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 [2008] (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 (a) is correct: \(-\frac{d[A]}{dt}=\frac{1}{2}\frac{d[B]}{dt}\).

Step 1: Understanding the Reaction

We are given a chemical reaction: \(\frac{1}{2} A \longrightarrow 2 B\). This indicates that half a mole of substance \(A\) produces 2 moles of substance \(B\). We aim to relate the rate of disappearance of \(A\) with the rate of appearance of \(B\).

Step 2: Writing the Rate Expressions

The rate of disappearance of \(A\) can be expressed as \(-\frac{1}{2} \frac{d[A]}{dt}\) because the coefficient of \(A\) is \(\frac{1}{2}\). The rate of appearance of \(B\) is expressed as \(\frac{d[B]}{dt}\) since 2 moles of \(B\) are produced.

Step 3: Setting Up the Relation

The relationship between the rate of disappearance of \(A\) and the rate of appearance of \(B\) based on stoichiometry is given by equating the rates: \(-\frac{1}{2} \frac{d[A]}{dt} = \frac{1}{2} \times \frac{d[B]}{dt}\). To find an expression without fractions, we rearrange as: \(-\frac{d[A]}{dt} = 2 \frac{d[B]}{dt}\).

Step 4: Determining the Correct Expression

The derived expression \(-\frac{d[A]}{dt} = 2 \frac{d[B]}{dt}\) matches with option (a): \(-\frac{d[A]}{dt} = \frac{1}{2} \frac{d[B]}{dt}\). Thus, option (a) correctly relates the rate of disappearance of \(A\) to the rate of appearance of \(B\).

Key Concepts

Rate of Reaction

The rate of reaction is a measure of how fast or slow a chemical reaction occurs. It tells us how quickly the reactants are consumed or the products are formed. Rates can vary greatly, with some reactions happening in a fraction of a second while others take years. Understanding the rate of reaction is crucial because it can help us control reactions to optimize processes in industrial settings.

A reaction rate usually depends on several factors:

• The concentration of reactants: Higher concentrations generally lead to faster reactions.
• The temperature: Increasing temperature typically speeds up a reaction.
• The presence of a catalyst: Catalysts can lower the energy barrier, thus accelerating the rate.

The unit for rate of reaction is typically moles per liter per second (mol/L/s), and it is determined by changes over time in the concentration of either the reactants or the products.

Stoichiometry

Stoichiometry is the branch of chemistry that involves calculating the relationships between the quantities of reactants and products in chemical reactions. It is based on the conservation of mass and the idea that matter cannot be created or destroyed.

In the given equation, the stoichiometry is shown in the reaction: \(\frac{1}{2} A \rightarrow 2 B\). This equation reveals that half a mole of \(A\) is consumed to produce two moles of \(B\). The coefficients in a chemical equation guide us in determining how much of each substance is involved. This relationship helps in converting between the moles of different substances to ensure that reactants and products are in balance.

Understanding stoichiometry is essential for:

• Predicting the amounts of products formed in a reaction.
• Determining the quantities of reactants needed for a reaction to proceed.
• Scaling up reactions for industrial applications.

Rate of Disappearance

The rate of disappearance refers to the speed at which a reactant is consumed in a chemical reaction. In our example, we are looking at the rate of disappearance of \(A\).

Mathematically, it is expressed as \(-\frac{d[A]}{dt}\), where the negative sign indicates the concentration of \(A\) is decreasing over time. The rate of disappearance depends on both the reaction rate and the stoichiometry of the reactant. In this case, the stoichiometry is \(\frac{1}{2}\), which is used to calculate how much \(A\) decreases as products form.

To determine the rate of disappearance accurately, we often need to consider:

• The concentration of the reactants.
• The specific rate constant, which varies between reactions.

Understanding how quickly a reactant disappears helps chemists manipulate conditions to control the progress of a reaction.

Rate of Appearance

The rate of appearance measures how quickly a product forms in a chemical reaction. For the reaction \(\frac{1}{2} A \rightarrow 2 B\), the rate of appearance of \(B\) is of interest.

This is expressed mathematically as \(\frac{d[B]}{dt}\), showing the concentration of \(B\) increasing over time. Because the stoichiometry of product \(B\) is \(2\), it forms at twice the rate at which \(A\) is disappearing.

Understanding the rate of appearance helps professionals in:

• Calculating how fast a reaction produces products.
• Designing chemical processes to maximize yields in lesser time.
• Predicting whether reactions reach completion in given time frames.

Rate of appearance, in conjunction with the rate of disappearance, provides valuable insights into the dynamics of chemical reactions.