Question

Bromoform has a normal freezing point of \(7.734^{\circ} \mathrm{C} / \mathrm{m}\) and it's \(K_{f}=14.4^{\circ} \mathrm{C} / \mathrm{m}\). A solution of \(2.60 \mathrm{~g}\) of an unknown in \(100 \mathrm{~g}\) of bromoform freezes at \(5.43^{\circ} \mathrm{C}\). What is the molecular weight of the unknown? (a) \(16.25\) (b) \(162.5\) (c) 100 (d) none of these

Short answer

The molecular weight of the unknown is 162.5.

Step-by-step solution

Determine the change in freezing point

The change in freezing point (\r\(\Delta T_f\)) is the difference between the normal freezing point of the solvent and the freezing point of the solution. Calculate it using the formula \r\(\Delta T_f = (T_{\text{fp,solvent}} - T_{\text{fp,solution}})\) where \(T_{\text{fp,solvent}}\) is the normal freezing point of bromoform and \(T_{\text{fp,solution}}\) is the freezing point of the solution.

Use the freezing point depression equation

Apply the formula for freezing point depression which is \(\Delta T_f = K_{f} \cdot m\), where \(\Delta T_f\) is the change in freezing point, \(K_{f}\) is the cryoscopic constant, and \(m\) is the molality of the solution.

Calculate the molality of the unknown substance

Rearrange the freezing point depression equation from step 2 to solve for the molality \(m\): \(m = \frac{\Delta T_f}{K_{f}}\).

Calculate the moles of unknown substance

Molality is defined as moles of solute per kilogram of solvent. We can calculate the moles of unknown substance (\(n\)) using the formula \(n = m \times mass_{\text{solvent in kg}}\). The mass of the solvent (bromoform) is given in grams, so convert it to kilograms before calculation.

Calculate the molecular weight of the unknown substance

Lastly, to find the molecular weight (MW) of the unknown, use the formula \(MW = \frac{mass_{\text{solute}}}{moles_{\text{solute}}}\).

Key concepts

Colligative Properties

Colligative properties are characteristics of solutions that depend solely on the number of dissolved particles and not on their identities. One of the principal colligative properties is the freezing point depression, which is the process where the freezing point of a liquid (a solvent) is lowered by adding another compound to it, thereby forming a solution.

For a given solvent, the extent of freezing point depression is directly proportional to the molality of the solute particles in the solution. This is a pivotal concept in chemistry because it reveals vital information about the solution's composition without relying on the nature of the substance dissolved in the solvent.

To quantify the freezing point depression, the cryoscopic constant (which we'll discuss in detail shortly) is used in calculations. JEE Chemistry students often utilize the concept of freezing point depression to calculate molecular weights of unknown compounds, which is a critical skill in analytical and physical chemistry.

Molecular Weight Determination

Molecular weight determination is a fundamental process in chemistry, essential for characterizing substances and understanding their properties. It is the calculation of the mass of one mole of a substance, usually expressed in grams per mole.

In the context of colligative properties such as freezing point depression, molecular weight determination of an unknown solute can be achieved by using the colligative property formula that relates the freezing point depression of the solution to the molality of the solute. The steps provided in the problem outline how to calculate the moles of solute and, subsequently, the molecular weight from the given data.

The mathematical relationship involved in determining the molecular weight is based on the concept that the number of moles is the ratio of the mass of the solute to its molecular weight, often represented as the formula \( n = \frac{mass_{\text{solute}}}{MW} \). By rearranging this relationship, we can solve for the molecular weight when the mass of the solute and the number of moles are known.

Cryoscopic Constant

The cryoscopic constant, symbolized by \(K_f\), is specific to each solvent and is a measure of how the freezing point of the solvent changes as solute is added. It is defined as the freezing point depression observed per molal concentration of a non-volatile, non-electrolyte solute.

The value of \(K_f\) can be found experimentally and is expressed in degrees Celsius per molal (°C/m). In the exercise provided, the cryoscopic constant of bromoform has been given as \(14.4^\circ C/m\). This constant is essential in calculating the molecular weight of an unknown substance by freezing point depression method.

Understanding the cryoscopic constant is crucial for JEE Chemistry students as it helps them to analyze and predict the effects of solute concentration on the freezing point of solutions. This understanding is not only academically beneficial but also practical in industries like pharmaceuticals where precise knowledge of chemical properties is paramount.