Problem 26 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

Consider a reaction \(a G+b H \rightarrow\) Products. When concentration of both the reactants \(G\) and \(H\) is doubled, the rate increases by eight times. However, when concentration of \(G\) is doubled keeping the concentration of \(H\) fixed, the rate is doubled. The overall order of the reaction is [2007] (a) 0 (b) 1 (c) 2 (d) 3

Step-by-Step Solution

Answer

The overall order of the reaction is 3.

Step 1: Formulate initial rate equation

For the given reaction \( aG + bH \rightarrow \text{Products} \), we can express the rate of reaction using the rate law: \( \text{Rate} = k[G]^x[H]^y \), where \( k \) is the rate constant, \( [G] \) and \( [H] \) are the concentrations of the reactants, and \( x \) and \( y \) are the orders with respect to \( G \) and \( H \) respectively.

Step 2: Analyze effect of changing both reactants

When the concentrations of both \( G \) and \( H \) are doubled, the rate increases by eight times. Mathematically, this can be expressed as:\[8 \times \text{Rate}_{\text{initial}} = k(2[G])^x(2[H])^y = k \cdot 2^x \cdot 2^y \cdot [G]^x \cdot [H]^y\]Simplifying, \( 8 = 2^{x+y} \). Thus, \( x + y = 3 \).

Step 3: Analyze effect of changing concentration of G only

When only the concentration of \( G \) is doubled, the rate is doubled. This can be written as:\[2 \times \text{Rate}_{\text{initial}} = k(2[G])^x[H]^y = k \cdot 2^x \cdot [G]^x \cdot [H]^y\]This implies \( 2 = 2^x \), leading to \( x = 1 \).

Step 4: Determine order of reaction with respect to H

Substituting \( x = 1 \) into the equation \( x + y = 3 \), we can solve for \( y \):\[1 + y = 3 \Rightarrow y = 2\]This shows the reaction is first order with respect to \( G \) and second order with respect to \( H \).

Step 5: Calculate overall order of the reaction

The overall order of the reaction is the sum of the orders with respect to each reactant. Therefore, the overall order is:\[x + y = 1 + 2 = 3\]

Key Concepts

Rate Law

The rate law is a mathematical expression that describes the speed of a chemical reaction with respect to the concentration of the reactants. It is central to determining how changes in concentration affect the rate of reaction. For a given reaction, the rate law can be written as

• \[\text{Rate} = k[G]^x[H]^y\]

here,

• \(k\) is the rate constant.
• \([G]\) and \([H]\) represent the concentrations of the reactants.
• \(x\) and \(y\) are the reaction orders.

The reaction orders \(x\) and \(y\) indicate how sensitive the rate is to changes in concentrations of \([G]\) and \([H]\).
For example, if a concentration doubles and the rate doubles too, it suggests a first-order dependence.
Likewise, if the rate increases by a factor of four when the concentration doubles, this indicates a second-order dependence.

The overall order of the reaction is obtained from the sum of all individual orders. It's critical in understanding the kinetics behind the reactions and predicting how the system will respond in different scenarios.

Reaction Kinetics

Reaction kinetics involves studying the rates of chemical reactions and the factors that affect them. It explores how reactant concentrations, temperature, and catalysts can influence the reaction speed. In this exercise, we focus on concentration changes.
The reaction kinetics for the given reaction indicates that when both \([G]\) and \([H]\) are doubled, the rate increases by eight times.
Mathematically, this is expressed using the equation:\[\text{Rate} = k (2[G])^x (2[H])^y\]Simplifying, we find that the rate increases eightfold when both are doubled, implying that the sum of the reaction orders \(x + y\) is three. These observations help in the identification of the order of the reaction for each reactant.
Reaction kinetics studies the steps (or mechanisms) through which reactants convert into products, helping chemists design more efficient reactions and understand natural processes.

Chemical Reaction Mechanism

In chemistry, understanding the mechanism of a reaction involves identifying the sequence of steps that leads reactants to products. Each step may have its own rate and order, contributing to the overall reaction order.
Chemical reaction mechanisms reveal the series of elementary reactions which occur at the molecular level.
In this exercise, the reaction mechanism can be analyzed by determining the order of reaction with respect to each reactant.
For \(G\), doubling the concentration doubles the rate, indicating that the reaction is first order with respect to \(G\).
For \(H\), we deduce it to be a second-order reaction since 1 with 2 gives the total order of 3 when both reactions are combined.

Understanding the chemical reaction mechanism is crucial as it helps predict rates and outcomes under varying conditions, and optimize processes in industrial chemistry or even inside living organisms.