Problem 113 - A Complete Resource Book in Chemistry for JEE Main

Question

\(3 \mathrm{~A} \longrightarrow 2 \mathrm{~B}\), rate of reaction \(+\frac{\mathrm{d}[\mathrm{B}]}{\mathrm{dt}}\) is equal to (a) \(-\frac{3}{2} \frac{\mathrm{d}[\mathrm{A}]}{\mathrm{dt}}\) (b) \(-\frac{2}{3} \frac{\mathrm{d}[\mathrm{A}]}{\mathrm{dt}}\) (c) \(-\frac{1}{3} \frac{\mathrm{d}[\mathrm{A}]}{\mathrm{dt}}\) (d) \(+2 \frac{\mathrm{d}[\mathrm{A}]}{\mathrm{dt}}\)

Step-by-Step Solution

Answer

Option (b) \(-\frac{2}{3} \frac{\mathrm{d}[\mathrm{A}]}{\mathrm{dt}}\) is the correct answer.

Step 1: Understand the Reaction

The given reaction is \(3A \rightarrow 2B\). This indicates that 3 moles of \(A\) react to form 2 moles of \(B\). Understanding this stoichiometry is fundamental to solving the problem.

Step 2: Rate of Reaction Definition

The rate of reaction can be associated with the change in concentration of reactants or products over time. For the product \(B\), the rate can be expressed as \(+\frac{d[B]}{dt}\). For the reactant \(A\), it can be expressed as \(-\frac{d[A]}{dt}\).

Step 3: Relate Rates Using Stoichiometry

According to the balanced equation \(3A \rightarrow 2B\), the disappearance of \(A\) is related to the formation of \(B\) by the stoichiometric coefficients. We use the relation \(\frac{\Delta [A]}{3} = \frac{\Delta [B]}{2}\).

Step 4: Set Up the Equation Based on Stoichiometry

Using stoichiometry, the relationship between the rates of disappearance and appearance is \(-\frac{1}{3}\frac{d[A]}{dt} = \frac{1}{2}\frac{d[B]}{dt}\). We want \(+\frac{d[B]}{dt}\), so rearrange the terms: \(+\frac{d[B]}{dt} = -\frac{2}{3}\frac{d[A]}{dt}\).

Step 5: Choose the Correct Option

By comparing the equation obtained \(+\frac{d[B]}{dt} = -\frac{2}{3}\frac{d[A]}{dt}\) with the given options, we see that option (b) matches precisely. Therefore, the solution to the exercise is option (b).

Key Concepts

Rate of Reaction

The rate of reaction is a key concept in chemical kinetics, which describes how quickly a reactant is consumed or a product is formed in a chemical process. It's measured by the change in concentration of either a reactant or a product over a specific period of time. In our exercise, the rate at which product \( B \) forms can be denoted as \( +\frac{d[B]}{dt} \), representing the increase in concentration of \( B \) over time.

For reactant \( A \), which is being consumed, we express the rate as \( -\frac{d[A]}{dt} \), using a negative sign because the concentration of \( A \) decreases over time. This negative sign is important in keeping track of the direction in which the reaction is proceeding.

Overall, understanding the rate of reactions helps chemists predict how fast a reaction proceeds, which is crucial in both laboratory and industrial settings. The rates are influenced by factors such as temperature, concentration, and the presence of catalysts. By mastering these concepts, students can better understand how various conditions affect chemical processes.

Stoichiometry

Stoichiometry is a fundamental concept in chemistry that involves using balanced chemical equations to determine the relationship between reactants and products in a reaction. It provides a quantitative way to calculate how much of each substance is involved in a chemical reaction. In our exercise, the equation is \( 3A \rightarrow 2B \).

This tells us that 3 moles of \( A \) react to form 2 moles of \( B \). From this stoichiometric relationship, we can relate the rates of disappearance of \( A \) to the appearance of \( B \). Specifically, we have \( \frac{\Delta [A]}{3} = \frac{\Delta [B]}{2} \).

Stoichiometry is crucial because it allows chemists to predict the quantities of products formed given a certain amount of reactants. It also helps in converting laboratory-scale reactions to industrial-scale processes, ensuring efficiency and minimal waste. For students, a firm grasp of stoichiometry provides the tools needed to navigate many aspects of chemical reactions and calculations.

Reaction Mechanism

A reaction mechanism is a detailed step-by-step description of how a chemical reaction occurs at the molecular level. It explains the sequence of elementary steps or stages, the changes that individual molecules undergo, and the resulting transition states and intermediates formed.

In our example reaction \( 3A \rightarrow 2B \), the equation provides a simple overview of the reactants and products, but not of the actual pathway or mechanism that the reaction follows. The mechanism can include multiple steps, each with its rate laws and intermediates, which collectively contribute to the overall transformation from reactants to products.

Understanding reaction mechanisms is essential because it helps chemists develop better catalysts, optimize reaction conditions, and propose new reactions. It also aids in explaining why certain reactions occur faster than others or why some reactions yield unexpected products. For students, grasping the concept of reaction mechanisms can provide insight into the dynamic world of chemical reactions beyond just the initial and final states.