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Chapter 13: Problem 93

If the reaction \(A \rightarrow B\) is first order in \(A\) and first order overall, does it occur in just one step?

Short Answer

Expert verified
#Answer# Yes, the reaction \(A \rightarrow B\) is a single-step reaction, as it is first order in \(A\) and first order overall with no intermediate species involved.

Step by step solution

01

Identify the given reaction order

The reaction \(A \rightarrow B\) is given as first order in \(A\) and first order overall. This means that the rate of the reaction depends solely on the concentration of \(A\).
02

Analyze the reaction

The reaction is a simple conversion of a reactant \(A\) into a product \(B\). There are no intermediate steps or species involved in the reaction. Therefore, the rate law for this reaction can be written as: \[Rate = k [A]\]
03

Determine if the reaction occurs in one step

Since the reaction's rate law is dependent only on the concentration of \(A\), and there are no intermediate species, we can conclude that the reaction occurs in one step. This is because there is a direct relationship between the reactant \(A\) and the product \(B\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Reaction Order
Understanding the concept of reaction order is vital in chemical kinetics. Reaction order refers to the power to which the concentration of a reactant is raised in the rate law equation. This reflects how changes in concentration impact the rate of the reaction. For example, if the reaction is first order with respect to a particular reactant, it means that the rate of reaction changes linearly with changes in the concentration of that reactant. In a first order reaction like the conversion of A to B, doubling the concentration of A doubles the rate of reaction as well.
This concept helps chemists predict how a reaction's rate will change with varying concentrations, which is essential for controlling reaction speeds in industrial processes and understanding reaction mechanisms.
Rate Law
The rate law of a chemical reaction is an equation that links the rate of reaction to the concentration of the reactants. For a simple reaction such as the conversion of A to B, the rate law is expressed as \(Rate = k [A]\). Here, \(k\) is the rate constant, a proportional factor that remains constant at a given temperature.
  • The rate law provides part of the picture needed to understand how quickly a reaction proceeds under specific conditions.
  • It offers clarity on which reactants influence the rate, helping to deduce the mechanism of the reaction.
The simplicity of the rate law in first order reactions suggests that the process occurs in a single step, without intermediate stages, creating a direct transformation from A to B.
First Order Reaction
A first order reaction involves reactants in which the reaction rate is directly proportional to the concentration of only one reactant. This is typified by the reaction \(A \rightarrow B\), where the rate equation \(Rate = k [A]\) indicates a dependence solely on the concentration of A.
The characteristic behavior of first order reactions is evident from their reaction kinetics:
  • The half-life of a first order reaction is constant, meaning it does not change with the concentration of reactants.
  • The rate of reaction decreases over time as the concentration of A decreases, following an exponential decay pattern.

These features are pivotal in predicting how the reaction progresses over time, making first order reactions a fundamental concept in chemical kinetic studies. Understanding such reactions helps in fields such as pharmacology, where drug dosages relate directly to how first order processes eliminate substances from the body.

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Most popular questions from this chapter

Why are the units of the rate constants different for reactions of different order?

Nitrous acid slowly decomposes to \(\mathrm{NO}, \mathrm{NO}_{2},\) and water in the following second-order reaction: $$ 2 \mathrm{HNO}_{2}(a q) \rightarrow \mathrm{NO}(g)+\mathrm{NO}_{2}(g)+\mathrm{H}_{2} \mathrm{O}(\ell) $$ a. Use the data in the table to determine the rate constant for this reaction at \(298 \mathrm{K}\) $$\begin{array}{cc} \text { Time (min) } & {\left[\mathrm{HNO}_{2}\right](\mu M)} \\ 0.0 & 0.1560 \\ \hline 1000.0 & 0.1466 \\ \hline 1500.0 & 0.1424 \\ \hline 2000.0 & 0.1383 \\ \hline 2500.0 & 0.1345 \\ \hline 3000.0 & 0.1309 \\ \hline \end{array}$$ b. Determine the half-life for the decomposition of \(\mathrm{HNO}_{2}.\)

The rate laws for the thermal and photochemical decomposition of \(\mathrm{NO}_{2}\) are different. Which of the following mechanisms are possible for the photochemical decomposition of \(\mathrm{NO}_{2}\) given that the rate \(=k\left[\mathrm{NO}_{2}\right] ?\) a. \(\mathrm{NO}_{2}(g)+\mathrm{NO}_{2}(g) \stackrel{\text { thaw }}{\longrightarrow} \mathrm{N}_{2} \mathrm{O}_{4}(g)\) \(\begin{aligned} \mathrm{N}_{2} \mathrm{O}_{4}(g) & \stackrel{\text { faits }}{\mathrm{cm}+} \mathrm{N}_{2} \mathrm{O}_{3}(g)+\mathrm{O}(g) \\\ \mathrm{N}_{2} \mathrm{O}_{3}(g)+\mathrm{O}(g) & \mathrm{N}_{2} \mathrm{O}_{2}(g)+\mathrm{O}_{2}(g) \\ \mathrm{N}_{2} \mathrm{O}_{2}(g) &\stackrel{(\mathrm{atut}}{\longrightarrow}) \mathrm{NO}(g) \end{aligned}\) b. \(\mathrm{NO}_{2}(g)+\mathrm{NO}_{2}(g) \stackrel{\text { that }}{\longrightarrow} \mathrm{NO}(g)+\mathrm{NO}_{3}(g)\) \(\mathrm{NO}_{3}(g) \stackrel{\text { fast }}{\longrightarrow} \mathrm{NO}(g)+\mathrm{O}_{2}(g)\) c. \(\mathrm{NO}_{2}(g) \stackrel{\text { slow. }}{\longrightarrow} \mathrm{N}(g)+\mathrm{O}_{2}(g)\) \(\begin{aligned} \mathrm{N}(g)+& \mathrm{NO}_{2}(g) \stackrel{\text { fist. }}{\mathrm{N}_{2} \mathrm{O}_{2}(g)} & \mathrm{N}_{2} \mathrm{O}_{2}(g) \\ & \stackrel{\pm m}{\longrightarrow} \mathrm{NO}(g) \end{aligned}\)

Power Plant Emissions Sulfur dioxide emissions in stack gases at power plants may react with carbon monoxide as follows: $$ \mathrm{SO}_{2}(g)+3 \mathrm{CO}(g) \rightarrow 2 \mathrm{CO}_{2}(g)+\operatorname{COS}(g) $$ Write an equation relating the rates for each of the following: a. The rate of formation of \(\mathrm{CO}_{2}\) to the rate of consumption of CO b. The rate of formation of \(\mathrm{COS}\) to the rate of consumption of \(\mathrm{SO}_{2}\) c. The rate of consumption of \(\mathrm{CO}\) to the rate of consumption of \(\mathrm{SO}_{2}\)

Each of the following reactions is first order in the reactants and second order overall. Which reaction is fastest if the initial concentrations of the reactants are the same? All reactions are at \(298 \mathrm{K}\) a. \(\mathrm{ClO}_{2}(g)+\mathrm{O}_{3}(g) \rightarrow \mathrm{ClO}_{3}(g)+\mathrm{O}_{2}(g)\) \(k=3.0 \times 10^{-19} \mathrm{cm}^{3} /(\text { molecule } \cdot \mathrm{s})\) b. \(\mathrm{ClO}_{2}(g)+\mathrm{NO}(g) \rightarrow \mathrm{NO}_{2}(g)+\mathrm{ClO}(g)\) \(k=3.4 \times 10^{-13} \mathrm{cm}^{3} /(\mathrm{molecule} \cdot \mathrm{s})\) c. \(\mathrm{ClO}(g)+\mathrm{NO}(g) \rightarrow \mathrm{Cl}(g)+\mathrm{NO}_{2}(g)\) \(k=1.7 \times 10^{-11} \mathrm{cm}^{3} /(\text { molecule } \cdot \mathrm{s})\) d. \(\mathrm{ClO}(g)+\mathrm{O}_{3}(g) \rightarrow \mathrm{ClO}_{2}(g)+\mathrm{O}_{2}(g)\) \(k=1.5 \times 10^{-17} \mathrm{cm}^{3} /(\text { molecule } \cdot \mathrm{s})\)
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