Problem 55 - A Complete Resource Book in Chemistry for JEE Main

Question

In a chemical equilibrium rate constant of forward reaction is \(7.5 \times 10^{-4}\) and the equilibrium constant is 1.5. The rate constant of backward reaction is (a) \(2.5 \times 10^{4}\) (b) \(5 \times 10^{-4}\) (c) \(2.5 \times 10^{-4}\) (d) \(5 \times 10^{4}\)

Step-by-Step Solution

Answer

The rate constant of the backward reaction is \(5 \times 10^{-4}\).

Step 1: Understand the Relationship Between Constants

The equilibrium constant \(K\) for a reaction is related to the forward rate constant \(k_f\) and the backward rate constant \(k_b\) by the formula \(K = \frac{k_f}{k_b}\). We are provided \(K = 1.5\) and \(k_f = 7.5 \times 10^{-4}\).

Step 2: Rearrange to Find the Backward Rate Constant

Rearrange the equation to solve for \(k_b\): \(k_b = \frac{k_f}{K}\). This means we plug in the given values to calculate \(k_b\).

Step 3: Plug in the Values and Calculate

Substitute the given values into the equation: \(k_b = \frac{7.5 \times 10^{-4}}{1.5}\). Perform the division to find \(k_b\).

Step 4: Simplify and Find the Correct Answer

Calculate \(\frac{7.5 \times 10^{-4}}{1.5} = 5 \times 10^{-4}\). This matches option (b).

Key Concepts

Equilibrium Constant

In chemistry, the equilibrium constant, denoted as \( K \), plays a crucial role in characterizing the balance between the forward and backward reactions in a chemical system. It provides a snapshot of the concentrations of reactants and products at equilibrium. The value of \( K \) is specific for every reaction and depends only on temperature, making it a powerful tool for understanding reaction dynamics.

• An equilibrium constant greater than 1 indicates that the products are favored at equilibrium.
• An equilibrium constant less than 1 indicates that the reactants are favored.
• A greater \( K \) value denotes a reaction that proceeds vigorously in the forward direction.

When we discuss the equilibrium constant in quantitative terms, we're essentially determining how far a reaction will proceed under a given set of conditions before reaching equilibrium.

Forward Rate Constant

The forward rate constant, symbolized as \( k_f \), is an essential factor that specifies the speed of the reaction proceeding from reactants to products. It provides a rate at which the chemical reaction approaches completion in the forward direction.

• A higher \( k_f \) means the forward reaction happens quickly, whereas a lower \( k_f \) suggests a slower forward reaction.
• Factors like temperature, pressure, and catalyst presence can significantly influence \( k_f \).

For the given exercise, the forward rate constant is \( 7.5 \times 10^{-4} \), which gives you an idea of how reactive the setup is under specific conditions. Understanding \( k_f \) helps in controlling and predicting the behavior of chemical reactions in various industries and natural processes.

Backward Rate Constant

The backward rate constant, or \( k_b \), is critical in determining how fast the reverse reaction occurs in a chemical system. It measures the rate at which products revert back to reactants, balancing the forward progression.

• A larger \( k_b \) value signifies a faster reverse reaction, while a smaller one indicates a slower reverse reaction.
• Physical attributes such as temperature can alter \( k_b \), similar to how \( k_f \) is affected.

In the context of the exercise, we find \( k_b \) by rearranging the equilibrium condition: \( k_b = \frac{k_f}{K} \). By placing this in perspective, the backward rate constant advances our comprehension of a reaction's dynamics and equilibrium.

Reaction Rate Equation

The reaction rate equation is central to understanding how rapidly reactions occur over time. It builds on the principles described by both the forward and backward rate constants, as it relates the concentrations of reactants and the rate at which they transform.

• A basic rate equation can be represented as \( \ ext{rate} = k_f [Reactants] \), where the rate is defined by the forward reaction constant and reactant concentration.
• Balanced equations are key to establishing correct proportional relationships in the rate equation.

By linking the rate equation to equilibrium, one can predict how shifting concentrations or conditions affect the reaction's course. Understanding this concept is pivotal for engineers and scientists to optimize reactions for industries, ranging from pharmaceuticals to energy production.