Problem 68 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

What is the molecular mass of a nonionizing solid if \(10 \mathrm{~g}\) of this solid, when dissolved in \(100 \mathrm{~g}\) of water, forms a solution, which freezes at \(-1.24^{\circ} \mathrm{C}\) ? \(K_{\mathrm{f}}\left(\mathrm{H}_{2} \mathrm{O}\right)=1.86^{\circ} \mathrm{C} \mathrm{kg} \mathrm{mol}^{-1}\) (a) 250 (b) 150 (c) 120 (d) 75

Step-by-Step Solution

Answer

The molecular mass of the nonionizing solid is 150 g/mol.

Step 1: Understand the Concept

To find the molecular mass, we need to use the freezing point depression formula: \(\Delta T_f = K_f \times m\), where \(\Delta T_f\) is the freezing point depression, \(K_f\) is the freezing point depression constant for water, and \(m\) is the molality of the solution. The molality is equal to the moles of solute per kilogram of solvent.

Step 2: Calculate the Freezing Point Depression

The freezing point depression (\(\Delta T_f\)) is the difference between the freezing point of pure water (0°C) and the freezing point of the solution. Thus, \(\Delta T_f = 0^\circ\mathrm{C} - (-1.24^\circ\mathrm{C}) = 1.24^\circ\mathrm{C}\).

Step 3: Calculate the Molality

With the freezing point depression and the freezing point depression constant, calculate the molality using the formula \(\Delta T_f = K_f \times m\): \(m = \frac{\Delta T_f}{K_f} = \frac{1.24^\circ\mathrm{C}}{1.86^\circ\mathrm{C}\,\mathrm{kg}\,/\,\mathrm{mol}}\).

Step 4: Convert Mass of Water to Kilograms

Convert the mass of water to kilograms to be used in the molality calculation. Since we have 100 g of water, this is equivalent to \(100\,\mathrm{g} \times \frac{1\,\mathrm{kg}}{1000\,\mathrm{g}} = 0.1\,\mathrm{kg}\).

Step 5: Calculate the Number of Moles of Solute

Use the molality and the mass of the solvent (in kilograms) to find the number of moles of solute: \(\text{moles of solute} = m \times \text{mass of solvent (kg)} = \frac{1.24}{1.86} \times 0.1\).

Step 6: Calculate the Molar Mass of the Solute

Finally, use the mass of the solute (10 g) and the number of moles of solute to calculate the molar mass (molecular mass) of the solute: \( \text{Molar Mass} = \frac{\text{Mass of solute (g)}}{\text{Moles of solute}}\).

Key Concepts

Colligative Properties

Colligative properties are properties of solutions that depend on the number of particles in a given amount of solvent and not on the nature of the chemical species present. The term colligative is derived from the Latin word 'colligatus' meaning bound together, signifying that the properties are linked to the number of solute particles rather than their identity.

One of the key colligative properties is freezing point depression, which is the phenomenon where the freezing point of a liquid (solvent) is lowered by adding another substance (solute). This occurs because the solute particles disrupt the solvent's ability to form a solid structure, hence freezing requires a lower temperature. Other colligative properties include boiling point elevation, vapor pressure lowering, and osmotic pressure.

Molality Calculation

Molality is a crucial concept for understanding colligative properties. It is defined as the number of moles of solute per kilogram of solvent. Unlike molarity, which is the number of moles of solute per liter of solution, molality is not affected by changes in temperature because masses do not change with temperature.

To calculate the molality \(m\), you need the mass of the solute in grams, the molar mass of the solute, and the mass of the solvent in kilograms. The formula for molality is:
\[ m = \frac{{\text{{moles of solute}}}}{{\text{{mass of solvent (kg)}}}} \]
This calculation is important when working with colligative properties like freezing point depression where the formula \( \Delta T_f = K_f \times m \) is used. Here, \( \Delta T_f \) represents the change in freezing point, and \(K_f\) stands for the freezing point depression constant of the solvent.

Molar Mass Determination

Determining the molar mass (molecular mass) of a solute is a fundamental step in various chemical calculations. Molar mass represents the mass of one mole of substance and has units of grams per mole (g/mol). It is also the key to calculating the number of moles when the mass of a substance is known.

To find the molar mass, you can rearrange the formula for molality:\[ Molar Mass = \frac{{\text{{Mass of solute (g)}}}}{{\text{{Moles of solute}}}} \]
When involving colligative properties like freezing point depression, you will typically be given the mass of the solute and the solvent, and the freezing point depression constant. You use these to find the moles of solute and ultimately the molar mass. This process is important in a wide variety of fields including chemistry, biology, and pharmacology as it helps to identify and quantify the solute in a mixture.