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 timel (d) (conce')vs time"

Short answer

Plot \( \frac{1}{[A]} \) vs. time for a straight line in a second-order reaction.

Step-by-step solution

Understand the Reaction Order

This is a second-order reaction described by the equation \(2 \text{A} \rightarrow \text{Product}\). In a second-order reaction, the rate is proportional to the square of the concentration of A; i.e., \( \text{rate} = k[A]^2\).

Identify the Integrated Rate Law

For a second-order reaction, the integrated rate law in terms of concentration \([A]\) and time \(t\) is given by \( \frac{1}{[A]} = kt + \frac{1}{[A]_0} \), where \([A]_0\) is the initial concentration.

Determine the Plot for a Straight Line

With the integrated rate law \( \frac{1}{[A]} = kt + \frac{1}{[A]_0} \), if we plot \( \frac{1}{[A]} \) (i.e., the reciprocal of the concentration) versus time \( t \), we should obtain a straight line. Therefore, option (d), which implies a plot of \( \frac{1}{[A]} \) vs time, is the correct one.

Key concepts

Reaction Kinetics

Reaction kinetics is the study of rates of chemical reactions and the factors that affect these rates. When it comes to a second-order reaction like \( 2 \text{A} \rightarrow \text{Product} \), understanding kinetics helps us know how quickly products are formed. The rate of this reaction is proportional to the square of the concentration of A, represented mathematically as \( \text{rate} = k[A]^2 \). This means that as the concentration of A increases, the rate at which the product is formed increases even more quickly than in a first-order reaction.

• In a second-order reaction, the rate law expresses how the rate depends on the concentration of one or more reactants.
• The rate constant \( k \) is crucial here as it doesn't change with concentration and provides the relationship between concentration and rate.
• Changes in concentration due to different conditions can dramatically influence the rate in second-order reactions.

Integrated Rate Law

To find out how concentrations change over time in a reaction, we use the integrated rate law. For a second-order reaction, the integrated rate law has a specific form: \( \frac{1}{[A]} = kt + \frac{1}{[A]_0} \). This equation provides a relationship between the concentration of A and time \( t \). It shows us how concentration decreases as the reaction progresses over time.

• \([A]_0\) represents the initial concentration of A, indicating where the reaction starts.
• The integrated rate law is useful for predicting concentrations at any point in time during the reaction.
• This mathematical form is specific to second-order reactions and differs from zero or first-order reactions.

Using these concepts, chemists can understand how and why the concentration of reactants decreases as reactions progress.

Plotting Reaction Data

Plotting reaction data can visually illustrate how a reaction proceeds over time. For a second-order reaction, the integrated rate law guides us on how to correctly plot the data for useful insights. According to the given equation \( \frac{1}{[A]} = kt + \frac{1}{[A]_0} \), plotting \( \frac{1}{[A]} \) against time \( t \) results in a straight line. This plot is crucial as it confirms the second-order nature of the reaction.

• A straight line trend confirms that your reaction follows second-order kinetics.
• The slope of this line is equal to the rate constant \( k \), providing valuable information about the reaction's speed.
• Correctly plotting your data helps in predicting how changes in conditions might affect future experiments or processes.

In essence, plotting helps bridge the gap between theoretical equations and practical observation, enabling a deeper understanding of reaction kinetics.