Problem 28 - A Complete Resource Book in Chemistry for JEE Main

Question

For a second-order reaction, \(2 \mathrm{~A} \longrightarrow\) Product, a straight line is obtained if we plot (a) concentration vs time (b) log (conc.) vs time (c) log (conc.) vs time \(^{1}\) (d) (concc')vs time"

Step-by-Step Solution

Answer

For a second-order reaction, plot \( \frac{1}{[A]} \) versus time to get a straight line (option d).

Step 1: Identify Reaction Order

For the given reaction \(2 \text{A} \rightarrow \text{Product}\), it's specified to be a second-order reaction. Second-order reactions have the general form \(A + A \rightarrow \ ext{Products}\).

Step 2: Understand Rate Law

The rate law for a second-order reaction is generally given by \( \text{Rate} = k[A]^2\), where \(k\) is the rate constant and \([A]\) is the concentration of \(A\).

Step 3: Derive Integrated Rate Law

For a second-order reaction, the integrated rate law can be expressed as \( \frac{1}{[A]} = kt + \frac{1}{[A]_0}\), where \([A]_0\) is the initial concentration of \(A\).

Step 4: Determine Linear Plot

The integrated rate law \( \frac{1}{[A]} = kt + \frac{1}{[A]_0} \) suggests that plotting \( \frac{1}{[A]} \) versus time \(t\) will yield a straight line for a second-order reaction. This corresponds to option (d) if \(\text{(conc')} = \frac{1}{[A]}\).

Step 5: Identify the Correct Plot

Option (d), where \((\text{conc'}) \) represents \( \frac{1}{[A]} \), matches the need for a straight-line plot for second-order reactions when plotted against time.

Key Concepts

Rate law

In the context of chemical kinetics, the rate law is a mathematical expression that describes the rate of a reaction in terms of the concentration of reactants. For a second-order reaction like the one given, where two molecules of the same species react to form products – represented as \(2 ext{A} \rightarrow \text{Products}\) – the rate law has a particular significance. It is formulated as \( ext{Rate} = k[A]^2 \).
This equation indicates that the reaction rate depends quadratically on the concentration \([A]\) of the reactant. As a result, doubling the concentration of \(A\) quadruples the rate of reaction. This distinct feature of second-order reactions allows for greater insight into how molecular interactions influence reaction speed and progression. The rate constant \(k\) is particularly crucial here, typically influenced by temperature and solvent conditions in which the reaction takes place.

Integrated rate law

The integrated rate law for a second-order reaction provides a relationship between concentration and time as the reaction progresses. For second-order reactions, the integrated rate law is expressed as:\[\frac{1}{[A]} = kt + \frac{1}{[A]_0}\]Here, \([A]_0\) represents the initial concentration of reactant \(A\), and \([A]\) is the concentration of \(A\) at any time \(t\).

• The formula is derived from the basic rate law by integrating over time, providing a clear way to predict reactant concentrations at different time intervals.
• The slope of the plot from this equation reveals the rate constant \(k\), enabling further analysis of the reaction's kinetics.

The integrated rate law is especially useful for its ability to describe concentration changes over time, allowing chemists to infer about reaction dynamics even when experimental setups don't allow for continuous monitoring of concentrations.

Reaction kinetics

Reaction kinetics is a fundamental domain in chemistry that studies the rates at which chemical processes occur. This area provides understanding into factors that influence reaction rates, such as concentration, temperature, and catalysts. For any reaction type, including second-order reactions, kinetics involves determining:

• The rate law for the reaction, which mathematically describes the rate.
• The order of the reaction, which tells us how concentration affects reaction rate.
• Activation energy, which indicates the energy barrier that must be overcome for a reaction to occur.

In second-order reactions like \(2 ext{A} \rightarrow \text{Products}\), reaction kinetics help clarify how and why reaction speeds change with concentration. Understanding the nuances of these principles is crucial for manipulating reaction conditions in industrial processes or predicting experimental outcomes in laboratory settings.

Concentration vs time plot

The relationship between concentration and time is often depicted graphically to interpret the kinetics of a reaction. For second-order reactions, a key strategy for visual analysis is plotting \( \frac{1}{[A]} \) against time \(t\). The formula for the integrated rate law \(\frac{1}{[A]} = kt + \frac{1}{[A]_0}\) implies that this plot should yield a straight line.

• This linear plot indicates that the reaction follows second-order kinetics, where the line's slope equals the rate constant \(k\).
• From this graph, one can easily assess how the concentration of \(A\) decreases more rapidly over time as the reaction progresses.

Understanding the concentration vs. time plot is vital for chemists, as it allows for accurate determination of reaction order, as well as assessment of the consistency of experimental data with theoretical models, paving the way for precise rate constant calculations.