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

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

Step 1: Understand the Reaction

The chemical equation given is \(3\, \mathrm{A} \rightarrow 2\, \mathrm{B}\). This means 3 moles of \(\mathrm{A}\) react to produce 2 moles of \(\mathrm{B}\). The change in concentration for \(\mathrm{A}\) and \(\mathrm{B}\) over time is related by their stoichiometry.

Step 2: Relate Rates of Product and Reactant

The rate of disappearance of a reactant and the rate of appearance of a product are related through their stoichiometric coefficients. For the reaction \(3\, \mathrm{A} \rightarrow 2\, \mathrm{B}\), it relates to: \[-\frac{1}{3}\frac{\mathrm{d}[\mathrm{A}]}{\mathrm{dt}} = \frac{1}{2} \frac{\mathrm{d}[\mathrm{B}]}{\mathrm{dt}}\]

Step 3: Solve for the Rate of Reaction of B

We need to express the rate of reaction for \(\mathrm{B}\), \(+\frac{\mathrm{d}[\mathrm{B}]}{\mathrm{dt}}\), in terms of \(-\frac{\mathrm{d}[\mathrm{A}]}{\mathrm{dt}}\). Rearranging the expression from Step 2 gives: \[\frac{\mathrm{d}[\mathrm{B}]}{\mathrm{dt}} = -\frac{2}{3} \frac{\mathrm{d}[\mathrm{A}]}{\mathrm{dt}}\]

Step 4: Match with Given Options

From the calculation, the correct relationship is \(+\frac{\mathrm{d}[\mathrm{B}]}{\mathrm{dt}} = -\frac{2}{3} \frac{\mathrm{d}[\mathrm{A}]}{\mathrm{dt}}\). This matches option (b).

Key Concepts

Stoichiometry

Stoichiometry is an integral concept in chemistry that deals with measuring and calculating the quantities of reactants and products in chemical reactions. Imagine it as a recipe in cooking—where ingredients need to be mixed in precise amounts to achieve the desired result. In chemical reactions, the amounts of reactants and products are related by their coefficients in the balanced chemical equation. The equation, in our exercise, is: \[3 \, \mathrm{A} \longrightarrow 2 \, \mathrm{B}\]This tells us that 3 molecules of substance A react to form 2 molecules of substance B. The coefficients (3 for \(\mathrm{A}\) and 2 for \(\mathrm{B}\)) are crucial as they show the exact proportions needed for the reaction to occur without leftovers. They give rise to molar relationships that form the basis of calculating how much of each substance is involved at any given time. This relationship is key in understanding how the rate of reaction is influenced by the stoichiometry of a reaction.

Reaction Rates

The reaction rate is a measure of how quickly a chemical reaction proceeds. It tells us the speed at which reactants are consumed or products are formed. Understanding reaction rates is essential in fields ranging from industrial manufacturing to environmental science. For the reaction given:\[3 \, \mathrm{A} \longrightarrow 2 \, \mathrm{B}\]The rate can be viewed from the perspective of either the disappearance of reactants or the appearance of products. The rate is typically expressed as a change in concentration over time, often denoted by \(\frac{\mathrm{d}[\text{concentration}]}{\mathrm{dt}}\). This differential notation emphasizes the dynamic nature of chemical reactions as they progress over time. In our example, the reaction rate for \(\mathrm{A}\) disappearing is not the same numerically as the rate of \(\mathrm{B}\) appearing because they have different stoichiometric coefficients. Hence, knowing the relationships between these rates is critical for predicting how fast a product can be formed or how fast a reactant will deplete.

Rate of Reaction Calculation

Calculating the rate of reaction involves connecting the rates of disappearance of reactants and appearance of products using their stoichiometric coefficients. This connection is mathematical and relies on understanding the proportional relationships dictated by the balanced equation.Consider:\[-\frac{1}{3}\frac{\mathrm{d}[\mathrm{A}]}{\mathrm{dt}} = \frac{1}{2} \frac{\mathrm{d}[\mathrm{B}]}{\mathrm{dt}}\]This equation states that for every three parts of \(\mathrm{A}\) consumed, two parts of \(\mathrm{B}\) appear. To find the rate of \(\mathrm{B}\)'s appearance \(+\frac{\mathrm{d}[\mathrm{B}]}{\mathrm{dt}}\), in terms of \(-\frac{\mathrm{d}[\mathrm{A}]}{\mathrm{dt}}\), rearrange the terms:\[\frac{\mathrm{d}[\mathrm{B}]}{\mathrm{dt}} = -\frac{2}{3} \frac{\mathrm{d}[\mathrm{A}]}{\mathrm{dt}}\]This solution indicates that for each unit increase in \(\mathrm{B}\), \(\mathrm{A}\) decreases by \(\frac{2}{3}\) units, showcasing a beautifully compact relationship between the reactant and product via stoichiometric analysis. Recognizing this relationship not only answers the question posed but enhances comprehension of reaction dynamics.