Problem 155 - A Complete Resource Book in Chemistry for JEE Main

Question

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 \(\pi=\mathrm{MRT}\), where \(\mathrm{M}\) 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

Step-by-Step Solution

Answer

Statement (a) is false.

Step 1: Analyze Statement (a)

Check the consistency of the freezing point depression rule for solutions. Freezing point depression depends on the molality and the properties of the solvent. Since different solvents have different freezing point depression constants, the freezing points can differ. Therefore, statement (a) is false.

Step 2: Examine Statement (b)

Verify the relation for osmotic pressure. The equation \(\pi = MRT\) correctly defines the osmotic pressure for a solution, where \M\ is molarity, \R\ is the universal gas constant, and \T\ is temperature in Kelvin. This is a true statement.

Step 3: Evaluate Statement (c)

Check the claim about Raoult's law. Raoult's law states that the partial vapor pressure of each component in a solution is directly proportional to its mole fraction. Thus, statement (c) is true.

Step 4: Assess Statement (d)

Determine the order of osmotic pressure based on van't Hoff factor \(i\). \(\mathrm{BaCl}_{2}\) disassociates into 3 ions \((i=3)\), \(\mathrm{KCl}\) into 2 ions \((i=2)\), \(\mathrm{CH}_{3} \mathrm{COOH}\)—a weak acid—partially dissociates more than sucrose which does not dissociate \((i=1)\). Therefore, the correct order should be \(\mathrm{BaCl}_{2} > \mathrm{KCl} > \mathrm{CH}_{3} \mathrm{COOH} > \text{sucrose}\). This statement is true.

Key Concepts

Freezing Point Depression

When a solute is added to a solvent, the freezing point of that mixture becomes lower than the freezing point of the pure solvent. This phenomenon, known as freezing point depression, occurs because the solute particles disrupt the formation of the solid phase of the solvent, effectively requiring a lower temperature to achieve solidification. The magnitude of freezing point depression is determined by the molal concentration of the solute and a unique constant related to the solvent, known as the freezing point depression constant ().

The formula for freezing point depression is:
\[ \Delta T_f = i  m \] where:

• \( \Delta T_f \) is the change in freezing point,
• \( i \) is the van't Hoff factor, representing the number of particles the solute breaks into,
• \(  \) is the freezing point depression constant for the solvent,
• \( m \) is the molality of the solution.

It's important to note that different solvents have different \(  \) values, which means that even if two solutions have the same molality, they can have different freezing point depressions if the solvents are different. This concept explains why statement (a) in the original exercise is false.

Osmotic Pressure

Osmotic pressure is an important colligative property that distinguishes itself from others by its dependence on the movement of solvent molecules through a semipermeable membrane. This pressure arises when there is a difference in solute concentration across the membrane. The movement of solvent molecules from a region of lower solute concentration to a higher concentration continues until equilibrium is reached, causing a measurable pressure called osmotic pressure.

The equation to calculate osmotic pressure is given by:\[\pi = MRT\]where:

• \( \pi \) is the osmotic pressure,
• \( M \) is the molarity of the solution,
• \( R \) is the universal gas constant,
• \( T \) is the temperature in Kelvin.

The osmotic pressure increases with higher solute concentrations and temperatures, making it crucial for phenomena like plant nutrient absorption and human kidney function. In the context of the original exercise, this correctly reflects statement (b), as the formula accurately calculates the osmotic pressure.

Raoult's Law

Raoult's Law plays a central role in understanding solutions, specifically in predicting how the presence of a solute affects the vapor pressure of a solvent. According to Raoult's Law, the partial vapor pressure of a solvent in a solution is directly proportional to the mole fraction of the solvent in the solution. This means that as you increase the concentration of the solvent, its vapor pressure increases. Conversely, introducing a non-volatile solute lowers the vapor pressure due to reduced solvent mole fraction.

The law is mathematically expressed as:\[P_i = x_i P_i^0\]where:

• \( P_i \) is the partial vapor pressure of the component in the solution,
• \( x_i \) is the mole fraction of the solvent,
• \( P_i^0 \) is the vapor pressure of the pure solvent.

This principle helps explain why solvents with non-volatile solutes are more stable and don't evaporate as quickly. It's also the basis for distillation processes. In the given exercise, statement (c) involving Raoult's law is true, as it accurately describes the relationship between vapor pressure and mole fraction.