Problem 60 - A Complete Resource Book in Chemistry for JEE Main

Question

The equilibrium constant for the following reaction will be \(3 \mathrm{~A}+2 \mathrm{~B} \rightleftharpoons \mathrm{C}\) (a) \(\frac{[3 \mathrm{~A}][2 \mathrm{~B}]}{[\mathrm{C}]}\) (b) \(\frac{[\mathrm{C}]}{[3 \mathrm{~A}][2 \mathrm{~B}]}\) (c) \(\frac{[\mathrm{C}]}{[\mathrm{A}]^{2}[\mathrm{~B}]^{2}}\) (d) \(\frac{[\mathrm{C}]}{[\mathrm{A}]^{3}[\mathrm{~B}]^{2}}\)

Step-by-Step Solution

Answer

The correct answer is (d).

Step 1: Understanding the Equilibrium Constant Expression

For the reaction \(3 \mathrm{A} + 2 \mathrm{B} \rightleftharpoons \mathrm{C}\), the equilibrium constant expression \(K\) is written as the concentration of products over the concentration of reactants, each raised to the power of their coefficients in the balanced equation. So, the expression is \(K = \frac{[\mathrm{C}]}{[\mathrm{A}]^3 [\mathrm{B}]^2}\).

Step 2: Evaluating Each Option

Now, let's compare each option with the correct expression, \(\frac{[\mathrm{C}]}{[\mathrm{A}]^3 [\mathrm{B}]^2}\). Option (a) is \(\frac{[3\mathrm{A}][2\mathrm{B}]}{[\mathrm{C}]}\), Option (b) is \(\frac{[\mathrm{C}]}{[3\mathrm{A}][2\mathrm{B}]}\), Option (c) is \(\frac{[\mathrm{C}]}{[\mathrm{A}]^2[\mathrm{B}]^2}\), and Option (d) is \(\frac{[\mathrm{C}]}{[\mathrm{A}]^3[\mathrm{B}]^2}\).

Step 3: Selecting the Correct Option

Among the options, Option (d) matches the correct equilibrium constant expression \(\frac{[\mathrm{C}]}{[\mathrm{A}]^3[\mathrm{B}]^2}\). Therefore, the correct answer is Option (d).

Key Concepts

Chemical Equilibrium

Chemical equilibrium is a state in a chemical reaction where the concentrations of reactants and products remain constant over time. This means that the rate at which the reactants are converted to products is equal to the rate at which products are converted back into reactants. It's important to remember that equilibrium doesn't mean the amounts of reactants and products are equal, just that their concentrations don't change.
This balance can be understood through the concept of the dynamic equilibrium. Even at equilibrium, both the forward and backward reactions are still occurring, but at equal rates. This happens in reversible reactions, which are represented by the double arrow symbol (\( \rightleftharpoons \)) in equations.
Recognizing chemical equilibrium is crucial as it tells us that no matter how much we continue a reaction, the system won't produce any more products beyond this equilibrium point without changing the conditions, like temperature or pressure.

Reaction Coefficients

Reaction coefficients are the numbers that appear before the molecules in a balanced chemical equation. They are essential because they tell us the ratio in which reactants are used up and products are formed during the reaction.
These coefficients also play a critical role in calculating the equilibrium constant as they become the exponents in the equilibrium constant expression. For the given reaction \(3\mathrm{A} + 2\mathrm{B} \rightleftharpoons \mathrm{C}\), the coefficients are 3 for A, 2 for B, and 1 for C. This means for every 3 molecules of A and 2 molecules of B reacting, one molecule of C is produced.
In equilibrium expressions and calculations, the concentrations of reactants and products are raised to the power of these coefficients. This ensures the expression accurately reflects the reaction's stoichiometry, which is the quantitative relationship of reactants and products in a chemical reaction.

Equilibrium Expressions

Equilibrium expressions are mathematical representations of the ratio of products to reactants in a reversible reaction at equilibrium. For any reaction at equilibrium, the equilibrium constant \(K\) quantifies the concentrations of products and reactants, each raised to the power of their respective reaction coefficients.
The general form of this expression is \( K = \frac{[\text{products}]}{[\text{reactants}]} \), where the concentrations of products and reactants are expressed in molarity (moles per liter). Using the initial reaction \(3 \mathrm{A} + 2 \mathrm{B} \rightleftharpoons \mathrm{C}\), the correct equilibrium expression is \( K = \frac{[C]}{[A]^3 [B]^2} \).
This equilibrium constant tells us about the mixture at equilibrium. A larger \(K\) value means that the equilibrium position is more towards the products, whereas a smaller \(K\) indicates a position towards the reactants. Equilibrium expressions are fundamental in predicting the outcome of reactions and understanding how different factors like pressure and concentration changes affect the equilibrium state.