Problem 9 - A Complete Resource Book in Chemistry for JEE Main

Question

\(2 \mathrm{SO}_{2}(\mathrm{~g})+\mathrm{O}_{2}(\mathrm{~g}) \rightleftharpoons 2 \mathrm{SO}_{3}(\mathrm{~g})\) in the above reaction \(\mathrm{K}_{\mathrm{p}}\) and \(\mathrm{K}_{\mathrm{c}}\) are related as (a) \(\mathrm{K}_{\mathrm{p}}=\mathrm{K}_{\mathrm{c}} \times(\mathrm{RT})\) (b) \(\mathrm{K}_{\mathrm{p}}=\mathrm{K}_{\mathrm{c}} \times(\mathrm{RT})^{-1}\) (c) \(\mathrm{K}_{\mathrm{c}}=\mathrm{K}_{\mathrm{p}} \times(\mathrm{RT})^{2}\) (d) \(\mathrm{K}_{0}=\mathrm{K}_{\mathrm{c}} \times(\mathrm{RT})^{-2}\)

Step-by-Step Solution

Answer

The correct relation is (b) \(\mathrm{K}_{\mathrm{p}} = \mathrm{K}_{\mathrm{c}} \times(\mathrm{RT})^{-1}\).

Step 1: Understanding the Equation

The given chemical reaction is \(2 \mathrm{SO}_{2}(\mathrm{~g})+\mathrm{O}_{2}(\mathrm{~g}) \rightleftharpoons 2 \mathrm{SO}_{3}(\mathrm{~g})\). We need to find how \(\mathrm{K}_{\mathrm{p}}\) and \(\mathrm{K}_{\mathrm{c}}\) are related.

Step 2: Identify Change in Moles

Determine the change in the number of moles of gas in the reaction: \(\Delta n = n_{\text{products}} - n_{\text{reactants}}\). For this reaction, \(\Delta n = 2 - (2+1) = 2 - 3 = -1\).

Step 3: Review the Relationship Formula

The relationship between \(\mathrm{K}_{\mathrm{p}}\) and \(\mathrm{K}_{\mathrm{c}}\) is given by the formula: \(\mathrm{K}_{\mathrm{p}} = \mathrm{K}_{\mathrm{c}} ( \mathrm{RT})^{\Delta n}\).

Step 4: Substitute in the Delta n

Substitute \(\Delta n = -1\) into the formula to get: \(\mathrm{K}_{\mathrm{p}} = \mathrm{K}_{\mathrm{c}} ( \mathrm{RT})^{-1}\).

Step 5: Match with Given Options

Compare the derived equation \(\mathrm{K}_{\mathrm{p}} = \mathrm{K}_{\mathrm{c}} (\mathrm{RT})^{-1}\) with the options. This matches option (b).

Key Concepts

Chemical Equilibrium

In chemical reactions, chemical equilibrium refers to a state where the rates of the forward and reverse reactions are equal. This doesn't mean the concentrations of reactants and products have to be equal, but rather that their ratios remain constant over time.
When a reaction reaches equilibrium, no further changes in concentrations occur unless external conditions are altered. Here are some key points about chemical equilibrium:

• The system's composition becomes stable.
• It is dynamic, where molecules continuously react and unreact.
• Concentration changes invoke shifts dictated by Le Chatelier's Principle.

Understanding chemical equilibrium is crucial because it allows chemists to predict the outcome and efficiency of reactions.

Reaction Quotient

The reaction quotient, denoted as Q, helps to understand the progress of a reaction concerning equilibrium. It is calculated similarly to the equilibrium constant, but with current concentrations at any point in the reaction.

• Q is a snapshot of the reaction at a specific moment.
• If Q = K, the system is at equilibrium.
• If Q < K, the reaction proceeds forward to reach equilibrium.
• If Q > K, the reaction prefers the reverse direction to regain equilibrium.

By comparing Q to the equilibrium constant (K), chemists can predict which direction a reaction will shift to reach equilibrium, ensuring efficient adjustments in industrial and laboratory settings.

Equilibrium Expression

An equilibrium expression is a mathematical representation of a chemical reaction at equilibrium. It ties together the concentrations of the reactants and products through the equilibrium constant (K).
The equilibrium expression for a general reaction \( aA + bB \rightleftharpoons cC + dD \) is given by:\[K = \frac{[C]^c [D]^d}{[A]^a [B]^b}\]Here are key features of equilibrium expressions:

• The expression depends only on the stoichiometry of the balanced equation.
• K indicates how far the reaction proceeds; a large K means more products.
• Activity or concentration forms can be used, where \([C], [D], [A], [B]\) are molarities or partial pressures.

Equilibrium expressions are foundational in determining the extent and favorability of chemical reactions within various environments.