Question

The thermal decomposition of \(\mathrm{N}_{2} \mathrm{O}_{5}\) obeys firstorder kinetics. At \(45^{\circ} \mathrm{C}\), a plot of \(\ln \left[\mathrm{N}_{2} \mathrm{O}_{5}\right]\) versus \(t\) gives a slope of \(-6.18 \times 10^{-4} \mathrm{~min}^{-1}\). What is the half-life of the reaction?

Short answer

The half-life of the reaction at \( 45 ^{\circ} \mathrm{C} \) is 1121 minutes.

Step-by-step solution

Understanding Given Information

Even before starting to solve the task, it's necessary to understand the given information. The exercise mentions that the thermal decomposition of \(\mathrm{N}_{2} \mathrm{O}_{5}\) follows first-order kinetics. Therefore, the rate law of this reaction can be represented as \( r = k[\mathrm{N}_{2} \mathrm{O}_{5}]\), where \( k \) is the rate constant. The task also provides information that the slope of the graph \(\ln \left[\mathrm{N}_{2}\mathrm{O}_{5}\right]\) versus \( t \) is \(-6.18 \times 10^{-4} \mathrm{~min}^{-1}\). In terms of first-order kinetics, this slope is equivalent to \( -k \). Thus, the rate constant \( k \) is \( 6.18 \times 10^{-4} \mathrm{~min}^{-1}\).

Apply First-Order Kinetic Formula to Find Half-Life

The formula to find the half-life of a first-order reaction is \( t_{1/2} = \frac{0.693}{k} \). Substituting the value of \( k \) in this formula gives the half-life. Therefore, the half-life \( t_{1/2} \) of this reaction is \( \frac{0.693}{6.18 \times 10^{-4} \mathrm{~min}^{-1}} \).

Perform Calculation

By proceeding with calculation, we get \( t_{1/2} = 1121 \mathrm{~min} \). Therefore, at \( 45 ^{\circ} \mathrm{C} \), the half-life of the reaction is 1121 minutes.

Key concepts

Half-life of reaction

In chemistry, the half-life of a reaction is the time it takes for half of a reactant to be consumed in a chemical reaction. This concept is crucial in understanding how long a particular reaction may last or how quickly it progresses. The half-life is especially useful in the study of first-order reactions, which are reactions where the rate depends on the concentration of one reactant.

For first-order reactions, like the thermal decomposition of \(\mathrm{N}_{2} \mathrm{O}_{5}\), the half-life is constant and does not depend on the initial concentration of the reactant.
This is a unique property of first-order reactions. The formula to calculate the half-life \(t_{1/2}\) of a first-order reaction is given by:

• \(t_{1/2} = \frac{0.693}{k}\)

Here, \(k\) is the rate constant of the reaction.

Knowing the half-life can help predict how a reaction will behave over time, making it a fundamental aspect in reactor design and chemical engineering.

Rate constant

The rate constant, denoted as \(k\), is a crucial parameter in the study of chemical kinetics. It determines the speed at which a reaction proceeds and directly influences the half-life of a reaction. In first-order kinetics, the rate of reaction is directly proportional to the concentration of the reactant.

In the thermal decomposition of \(\mathrm{N}_{2} \mathrm{O}_{5}\), the relationship can be expressed as:

• \( r = k[\mathrm{N}_{2} \mathrm{O}_{5}] \)

Here, \(r\) is the rate of the reaction, and \([\mathrm{N}_{2} \mathrm{O}_{5}]\) is the concentration of \(\mathrm{N}_{2} \mathrm{O}_{5}\).

The slope of the plot \(\ln [\mathrm{N}_{2} \mathrm{O}_{5}] \) versus \(t\) provides us with the rate constant \(k\), which in this case is found to be \(6.18 \times 10^{-4} \mathrm{~min}^{-1}\). The negative sign in the slope indicates a decrease in concentration over time.

Understanding the rate constant is essential for predicting how fast a reaction will occur and for controlling the conditions to optimize the reaction's progress.

Thermal decomposition

Thermal decomposition is a type of reaction where a compound is broken down into simpler substances when heated. This process is crucial in many industrial applications, such as producing energy or synthesizing new chemicals.

In the context of \(\mathrm{N}_{2} \mathrm{O}_{5}\), thermal decomposition involves the compound breaking down into \(\mathrm{NO}_2\) and \(\mathrm{O}_2\) when exposed to a certain temperature, in this case, \(45^{\circ} \mathrm{C}\).

During thermal decomposition, the reaction follows first-order kinetics, making it predictable and calculable with the relationships provided by its kinetic parameters, such as the rate constant and the half-life.

This predictability is advantageous for both laboratory studies and industrial processes, as it enables chemists and engineers to plan and execute reactions under controlled conditions, optimizing product yield and maintaining safety standards.

• Temperature: Increases in temperature generally increase the decomposition rate.
• Reaction Order: Follows first-order kinetics, simplifying the mathematical modeling of the reaction.

Overall, understanding the principles of thermal decomposition allows for better design and execution of chemical reactions that are temperature-sensitive.