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

Question

The equilibrium constant for the following reaction will be \(3 \mathrm{~A}+2 \mathrm{~B}=\mathrm{C}\) (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 option is (d) \( \frac{[\text{C}]}{[\text{A}]^{3}[\text{B}]^{2}} \).

Step 1: Understanding Chemical Equilibrium

In a chemical equilibrium, the rate of the forward reaction equals the rate of the reverse reaction, and the concentrations of the reactants and products remain constant. The equilibrium constant, denoted as \( K \), expresses the ratio of product concentrations to reactant concentrations, each raised to the power of their coefficients in the balanced reaction.

Step 2: Equation for the Reaction

The equilibrium reaction given is \(3 \, \text{A} + 2 \, \text{B} = \text{C}\). In this reaction, A and B are the reactants, and C is the product. The coefficients next to the chemical species are 3 for A, 2 for B, and 1 for C.

Step 3: Writing the Expression for the Equilibrium Constant

The expression for the equilibrium constant \( K \) for the reaction \(3 \, \text{A} + 2 \, \text{B} = \text{C}\) is given by the formula: \[ K = \frac{[\text{C}]}{[\text{A}]^3[\text{B}]^2} \] Here, \([\text{C}]\), \([\text{A}]\), and \([\text{B}]\) denote the equilibrium concentrations of \(\text{C}\), \(\text{A}\), and \(\text{B}\), respectively.

Step 4: Matching the Correct Option

From the options provided: - (a) \( \frac{[3 \, \text{A}][2 \, \text{B}]}{[\text{C}]} \) is incorrect, as it does not match the equilibrium constant expression.- (b) \( \frac{[\text{C}]}{[3 \, \text{A}][2 \, \text{B}]} \) is incorrect due to misuse of stoichiometric coefficients.- (c) \( \frac{[\text{C}]}{[\text{A}]^{2}[\text{B}]^{2}} \) is incorrect because the exponent of \([\text{A}]\) should be 3.- (d) \( \frac{[\text{C}]}{[\text{A}]^{3}[\text{B}]^{2}} \) correctly matches the expression for the equilibrium constant.

Key Concepts

Equilibrium Constant

In chemistry, the concept of the equilibrium constant, denoted as \( K \), is fundamental for understanding how reactions behave when they reach a state of balance. At equilibrium, the forward and reverse reactions proceed at the same rate. This does not mean the reactions stop, but rather that the amounts of reactants and products remain stable over time.
The equilibrium constant is a special number that tells us the ratio of the concentrations of the products to the reactants, with each concentration raised to the power of its coefficient from the balanced chemical equation. It is a snapshot of the balance between reactants and products at a given temperature. This helps chemists predict how a reaction mixture will behave when slightly disturbed. If \( K \) is large, it means the products are favored at equilibrium. Conversely, a small \( K \) indicates the reactants are favored.
Remember, \( K \) is unitless. It provides valuable insight into the reactive dynamics and the stability of the chemical system.

Reaction Quotient

The reaction quotient, represented as \( Q \), is a crucial tool in chemistry for predicting the direction of a reaction. While the equilibrium constant \( K \) describes the system at equilibrium, \( Q \) can be calculated at any point in time to check how close a reaction is to reaching equilibrium. To calculate \( Q \), we use the same formula as for \( K \): the ratio of product concentrations to reactant concentrations, each raised to their balanced equation coefficients.

• If \( Q = K \), the reaction is at equilibrium.
• If \( Q < K \), the reaction will proceed forward, favoring product formation.
• If \( Q > K \), the reaction will proceed in reverse, favoring reactant formation.

By comparing \( Q \) to \( K \), chemists can determine how a reaction will shift to achieve equilibrium, making it a powerful concept in reaction analysis.

Stoichiometry

Stoichiometry is a cornerstone of chemistry, focusing on the quantitative relationships in chemical reactions. It helps predict the quantities of reactants needed or products formed. Understanding stoichiometry allows chemists to scale reactions efficiently in a controlled manner.
In the context of chemical equilibrium, stoichiometry becomes particularly important when calculating equilibrium expressions. The coefficients from a balanced chemical equation are crucial, as they define how many molecules of each component are involved in the reaction. These coefficients also determine how we raise each concentration in the expression for \( K \). For instance, in the reaction \(3 \, \text{A} + 2 \, \text{B} = \text{C}\), the coefficients 3, 2, and 1 are used to write the equilibrium constant as \( K = \frac{[\text{C}]}{[\text{A}]^3[\text{B}]^2} \).
Stoichiometry is essential for ensuring reactants are mixed in the right proportions, influencing the yield and rate of reactions.

Balanced Chemical Equation

A balanced chemical equation is indispensable in chemical reactions as it ensures that all atoms involved are accounted for both before and after a reaction takes place. It confirms the law of conservation of mass, stating that matter cannot be created or destroyed in a closed system.
For chemical equilibrium, a balanced equation provides the coefficients needed to set up the equilibrium expression. In the reaction \(3 \, \text{A} + 2 \, \text{B} = \text{C}\), balancing tells us we need 3 parts of \( \text{A} \) and 2 parts of \( \text{B} \) to make 1 part of \( \text{C} \). Thus, these coefficients become exponents in the calculation of \( K \) and \( Q \) under the equilibrium scenario.
This balanced equation is not just about numbers; it represents the actual interchange of reactants and products at the molecular level and guides chemists in understanding the stoichiometric relationships within the reaction.