Problem 61 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

While studying the decomposition of gaseous \(\mathrm{N}_{2} \mathrm{O}_{5}\), it is observed that a plot of logarithm of its partial pressure versus time is linear. What kinetic parameters can be obtained from this observation? [1985-2 Marks|

Step-by-Step Solution

Answer

The rate constant \(k\) for the first-order reaction is obtained from the slope of the plot.

Step 1: Understanding the Given Information

The problem states that a plot of the logarithm of the partial pressure of gaseous \(\mathrm{N}_{2} \mathrm{O}_{5}\) versus time is linear. This suggests that the reaction follows first-order kinetics, where the concentration of reactants decreases exponentially over time.

Step 2: Recalling First-Order Reaction Characteristics

In a first-order reaction, the rate of reaction is proportional to the concentration of a single reactant. The mathematical expression for a first-order reaction is given by \(\ln [A] = -kt + \ln [A]_0\), where \([A]\) is the concentration of the reactant, \(k\) is the rate constant, and \([A]_0\) is the initial concentration.

Step 3: Identifying Kinetic Parameters from the Graph

From the linear graph of \(\ln [A]\) versus time \(t\), we can infer that the slope of the line is equal to \(-k\). Thus, the rate constant \(k\) can be determined from the slope of the line. The intercept corresponds to \(\ln [A]_0\), which provides information about the initial concentration, but this problem specifically asks for kinetic parameters.

Step 4: Conclusion on Kinetic Parameters

The kinetic parameter that can be obtained from this observation is the rate constant \(k\) for the decomposition of \(\mathrm{N}_{2} \mathrm{O}_{5}\), which can be calculated from the slope of the plot. This confirms that the reaction is first-order in nature.

Key Concepts

First-Order Reaction

In chemical kinetics, a first-order reaction is one where the rate depends linearly on the concentration of a single reactant. This means that as the concentration of the reactant decreases, the speed of the reaction decreases proportionally. The hallmark of a first-order reaction is that the decrease in concentration over time forms a straight line when you plot the natural logarithm of concentration against time.
This is expressed mathematically by the equation: \[\ln [A] = -kt + \ln [A]_0\]where

• \([A]\) is the concentration of the reactant at time \(t\)
• \([A]_0\) is the initial concentration of the reactant
• \(k\) is the rate constant

Understanding this principle helps identify that any linear plot of \(\ln [A]\) versus time indicates a first-order reaction.

Rate Constant

The rate constant, denoted as \(k\), is a crucial factor in determining the speed of a chemical reaction. For reactions following first-order kinetics, the rate constant can be directly extracted from the slope of the line on a plot of the natural logarithm of concentration versus time. This slope is equal to \[-k\]indicating that the larger the absolute value of \(k\), the faster the reaction proceeds.

• If \(k\) is large, the reactant concentration decreases rapidly.
• If \(k\) is small, the reactant concentration decreases slowly.

The units of the rate constant in a first-order reaction are \(s^{-1}\), indicating that it represents the fraction of concentration lost per second. Knowing \(k\) allows chemists to predict how quickly a reaction will proceed under given conditions.

Gaseous Decomposition

Gaseous decomposition is a type of reaction where a gaseous compound breaks down into simpler substances. In the context of the exercise, the gaseous decomposition of \(\mathrm{N}_{2} \mathrm{O}_{5}\) is observed. This type of reaction is particularly interesting because it often follows specific kinetic patterns, such as first-order kinetics.
In decomposing, \(\mathrm{N}_{2} \mathrm{O}_{5}\) breaks down into smaller molecules, often driven by conditions such as temperature and pressure. Understanding the kinetics of gaseous decomposition helps predict how fast these reactions occur, which is vital in many industrial and laboratory processes.

Kinetic Parameters

Kinetic parameters are the constants or coefficients that define the speed and behavior of a chemical reaction over time. In the study of reaction kinetics, key parameters include the rate constant \(k\) and the reaction order.

• The rate constant \(k\) indicates how quickly a reaction proceeds under given conditions.
• The reaction order provides insight into how reactant concentrations affect the reaction rate.

By analyzing the plot of \(\ln [A]\) versus time, chemists can learn not only about the order of the reaction but also determine the precise value of \(k\). This allows for more accurate modeling and prediction of the reaction's progress, making it a fundamental aspect of chemical kinetics. Understanding these parameters is crucial for optimizing reaction conditions in both research and industrial applications.