Which one of the following statements is false? (a) Two sucrose solutions of same molality prepared in different solvents will have the same freezing point depression (b) The osmotic pressure (\pi) of a solution is given by the equation \(\pi=M R T\), where \(M\) is the molarity of the solution (c) Raoult's law states that the vapour pressure of a component over a solution is proportional to its mole fraction (d) The correct order of osmotic pressure for \(0.01 \mathrm{M}\) aqueous solution of each compound is \(\mathrm{BaCl}_{2}>\) \(\mathrm{KCl}>\mathrm{CH}_{3} \mathrm{COOH}>\) Sucrose

Understanding freezing point depression is crucial when exploring colligative properties of solutions. It describes how the freezing point of a liquid is lowered when a solute is dissolved in it. The more particles of solute present, the greater the depression of the freezing point. This phenomenon is governed by the formula \(\Delta T_f = k_f \cdot m\), where \(\Delta T_f\) is the change in freezing point, \(k_f\) is the cryoscopic constant specific to the solvent, and \(m\) is the molality of the solution.

This concept is important in real-world applications, such as antifreeze in car radiators, where the substance added lowers the freezing point to prevent the engine's coolant from freezing in cold climates. In the given exercise, statement (a) suggested that the molality alone determines freezing point depression. However, the solvent's cryoscopic constant also plays a significant role, making the statement false unless specifying the same solvent with the same \(k_f\).

Osmotic pressure is another colligative property and can be thought of as the 'sucking' pressure generated as water moves into a solution through a semipermeable membrane. The higher the concentration of dissolved particles in a solution, the higher its osmotic pressure. It's described mathematically by the van't Hoff equation: \(\pi = MRT\), where \(\pi\) is the osmotic pressure, \(M\) is molarity, \(R\) is the ideal gas constant, and \(T\) represents the temperature in Kelvin.

The principle of osmotic pressure is vital in biological systems, such as the transfer of water into and out of cells. In the exercise, option (b) accurately represents this formula and correctly suggests that osmotic pressure of a solution can be calculated if the molarity, temperature, and the gas constant are known.

Raoult's Law plays a fundamental part in the study of solutions, especially with regard to vapor pressure. It states that the partial vapor pressure of each component in an ideal solution is directly proportional to its mole fraction in that solution. This is expressed as \(P = P^* \cdot X\), where \(P\) is the partial vapor pressure of the component, \(P^*\) is the vapor pressure of the pure component, and \(X\) is the mole fraction of the component in the solution.

This law explains why adding a non-volatile solute to a solvent decreases the vapor pressure of the resulting solution. In the context of the exercise, statement (c) correctly describes the proportional relationship between vapor pressure and mole fraction as stated by Raoult's law for ideal solutions.

The van't Hoff equation is vital for understanding how solutes affect the behavior of solutions. It not only connects to osmotic pressure as shown by \(\pi = MRT\), but also links to other colligative properties through the van't Hoff factor, \(i\), which accounts for the degree of dissociation or ionization of a solute in solution. The extended form becomes \(\pi = i \cdot MRT\), where \(i\) represents the number of particles the solute produces in solution.

The factor \(i\) is significant because it captures the effect of electrolytes that dissociate into ions and therefore, can alter the colligative properties more than their non-electrolyte counterparts. Statement (d) in the exercise exemplifies this, as it shows the correct order of osmotic pressure for various solutes based on their dissociation into ions.

Dissociation in solution refers to the process by which molecules or ionic compounds separate into smaller particles, usually ions, when dissolved in a solvent. The extent of dissociation impacts the number of solute particles in solution, therefore, influencing colligative properties such as vapor pressure, boiling point elevation, freezing point depression, and osmotic pressure.

Ionic compounds like \(BaCl_2\) and \(KCl\) dissociate completely in aqueous solutions, producing multiple particles per formula unit, while molecular compounds like sucrose do not dissociate. The degree of dissociation is particularly important when calculating the van't Hoff factor (\(i\)), which helps in determining the exact magnitude of colligative effects. Option (d) in the problem statement correctly considers the dissociation in ranking the osmotic pressures of different solutions.