \(\text{i}\). Thus, for 0.01 m NaBr, \(\text{Δ}T_f\) will be proportional to 2, and for 0.01 m MgCl\text{2}, \(\text{Δ}T_f\) will be proportional to 3.

Comparative Analysis

A higher \(\text{i}\) means a greater depression of the freezing point. Hence, the freezing point of 0.01 m MgCl\text{2} (with i = 3) will be lower than that of 0.01 m NaBr (with i = 2).

Concluding Which Solution Matches Predicted Value

0.01 m NaBr, having a lower \(\text{i}\), will have a freezing point depression effect that is closer to the predicted value in the problem.

Key concepts

van't Hoff Factor

The van't Hoff factor, denoted as \(\text{i}\), is a crucial component in understanding colligative properties such as freezing point depression. It represents the number of particles into which a solute dissociates when dissolved in a solvent. For instance, when sodium bromide (NaBr) dissolves in water, it dissociates into two ions: Na\(^{+}\) and Br\(^{-}\). Therefore, \(\text{i} = 2\) for NaBr.
Similarly, magnesium chloride (MgCl\( _ 2 \)) dissociates into one magnesium ion (Mg\(^{2+}\)) and two chloride ions (2 Cl\( ^{-}\)). This gives us \(\text{i} = 3\) for MgCl\( _ 2 \). Higher \(\text{i}\) means more particles in the solution, leading to a greater impact on properties like freezing point.
Understanding the van't Hoff factor helps predict how substances will behave when dissolved, which is essential in fields ranging from chemistry to biology.

Molality

Molality (m) measures the concentration of a solution by counting the moles of solute per kilogram of solvent. It is different from molarity, which considers the volume of the solution instead of the mass.
For freezing point depression calculations, molality is particularly useful because it remains unaffected by temperature changes. This stability is important because temperature variations can alter the volume of a solution, but not its mass.
In the given exercise, both solutions have a molality of 0.01 \(\text{m}\), indicating that each contains 0.01 moles of solute per kilogram of water. This uniformity allows us to isolate the effects of the van't Hoff factor on the freezing point depression.

Aqueous Solutions

Aqueous solutions are those in which water acts as the solvent. Water is a universal solvent due to its polarity and ability to form hydrogen bonds, making it capable of dissolving a wide range of substances.
In chemistry, the behavior of solutes in aqueous solutions is particularly significant. The dissociation of ionic compounds like NaBr and MgCl\( _ 2 \) into their respective ions is an excellent example. These ions disrupt the hydrogen bonding network of water, leading to phenomena like freezing point depression.
Understanding aqueous solutions helps predict how solutes behave in biological and environmental systems, where water is the primary solvent. For example, the added ions can influence everything from cellular processes to the behavior of natural bodies of water in different seasons.