Problem 104 - A Complete Resource Book in Chemistry for JEE Main

Question

A decimolar solution of potassium ferrocyanide is \(50 \%\) dissociated at \(300 \mathrm{~K}\). Calculate the osmotic pressure of the solution. \(\left(\mathrm{R}=8.314 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}\right)\) (a) \(0.07389 \mathrm{~atm}\) (b) \(7.389 \mathrm{~atm}\) (c) \(738.89 \mathrm{~atm}\) (d) \(73.89 \mathrm{~atm}\)

Step-by-Step Solution

Answer

The osmotic pressure of the solution is approximately 7.389 atm (option b).

Step 1: Understanding the Problem

We are given a decimolar solution of potassium ferrocyanide, which is 50% dissociated, and asked to calculate the osmotic pressure at 300 K using the gas constant \( R = 8.314 \) J/mol·K. We need to apply concepts of colligative properties, specifically osmotic pressure, which is influenced by the degree of dissociation.

Step 2: Recall the Formula for Osmotic Pressure

Osmotic pressure is given by the van 't Hoff equation: \( \Pi = iCRT \), where \( \Pi \) is the osmotic pressure, \( i \) is the Van 't Hoff factor, \( C \) is the molar concentration, \( R \) is the universal gas constant, and \( T \) is the temperature in Kelvin.

Step 3: Determine the Van't Hoff Factor

The Van 't Hoff factor, \( i \), accounts for the dissociation of the electrolyte. For potassium ferrocyanide, \( K_4[Fe(CN)_6] \), which dissociates into 4 \( K^+ \) ions and 1 \( [Fe(CN)_6]^{4-} \) ion, resulting in 5 particles. Since the solution is 50% dissociated, \( i = 1 + 0.5 \times (5 - 1) = 1 + 0.5 \times 4 = 3 \).

Step 4: Calculate the Osmotic Pressure

Substitute the values into the osmotic pressure formula: \( \Pi = iCRT = 3 \times 0.1 \times 8.314 \times 300 \). First, calculate \( iCRT = 3 \times 0.1 \times 8.314 \times 300 = 747.06 \) J/L. To convert to atm, divide by 101.325 (since \( 1 \) atm = \( 101.325 \) J/L), resulting in \( \Pi = \frac{747.06}{101.325} = 7.373 \) atm.

Step 5: Round and Compare with Options

Round \( 7.373 \) atm to the nearest value given in the options. The closest answer is \( 7.389 \) atm, which matches option (b).

Key Concepts

Colligative Properties

Colligative properties are a set of properties that depend on the number of solute particles in a solution, rather than the type of particles. These include vapor pressure lowering, boiling point elevation, freezing point depression, and osmotic pressure.

Osmotic pressure is the focus of our exercise. It occurs when a solvent moves through a semipermeable membrane to the side with a higher solute concentration. This movement continues until equilibrium is reached. The key takeaway is that osmotic pressure is determined by how many particles are in the solution. Even if the solute particles have different masses or sizes, only their number matters for these properties.

• Number of particles affects the property, not the identity.
• Ideal for understanding effects of electrolytes in solutions.
• Related to phenomena like plants absorbing water through roots.

Colligative properties are powerful tools in chemistry since they allow us to predict how solutes affect a solution's behavior.

van 't Hoff Factor

The van 't Hoff factor, denoted as \( i \), is a crucial concept when dealing with solutions of electrolytes. It reflects how many particles a solute separates into in solution. For non-electrolytes that do not dissociate, \( i = 1 \). But for electrolytes, this factor can be greater than one, because they dissociate into multiple ions.

For example, potassium ferrocyanide \( K_4[Fe(CN)_6] \) dissociates into five different particles (four potassium ions and one ferrocyanide ion). Therefore, if the solution is fully dissociated, \( i = 5 \). However, if it's only partially dissociated like in our exercise where it is 50% dissociated, we calculate \( i \) as follows:

• Initial particles count: 1 formula unit.
• Final particles count if fully dissociated: 5 particles total.
• Given 50% dissociation: \( i = 1 + 0.5 \times (5-1) = 3 \).

The van 't Hoff factor helps us understand how significant the effect of a solute will be on colligative properties.

Electrolyte Dissociation

Electrolyte dissociation refers to the process by which ionic compounds dissolve in a solvent, typically water, breaking into ions. These ions can conduct electricity, which is why solutions containing them are known to be 'electrolytic.'

The degree of dissociation indicates what fraction of the solute dissociates into ions. In the case of potassium ferrocyanide in our exercise, we see a 50% dissociation rate. This means that half of the salt dissolves into its constituent ions, while the other half remains as intact units in solution.

• Electrolytes can be strong (complete dissociation) or weak (partial dissociation).
• Electrolyte solutions are used in many applications, like batteries and biological systems.
• The extent of dissociation directly affects properties such as osmotic pressure and conductivity.

Understanding electrolyte dissociation is essential for many fields, including chemistry and biology, as it impacts how substances behave in solution, influencing reactions and processes in natural and industrial settings.