Question

When a molar mass is determined from freezing-point depression, it is possible to make each of the following errors (among others). In each case, predict whether the error would cause the reported molar mass to be greater or less than the actual molar mass. (a) There was dust on the balance, causing the mass of solute to appear greater than it actually was. (b) The water was measured by volume, assuming a density of \(1.00 \mathrm{~g}-\mathrm{cm}^{-3}\), but the water was warmer and less dense than assumed. (c) The thermometer was not calibrated accurately, and so the temperature of the freezing point was actually \(0.5^{\circ} \mathrm{C}\) higher than recorded. (d) The solution was not stirred sufficiently, and so not all the solute dissolved.

Short answer

The reported molar mass would be (a) greater, (b) less, (c) greater, and (d) greater than the actual molar mass due to the respective errors described.

Step-by-step solution

Analyze Impact of Dust on Balance

When there is dust on the balance, it adds extra mass to the reading. Therefore, the mass of the solute will be overestimated. Since molar mass is calculated using the mass of the solute, an overestimated mass of solute would lead to a calculated molar mass that is higher than the actual molar mass.

Evaluate Effect of Water Density Error

If the water is warmer and therefore less dense than assumed, the volume measurement would yield less mass than if the water were at the assumed density of 1.00 g/cm^3. This means that for a given volume, the actual mass of water would be less. Since the molar mass calculation depends on the mass of the solvent, an underestimated mass would lead to a calculated molar mass that is lower than it actually is.

Determine Effect of Inaccurate Thermometer Calibration

If the freezing point is actually 0.5°C higher than recorded, the freezing point depression is less than what would be measured accurately. Since the freezing point depression is used to calculate the molar mass, a smaller depression would result in a calculated molar mass that is greater than the actual molar mass.

Assess the Consequence of Incomplete Solute Dissolution

If the solution is not stirred sufficiently and not all the solute dissolves, the undissolved portion does not contribute to the freezing point depression. This would lead to an underestimation of the freezing point depression, causing the calculation to yield a higher molar mass than the real value.

Key concepts

Freezing-Point Depression

Understanding the concept of freezing-point depression is crucial in determining the molar mass of a substance. When a solute is dissolved in a solvent, the resulting solution has a lower freezing point than the pure solvent. This phenomenon is known as freezing-point depression. It is a colligative property, which means that it depends on the number of solute particles in the solution, not on their chemical identity.

In the context of the exercise, the depression in the freezing point is used to calculate the molar mass of the dissolved solute. The bigger the molar mass of the solute, the less it depresses the freezing point of the solvent for a given amount of solute particles. Conversely, for a certain mass of solute, if the actual molar mass is larger than calculated, it means the number of moles (and thus particles) is smaller than estimated, resulting in a less significant depression of the freezing point.

To accurately determine the molar mass from freezing-point depression, one must ensure that the measurement of the depression is precise. Any errors in measuring the temperature of the freezing point, such as from an uncalibrated thermometer, can cause significant discrepancies in the calculated molar mass, as seen in the exercise.

Colligative Properties

Colligative properties are properties of solutions that depend on the ratio of the number of solute particles to the number of solvent molecules in a solution, and not on the nature of the chemical species present. Other colligative properties include boiling-point elevation, vapor-pressure lowering, and osmotic pressure.

The principle behind colligative properties is important in explaining why an error in measuring the mass of the solute or solvent (due to dust or inaccurate water density) impacts the determination of molar mass. If the mass of the solute is overestimated due to external contamination like dust, there will be an illusion of more solute particles than there actually are, leading to an erroneous calculation of a higher molar mass. On the other hand, underestimating the mass of the solvent will give a false indication of a higher concentration of solute, hence a higher calculated depression of freezing point and a lower molar mass.

For a student to grasp the relevance of accurate measurements, it's essential to understand that the subtle changes in mass or temperature can significantly impact the calculations due to the dependency on the ratio of solute to solvent in colligative properties.

Solution Preparation

Solution preparation is a fundamental part of many chemical processes, and meticulous attention must be paid to achieve accurate results in any experiment, including the determination of molar mass through freezing-point depression. To prepare a solution, one typically dissolves a known mass of solute in a known volume of solvent. Any errors in this process can lead to incorrect conclusions.

An integral aspect of solution preparation is ensuring that the entire solute dissolves. In the exercise, insufficient stirring leads to incomplete dissolution of the solute, causing an inaccurate measure of freezing-point depression and, consequently, an overestimation of the molar mass. In practice, students must ensure thorough mixing to allow for the expected interactions between solute and solvent that affect colligative properties.

Another critical step is the accurate measurement of both solute and solvent. This includes using a clean and properly calibrated balance for the solute and accounting for temperature effects on solvent density. As demonstrated in the exercise, overlooking these steps can result in errors in determining the solution's concentration, further affecting the accuracy of calculated molar mass through colligative properties such as freezing-point depression.