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 is \(7.389 \text{atm}\).

Step 1: Understand the formula for osmotic pressure

The osmotic pressure (\(\Pi\)) of a solution can be calculated using the formula: \[\Pi = iCRT\]where \(i\) is the van't Hoff factor, \(C\) is the molar concentration of the solution, \(R\) is the universal gas constant, and \(T\) is the temperature in Kelvin.

Step 2: Determine the van't Hoff factor (i)

The van't Hoff factor \(i\) is determined by the degree of dissociation. Potassium ferrocyanide dissociates into two ions (\(K_4[Fe(CN)_6] \rightarrow 4K^+ + [Fe(CN)_6]^{4-}\)).Given it is \(50\%\) dissociated:\[i = 1 + \alpha(n-1)\]\(\alpha = 0.5\) and \(n = 5\) because the initial compound dissociates into 5 ions in total (4 potassium ions and 1 ferrocyanide ion). Thus,\[i = 1 + 0.5(5 - 1) = 1 + 2 = 3\]

Step 3: Calculate the osmotic pressure

Substitute the values of \(i\), \(C\), \(R\), and \(T\) in the osmotic pressure formula:\[\Pi = iCRT\]\(i = 3\), \(C = 0.1\, \text{mol/L}\), \(R = 8.314\, \text{J K}^{-1} \text{mol}^{-1}\), \(T = 300\,\text{K}\).Convert \(R\) to \(\text{L atmK}^{-1} \text{mol}^{-1}\) using the conversion: \(1 \,\text{J} = 0.0082 \,\text{L atm}\).Now, \(R = 0.0821 \, \text{L atmK}^{-1} \text{mol}^{-1}\).Substituting values gives:\[\Pi = 3 \times 0.1 \times 0.0821 \times 300\, = 7.389 \text{atm}\]

Key Concepts

van't Hoff factor

To understand the van't Hoff factor, we need to grasp the concept of dissociation in a solution. The van't Hoff factor, represented as \( i \), helps us understand how many particles a solute produces when it dissolves in a solution. It provides insight into the effect of the solute on properties like osmotic pressure.

Here’s what you need to know:

• It is a measure of the number of moles of particles formed in solution for every mole of solute dissolved.
• A compound that does not dissociate in solution will have an \( i \) value of 1. However, if it dissociates, \( i \) will be greater than 1.

In our case, potassium ferrocyanide dissociates into 5 ions. If dissociation is complete, \( i \) would be 5. However, given the dissociation is 50%, we use:
\[ i = 1 + \alpha(n-1) \] where \( \alpha \) is the degree of dissociation (0.5 here), and \( n \) is the number of ions formed (5 here). Thus,\[ i = 1 + 0.5(5-1) = 3 \]Therefore, due to partial dissociation, the van't Hoff factor for potassium ferrocyanide equals 3.

degree of dissociation

The degree of dissociation, denoted by \( \alpha \), indicates the fraction of the solute that dissociates into ions in solution. It ranges from 0 to 1, where 0 indicates no dissociation and 1 indicates complete dissociation.

Let’s break it down:

• The degree of dissociation helps in calculating properties like osmotic pressure because it impacts the number of particles present in a solution.
• In our exercise, potassium ferrocyanide has a degree of dissociation of 50%, or 0.5.

This means half of the original formula units dissociate into ions. So for every 2 units of potassium ferrocyanide, 1 unit dissociates entirely. That's why, despite starting with a single type of formula unit, the dissociation into multiple ions effectively increases the number of solute particles in the solution.

Understanding \( \alpha \) is crucial, especially when you're tackling problems that involve changes in colligative properties.

potassium ferrocyanide dissociation

Potassium ferrocyanide, with the chemical formula \( K_4[Fe(CN)_6] \), is a compound that can dissociate into ions when dissolved in water. Understanding its dissociation is key to solving problems associated with osmotic pressure.

Here’s what happens when it dissociates:

• It breaks down into four potassium ions \( K^+ \) and one ferrocyanide ion \( [Fe(CN)_6]^{4-} \).
• In terms of stoichiometry, this results in five ions from each formula unit of potassium ferrocyanide.

Importantly, not all the formula units fully dissociate. At 50% dissociation, only half of the initial potassium ferrocyanide results in these free-floating ions. Yet, such dissociation is significant in determining the corresponding osmotic pressure.

By understanding this dissociation, the number of particles involved in calculations becomes clearer, enabling precise computation of colligative properties. This understanding, in conjunction with the van't Hoff factor and degree of dissociation, helps in predicting and calculating the osmotic pressure.