Problem 26 - A Complete Resource Book in Chemistry for JEE Main

Question

If for the reaction given below \(2 \mathrm{PQ} \rightleftharpoons \mathrm{P}_{2}+\mathrm{Q}_{2} \mathrm{~K}_{1}=2.5 \times 10^{5}\) \(\mathrm{PQ}+\frac{1}{2} \mathrm{R}_{2} \rightleftharpoons \mathrm{PQR} \mathrm{K}_{2}=5 \times 10^{-3}\) find \(K_{3}\) for the reaction \(\frac{1}{2} P_{2}+\frac{1}{2} Q_{2}+\frac{1}{2} R_{2} \rightleftharpoons P Q R\) (a) \(2.5 \times 10^{-3}\) (b) \(2.5 \times 10^{3}\) (c) \(1 \times 10^{-3}\) (d) \(5 \times 10^{-3}\)

Step-by-Step Solution

Answer

The equilibrium constant \(K_3\) is \(1 \times 10^{-3}\), which is option (c).

Step 1: Understand the Individual Reactions

Analyze the given reactions:\[ 2 \mathrm{PQ} \rightleftharpoons \mathrm{P}_2 + \mathrm{Q}_2 \quad \text{with } K_1 = 2.5 \times 10^5 \]\[ \mathrm{PQ} + \frac{1}{2} \mathrm{R}_2 \rightleftharpoons \mathrm{PQR} \quad \text{with } K_2 = 5 \times 10^{-3} \] These are equilibrium reactions with their respective equilibrium constants \(K_1\) and \(K_2\).

Step 2: Calculate K for Intermediate Reaction

First, consider the reverse of the first reaction: \(\mathrm{P}_2 + \mathrm{Q}_2 \rightleftharpoons 2 \mathrm{PQ}\). For this reverse reaction, the equilibrium constant \(K'_1 = \frac{1}{K_1} = \frac{1}{2.5 \times 10^5}\).

Step 3: Manipulate the Reactions

Now, we need to transform the reversed reaction so that it combines with the second reaction to yield the desired equation. Halve the reversed first reaction: \[ \frac{1}{2} \mathrm{P}_2 + \frac{1}{2} \mathrm{Q}_2 \rightleftharpoons \mathrm{PQ} \]\[ K''_1 = \sqrt{K'_1} = \sqrt{\frac{1}{2.5 \times 10^5}} = \frac{1}{\sqrt{2.5 \times 10^5}} \]

Step 4: Combine Reactions to Obtain Target

Combine the halved reversed first reaction with the second reaction: \[ \frac{1}{2} \mathrm{P}_2 + \frac{1}{2} \mathrm{Q}_2 +\frac{1}{2} \mathrm{R}_2 \rightleftharpoons \mathrm{PQR} \] The equilibrium constant for this combined reaction \(K_3\), is the product of the constants for these two half-reactions: \[ K_3 = K''_1 \times K_2 = \frac{1}{\sqrt{2.5 \times 10^5}} \times 5 \times 10^{-3} \]

Step 5: Simplify the Mathematical Expression

Calculate the numeric value: \[ \sqrt{2.5 \times 10^5} = 500 \] Thus: \[ K''_1 = \frac{1}{500} \] And so: \[ K_3 = \frac{1}{500} \times 5 \times 10^{-3} = 1 \times 10^{-3} \]

Step 6: Match with Provided Options

The result \(1 \times 10^{-3}\) corresponds to option (c).

Key Concepts

Chemical Equilibrium

Chemical equilibrium describes a state in a reversible chemical reaction where the rates of the forward and reverse reactions are equal, resulting in no net change in the concentration of reactants and products. This doesn't mean that the reactants and products are present in equal amounts, rather that their concentrations stabilize at a constant ratio. Chemical reactions reach equilibrium because the forward reaction and its reverse eventually balance each other out. When this happens, the system is said to be at equilibrium.

Equilibrium can be influenced by several factors:

• Concentration: Changing the concentration of reactants or products can shift the equilibrium according to Le Chatelier's principle.
• Temperature: Typically, increasing the temperature favors the endothermic direction of a reaction and can shift the equilibrium position.
• Pressure: In reactions involving gases, changing the pressure can shift equilibrium by favoring the side with fewer moles of gas.

Understanding chemical equilibrium is essential for calculating equilibrium constants, which describe the ratio of concentrations at equilibrium and help predict the direction of a reaction under various conditions.

Equilibrium Reaction Manipulation

Manipulating equilibrium reactions involves adjusting the reactions to derive new reactions and their respective equilibrium constants. This often requires reversing reactions and changing their stoichiometric coefficients. These manipulations can significantly alter the value of the equilibrium constant, denoted as \(K\), and necessitate careful recalculations.

Let's explore each manipulation technique:

• Reversing Reactions: When a reaction is reversed, the equilibrium constant for the reverse reaction \(K'\) is the reciprocal of that for the original reaction, such that \(K' = \frac{1}{K}\).
• Halving or Doubling Reactions: If a reaction is scaled by halving all coefficients, the equilibrium constant for the new reaction \(K''\) is the square root of the original constant \((K'' = \sqrt{K})\). If coefficients are doubled, \(K''\) is the square of the original constant \((K'' = K^2)\).
• Combining Reactions: When combining two reactions to form a new reaction, the resulting equilibrium constant is the product of the constants for the individual reactions (\(K_{combined} = K_1 \times K_2\)).

These manipulations are crucial for solving complex equilibrium problems and finding the equilibrium constants of new, derived reactions.

Equilibrium Constant Calculation

Calculating equilibrium constants involves using known constants from given reactions and manipulating the reactions to derive the desired target reaction. An equilibrium constant, \(K\), quantifies the ratio of concentrations of products to reactants at equilibrium at a given temperature. It is a crucial parameter that helps determine the extent of the reaction and predicts how changes in conditions can shift the reaction.

Here’s how to calculate an equilibrium constant:1. **Identify Known Reactions:** Start with the given reactions and their \(K\) values, such as \(K_1\) and \(K_2\).2. **Manipulate Reactions:** Reverse, halve, or combine reactions to transform the known reactions into the desired target reaction. Each manipulation impacts the equilibrium constant.3. **Calculate the Equilibrium Constant for the New Reaction:** For the reaction transformations, find the equilibrium constant for the resulting equation. In this case study: - Reversal: \(K'_1 = \frac{1}{K_1}\) - Halving: \(K''_1 = \sqrt{K'_1}\) - Final Combination: \(K_3 = K''_1 \times K_2\)4. **Simplify the Calculations:** Solve the numerical expression to find \(K_3\), which is the final equilibrium constant for the desired reaction.

This systematic approach to equilibrium constant calculation enables accurate predictions and conclusions about a reaction's behavior at equilibrium.