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Chapter 14: Problem 87

The reaction \(2 \mathrm{NO}_{2} \longrightarrow 2 \mathrm{NO}+\mathrm{O}_{2}\) has the rate constant \(k=0.63 \mathrm{M}^{-1} \mathrm{~s}^{-1}\). Based on the units for \(k\), is the reaction first or second order in \(\mathrm{NO}_{2} ?\) If the initial concentration of \(\mathrm{NO}_{2}\) is \(0.100 \mathrm{M}\), how would you determine how long it would take for the concentration to decrease to \(0.025 \mathrm{M} ?\)

Short Answer

Expert verified
The reaction is second-order concerning \(\mathrm{NO}_{2}\). To determine the time it takes for the concentration of \(\mathrm{NO}_{2}\) to decrease from \(0.100 \mathrm{M}\) to \(0.025 \mathrm{M}\), we use the integrated rate law for a second-order reaction: \(\frac{1}{[\mathrm{A}]_t} - \frac{1}{[\mathrm{A}]_0} = kt\). Plugging in the given values, we find that it takes approximately 47.62 seconds for the concentration to decrease.

Step by step solution

01

Determine the reaction order concerning \(\mathrm{NO}_{2}\)

The unit for the rate constant \(k\) is \(\mathrm{M}^{-1}\mathrm{~s}^{-1}\), which indicates that the reaction is second-order concerning \(\mathrm{NO}_{2}\). The second-order reaction rate law is given by: Rate \(=\) \(k[\mathrm{NO}_{2}]^{2}\)
02

Determine the integrated rate law for a second-order reaction

For a second-order reaction, the integrated rate law is given by: \(\frac{1}{[\mathrm{A}]_t} - \frac{1}{[\mathrm{A}]_0} = kt\) where \([\mathrm{A}]_t\) refers to the concentration of \(\mathrm{NO}_{2}\) at time \(t\), \([\mathrm{A}]_0\) is the initial concentration of \(\mathrm{NO}_{2}\), \(k\) is the rate constant, and \(t\) is the time.
03

Plug in the given values and solve for time (\(t\))

We are given that the initial concentration of \(\mathrm{NO}_{2}\) is \(0.100 \mathrm{M}\) and it eventually decreases to \(0.025 \mathrm{M}\). We can now plug these values, as well as the rate constant, into the integrated rate law equation: \(\frac{1}{0.025} - \frac{1}{0.100} = (0.63 \mathrm{M}^{-1} \mathrm{~s}^{-1})(t)\) Now, solve for \(t\): \(\frac{1}{0.025 \mathrm{M}} - \frac{1}{0.100 \mathrm{M}} = (0.63 \mathrm{M}^{-1} \mathrm{~s}^{-1})(t)\) \(40 \mathrm{M}^{-1} - 10 \mathrm{M}^{-1} = 0.63 \mathrm{M}^{-1} \mathrm{~s}^{-1} \cdot t\) \(30 \mathrm{M}^{-1} = 0.63 \mathrm{M}^{-1} \mathrm{~s}^{-1} \cdot t\) \(t = \frac{30 \mathrm{M}^{-1}}{0.63 \mathrm{M}^{-1} \mathrm{~s}^{-1}}\) \(t \approx 47.62 \mathrm{~s}\) So, it takes approximately 47.62 seconds for the concentration of \(\mathrm{NO}_{2}\) to decrease from \(0.100 \mathrm{M}\) to \(0.025 \mathrm{M}\).

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Key Concepts

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

Rate Laws
Rate laws are mathematical expressions that relate the rate of a chemical reaction to the concentration of its reactants. They provide critical information about how a reaction proceeds and help predict the speed of the reaction under various conditions.
In general, the rate law can be expressed as:
  • Rate = k[ Reactant]
    Superscript represents the order of reaction with respect to that reactant.
For a second-order reaction, like the one in our example, the rate depends on the concentration of NO 2 squared:
  • Rate = k[ NO 2 ]
    Superscript = 2
  • Here, k is the rate constant, and it is crucial for determining the reaction rate.
Understanding rate laws helps in figuring out how the reaction speed changes if the concentrations of reactants are altered.
Integrated Rate Law
The integrated rate law is used to link the concentration of reactants to time, offering a way to calculate the time needed for a certain concentration change under a given set of conditions.
For a second-order reaction, the integrated rate law formula is expressed as:
  • \(\frac{1}{[A]_t} - \frac{1}{[A]_0} = kt\)
Here:
  • \([A]_t\) is the concentration at time \(t\)
  • \([A]_0\) is the initial concentration
  • \(k\) is the rate constant
  • \(t\) represents time
This formula is invaluable for calculating the time required for a reactant concentration to change from one value to another, as seen in our exercise where the concentration of NO2 is desired to change from 0.100 M to 0.025 M.
Reaction Order
Reaction order is an essential concept that defines how the rate of a chemical reaction depends on the concentration of its reactants. In our example, the reaction is second order concerning NO2.
The reaction order is determined based on:
  • The exponent of the concentration term in the rate law equation.
  • For second-order reactions, the concentration is squared.
  • The units of the rate constant \(k\), for a second-order reaction, are typically \(M^{-1}s^{-1}\).
Knowing the reaction order is crucial as it helps predict how changes in concentration will affect the reaction rate, and dictates which integrated rate law applies.
Rate Constant
The rate constant, often denoted as \(k\), is a pivotal factor in the rate equation that remains unchanged under identical conditions. It determines the speed of the reaction at a given concentration.
The role and characteristics of the rate constant include:
  • Several factors affecting it include temperature and the presence of a catalyst.
  • For second-order reactions, \(k\) typically has the unit \(M^{-1}s^{-1}\), which helps identify the reaction order.
  • Understanding \(k\) allows chemists to manipulate reaction conditions to achieve desired outcomes.
The consistent value of \(k\) under unchanged conditions allows prediction of how the concentration of reactants will vary over time within the framework of the integrated rate law.

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Most popular questions from this chapter

For each of the following gas-phase reactions, write the rate expression in terms of the appearance of each product or disappearance of each reactant: (a) \(2 \mathrm{H}_{2} \mathrm{O}(g) \longrightarrow 2 \mathrm{H}_{2}(g)+\mathrm{O}_{2}(g)\) (b) \(2 \mathrm{SO}_{2}(g)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{SO}_{3}(g)\) (c) \(2 \mathrm{NO}(g)+2 \mathrm{H}_{2}(g) \longrightarrow \mathrm{N}_{2}(g)+2 \mathrm{H}_{2} \mathrm{O}(g)\)

The following mechanism has been proposed for the gas-phase reaction of chloroform \(\left(\mathrm{CHCl}_{3}\right)\) and chlorine: Step 1: \(\mathrm{Cl}_{2}(g) \underset{k_{1}}{\stackrel{k_{1}}{\rightleftharpoons}} 2 \mathrm{Cl}(g) \quad\) (fast) Step 2: \(\mathrm{Cl}(g)+\mathrm{CHCl}_{3}(g) \stackrel{k_{3}}{\longrightarrow} \mathrm{HCl}(g)+\mathrm{CCl}_{3}(g) \quad\) (slow) Step 3: \(\mathrm{Cl}(g)+\mathrm{CCl}_{3}(g) \stackrel{k_{2}}{\longrightarrow} \mathrm{CCl}_{4} \quad\) (fast) (a) What is the overall reaction? (b) What are the intermediates in the mechanism? (c) What is the molecularity of each of the elementary reactions? (d) What is the rate-determining step? (e) What is the rate law predicted by this mechanism? (Hint: The overall reaction order is not an integer.)

Enzymes are often described as following the two-step mechanism: $$ \begin{aligned} \mathrm{E}+\mathrm{S} & \rightleftharpoons \mathrm{ES} \text { (fast) } \\ \mathrm{ES} & \ldots \mathrm{E}+\mathrm{P} \text { (slow) } \end{aligned} $$ Where \(\mathrm{E}=\) enzyme, \(\mathrm{S}=\) substrate, and \(\mathrm{P}=\) product. If an enzyme follows this mechanism, what rate law is expected for the reaction?

A flask is charged with \(0.100 \mathrm{~mol}\) of \(\mathrm{A}\) and allowed to react to form \(B\) according to the hypothetical gas-phase reaction \(\mathrm{A}(\mathrm{g}) \longrightarrow \mathrm{B}(\mathrm{g})\). The following data are collected: \begin{tabular}{lccccc} \hline Time (s) & 0 & 40 & 80 & 120 & 160 \\ \hline Moles of A & \(0.100\) & \(0.067\) & \(0.045\) & \(0.030\) & \(0.020\) \\ \hline \end{tabular} (a) Calculate the number of moles of \(\mathrm{B}\) at each time in the table. (b) Calculate the average rate of disappearance of \(\mathrm{A}\) for each \(40-\mathrm{s}\) interval, in units of \(\mathrm{mol} / \mathrm{s}\). (c) What additional information would be needed to calculate the rate in units of concentration per time?

(a) Most heterogeneous catalysts of importance are extremely finely divided solid materials. Why is particle size important? (b) What role does adsorption play in the action of a heterogeneous catalyst?
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