Problem 216 - A Complete Resource Book in Chemistry for JEE Main

Question

For a reaction \(1 / 2 \mathrm{~A} \longrightarrow 2 \mathrm{~B}\), rate of disappearance of ' \(\mathrm{A}\) ' is related to the rate of appearance of ' \(\mathrm{B}\) ' by the expression (a) \(-\frac{\mathrm{d}[\mathrm{A}]}{\mathrm{dt}}=\frac{1}{2} \frac{\mathrm{d}[\mathrm{B}]}{\mathrm{dt}}\) (b) \(-\frac{\mathrm{d}[\mathrm{A}]}{\mathrm{dt}}=\frac{1}{4} \frac{\mathrm{d}[\mathrm{B}]}{\mathrm{dt}}\) (c) \(-\frac{\mathrm{d}[\mathrm{A}]}{\mathrm{dt}}=\frac{\mathrm{d}[\mathrm{B}]}{\mathrm{dt}}\) (d) \(-\frac{\mathrm{d}[\mathrm{A}]}{\mathrm{dt}}=4 \frac{\mathrm{d}[\mathrm{B}]}{\mathrm{dt}}\)

Step-by-Step Solution

Answer

Option (b) is the correct relationship: \(-\frac{\mathrm{d}[\mathrm{A}]}{\mathrm{dt}} = \frac{1}{4} \frac{\mathrm{d}[\mathrm{B}]}{\mathrm{dt}}\).

Step 1: Understand the Reaction

For the reaction \(\frac{1}{2} \mathrm{A} \rightarrow 2 \mathrm{B}\), we have a stoichiometric relation indicating that 1 mole of \(\mathrm{A}\) produces 4 moles of \(\mathrm{B}\), since \(\frac{1}{2} \mathrm{A}\) implies that 2 moles of \(\mathrm{B}\) are formed from 1 mole of \(\mathrm{A}/2\).

Step 2: Write Rate Expression for A

The rate of disappearance of \(\mathrm{A}\) can be expressed as \(-\frac{\mathrm{d}[\mathrm{A}]}{\mathrm{dt}}\). Since \(\frac{1}{2}\) mole of A is used up each time when 1 reaction occurs, this implies that the rate of disappearance of \(\mathrm{A}\) needs to be halved compared to the standard stoichiometry.

Step 3: Write Rate Expression for B

For \(\mathrm{B}\), the rate of appearance is \(\frac{\mathrm{d}[\mathrm{B}]}{\mathrm{dt}}\). Here, 2 moles of \(\mathrm{B}\) appear when this reaction happens once.

Step 4: Connect the Rates

Since the stoichiometry is \(\frac{1}{2} \mathrm{A}\) to \(2\mathrm{B}\), we interrelate the rates. Thus, \(-\frac{\mathrm{d}[\mathrm{A}]}{\mathrm{dt}}\) is half \(\frac{\mathrm{d}[\mathrm{B}]}{\mathrm{dt}}\). Thus, the rate relation is \(-\frac{\mathrm{d}[\mathrm{A}]}{\mathrm{dt}} = \frac{1}{4} \frac{\mathrm{d}[\mathrm{B}]}{\mathrm{dt}}\).

Step 5: Choose the Correct Option

Matching the derived rate relation \(-\frac{\mathrm{d}[\mathrm{A}]}{\mathrm{dt}} = \frac{1}{4} \frac{\mathrm{d}[\mathrm{B}]}{\mathrm{dt}}\) with the given options, the correct choice is option (b).

Key Concepts

Stoichiometry in Chemical Reactions

Stoichiometry is a fundamental concept in chemistry that refers to the quantitative relationships between the reactants and products in a chemical reaction. When we look at a chemical equation like \( \frac{1}{2} \text{A} \rightarrow 2 \text{B} \), stoichiometry tells us how much of each reactant is consumed and how much product is formed. In this particular reaction, the stoichiometric coefficients show that half a mole of substance A is needed to produce 2 moles of substance B.

Understanding these molar ratios is crucial to predicting how different quantities of reactants will react. It helps chemists design reactions that use the ideal amount of reactants to minimize waste and maximize the desired product. Stoichiometry underpins many calculations in chemistry, guiding how much of each chemical is required or produced in a reaction.

In more complex reactions, stoichiometry involves balancing chemical equations and using these balanced equations to calculate unknown quantities, such as the concentration of a reactant after a certain time or the total yield of a product.

Understanding Rate of Disappearance

The rate of disappearance is a measure of how quickly a reactant is used up in a chemical reaction. For the reaction \( \frac{1}{2} \text{A} \rightarrow 2 \text{B} \), the rate of disappearance of \( \text{A} \) is expressed as \( -\frac{\mathrm{d}[\text{A}]}{\mathrm{dt}} \). The negative sign indicates that the concentration of \( \text{A} \) is decreasing over time.

This measure is crucial in chemical kinetics, as it helps to quantify the speed of a reaction. By adjusting conditions like temperature or concentration, scientists can control the rate of disappearance to optimize the reaction's efficiency. Understanding this concept allows chemists to predict how a reaction proceeds, helping them to develop better industrial processes or synthetic pathways that are more efficient.

In the given exercise, the stoichiometric relationship between \( \text{A} \) and \( \text{B} \) implies that for every half a mole of \( \text{A} \) that disappears, 2 moles of \( \text{B} \) appear, which affects how we express their rates mathematically.

Comprehending Rate of Appearance

The rate of appearance, on the other hand, tells us how rapidly a product forms in a chemical reaction. In our example \( \frac{1}{2} \text{A} \rightarrow 2 \text{B} \), the rate of appearance of \( \text{B} \) is expressed as \( \frac{\mathrm{d}[\text{B}]}{\mathrm{dt}} \).

This measure is vital for evaluating how a new substance forms over time. It is directly influenced by the rate of disappearance of the reactants and their stoichiometric coefficients from the balanced chemical equation. The rate at which \( \text{B} \) appears is influenced by how quickly its precursor, \( \text{A} \), disappears.

In practical scenarios, calculating the rate of appearance is important for determining how long it will take to produce a certain amount of product or to ensure safety in reactions that may release gas or heat. Understanding this rate enables chemists to design and control experiments precisely, ensuring the desired outcomes are achieved efficiently.

Chemical Kinetics and Reaction Rates

Chemical kinetics is the study of rates of chemical processes. It involves understanding how different factors affect the speed of a reaction, such as concentration, temperature, and presence of a catalyst. For the given reaction, the rate expressions for disappearance and appearance are integral to kinetic studies.

When dealing with kinetics, it's crucial to monitor how the concentration of reactants or products changes over time. The stoichiometry of \( \frac{1}{2} \text{A} \rightarrow 2 \text{B} \) provides a basis for these measurements, helping to establish rate expressions such as \( -\frac{\mathrm{d}[\text{A}]}{\mathrm{dt}} \) and \( \frac{\mathrm{d}[\text{B}]}{\mathrm{dt}} \).

Chemical kinetics also explores factors like activation energy and reaction mechanisms to help in predicting and controlling reactions' speed. By mastering kinetics, chemists can optimize reactions for industrial applications, ensure the safety of chemical processes, and contribute to the creation of new products and materials.