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The isomerization reaction, \(\mathrm{CH}_{3} \mathrm{NC} \rightarrow \mathrm{CH}_{3} \mathrm{CN}\), is first order and the rate constant is equal to \(0.46 \mathrm{~s}^{-1}\) at \(600 \mathrm{~K}\). What is the concentration of \(\mathrm{CH}_{3} \mathrm{NC}\) after \(0.20\) minutes if the initial concentration is \(0.10 \mathrm{M}\) ? a. \(14.0 \times 10^{-4} \mathrm{M}\) b. \(4.0 \times 10^{-4} \mathrm{M}\) c. \(2.4 \times 10^{-4} \mathrm{M}\) d. \(6.4 \times 10^{-4} \mathrm{M}\)

Short Answer

Expert verified
The concentration of \(\mathrm{CH}_{3} \mathrm{NC}\) after 0.20 minutes is \(4.0 \times 10^{-4} \, \mathrm{M}\), choice b.

Step by step solution

01

Identify the equation for first-order reactions

The concentration for a first-order reaction is given by the formula:\[[A] = [A]_0 \times e^{-kt}\] where \([A]\) is the concentration after time \(t\), \([A]_0\) is the initial concentration, \(k\) is the rate constant, and \(t\) is the time in seconds.
02

Convert time from minutes to seconds

Since the rate constant is given in seconds (\(0.46 \, \text{s}^{-1}\)), we need to convert the time from minutes to seconds.\(0.20 \, \text{minutes} = 0.20 \times 60 \, \text{seconds} = 12 \, \text{seconds}\).
03

Calculate the concentration after 12 seconds

Use the first-order reaction formula:\[[A] = [A]_0 \times e^{-kt}\]Substitute the values:- \([A]_0 = 0.10 \, \text{M}\)- \(k = 0.46 \, \text{s}^{-1}\)- \(t = 12 \, \text{seconds}\)Calculate:\[[A] = 0.10 \, \text{M} \times e^{-(0.46 \times 12)}\]\[[A] = 0.10 \, \text{M} \times e^{-5.52}\]\[[A] \approx 0.10 \, \text{M} \times 0.0040 = 0.00040 \, \text{M}\]
04

Compare with the answer choices

The calculated concentration \(0.00040 \, \text{M}\) can be rewritten as \(4.0 \times 10^{-4} \, \text{M}\), which matches option (b).

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Key Concepts

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

Isomerization Reaction
An isomerization reaction involves the transformation of a molecule into another molecule with the same molecular formula but a different arrangement of atoms. This reaction changes the molecule's properties while maintaining the same elemental composition.
In the context of our exercise, the isomerization reaction \((\mathrm{CH}_{3} \mathrm{NC} \rightarrow \mathrm{CH}_{3} \mathrm{CN})\), demonstrates the transformation of methyl isocyanide into methyl cyanide.
This change occurs at a molecular level, affecting the atoms' spatial arrangement within the molecule without altering the types of atoms present. Isomerization reactions are common in chemistry and can significantly impact a compound's reactivity and stability. They play an essential role in various chemical processes such as drug synthesis and the polymerization of plastics.
Rate Constant
The rate constant, often symbolized by \(k\), is a crucial factor in chemical kinetics. It measures the speed of a chemical reaction's progress under specific conditions.
The rate constant's value depends on factors such as temperature and pressure, and it can vary from one reaction to another. In our exercise, the rate constant for the isomerization reaction is given as \(0.46\, \text{s}^{-1}\) at \(600 \text{ K}\). This tells us how quickly the reaction proceeds at this temperature.

Importance in First-order Reactions

The value and units of \(k\) help us determine the reaction order. For first-order reactions, the rate constant has units of \(\text{s}^{-1}\), indicating that the rate of the reaction is directly proportional to the concentration of one reactant.
Understanding the rate constant allows us to predict how fast the reaction occurs and calculate important quantities such as reactant concentrations over time.
Exponential Decay
Exponential decay describes how quantities decrease at a rate proportional to their current value. This concept is commonly seen in chemical reactions, particularly first-order reactions.
In these reactions, the concentration of a reactant decreases exponentially over time, following the equation: \([^]-kt]\), where \([A]_0\) is the initial concentration, \(k\) is the rate constant, and \(t\) is time.

Application in First-order Reactions

This exponential function yields a rapidly decreasing curve that appears steeper at the beginning of the reaction.
For instance, in the problem we're solving, \([A] = 0.10 \, \text{M} \times e^{-5.52}\) results in a much lower concentration of \([A] = 0.00040 \, \text{M}\) after 12 seconds. This sharp reduction illustrates the nature of exponential decay in first-order processes.
Chemical Kinetics
Chemical kinetics is the branch of chemistry that studies the speed or rate of chemical reactions. It provides insights into reaction mechanisms and factors influencing the rate of chemical processes.
By understanding kinetics, chemists can manipulate conditions to control reactions' speed, making it an essential aspect of designing industrial processes and chemical experiments.

Linking Concepts

In our example exercise, kinetics helps us predict how quickly \(\mathrm{CH}_{3} \mathrm{NC}\) converts to \(\mathrm{CH}_{3} \mathrm{CN}\) at a given temperature, considering its first-order nature and specific rate constant.
By applying the appropriate equations and understanding kinetic principles, we are able to calculate reactant concentrations at different times, demonstrating the synergy between theoretical chemistry and practical applications. A grasp of chemical kinetics allows scientists and engineers to develop processes that maximize efficiency and yield desirable products.

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Most popular questions from this chapter

The rate law has the form; Rate \(=\mathrm{k}[\mathrm{A}][\mathrm{B}]^{3 / 2}\), can the reaction be an elementary process? a. yes b. no c. may be yes or no d. can not be predicted

The following set of data was obtained by the method of initial rates for the reaction: $$ \begin{aligned} &\mathrm{S}_{2} \mathrm{O}_{8}^{2-}(\mathrm{aq})+3 \mathrm{I}^{-}(\mathrm{aq}) \rightarrow \\ &2 \mathrm{SO}_{4}^{2-}(\mathrm{aq})+\mathrm{I}_{3}-(\mathrm{aq}) \end{aligned} $$ What is the rate law for the reaction? $$ \begin{array}{lll} \hline\left[\mathrm{S}_{2} \mathrm{O}_{8}^{2-}\right], \mathrm{M} & {[\mathrm{I}-], \mathrm{M}} & \text { Initial rate, } \mathrm{M} \mathrm{s}^{-1} \\ \hline 0.25 & 0.10 & 9.00 \times 10^{-3} \\ 0.10 & 0.10 & 3.60 \times 10^{-3} \\ 0.20 & 0.30 & 2.16 \times 10^{-2} \\ \hline \end{array} $$ a. Rate \(=\mathrm{k}\left[\mathrm{S}_{2} \mathrm{O}_{8}^{2-}\right]\left[\mathrm{I}^{-}\right]^{2}\) b. Rate \(=\mathrm{k}\left[\mathrm{S}_{2} \mathrm{O}_{8}^{2-}\right]^{2}\left[\mathrm{I}^{-}\right]\) c. Rate \(=\mathrm{k}\left[\mathrm{S}_{2} \mathrm{O}_{8}^{2-}\right]\left[\mathrm{I}^{-}\right]\) d. Rate \(=\mathrm{k}\left[\mathrm{S}_{2} \mathrm{O}_{8}^{2-}\right]\left[\mathrm{I}^{-}\right]^{5}\)

In the following question two statements Assertion (A) and Reason (R) are given Mark. a. If \(\mathrm{A}\) and \(\mathrm{R}\) both are correct and \(\mathrm{R}\) is the correct explanation of \(\mathrm{A}\); b. If \(A\) and \(R\) both are correct but \(R\) is not the correct explanation of \(\mathrm{A}\); c. \(\mathrm{A}\) is true but \(\mathrm{R}\) is false; d. \(\mathrm{A}\) is false but \(\mathrm{R}\) is true, e. \(\mathrm{A}\) and \(\mathrm{R}\) both are false. (A): A catalyst does not alter the heat of reaction. (R): Catalyst increases the rate of reaction.

Which of the following is/are examples of unimolecular reactions? a. \(2 \mathrm{NO}+\mathrm{Cl}_{2} \rightarrow 2 \mathrm{NOCl}\) b. \(\mathrm{O}_{3} \rightarrow \mathrm{O}_{2}+\mathrm{O}\) c. C=CCCCC d. \(\mathrm{NO}+\mathrm{O}_{3} \rightarrow \mathrm{NO}_{2}+\mathrm{O}_{2}\)

In the following question two statements Assertion (A) and Reason (R) are given Mark. a. If \(\mathrm{A}\) and \(\mathrm{R}\) both are correct and \(\mathrm{R}\) is the correct explanation of \(\mathrm{A}\); b. If \(A\) and \(R\) both are correct but \(R\) is not the correct explanation of \(\mathrm{A}\); c. \(\mathrm{A}\) is true but \(\mathrm{R}\) is false; d. \(\mathrm{A}\) is false but \(\mathrm{R}\) is true, e. \(\mathrm{A}\) and \(\mathrm{R}\) both are false. (A): In rate laws, unlike in the expression for equilibrium constants, the exponents for concentrations do not necessarily match stoichiometric coefficients. (R): It is the mechanism and not the balanced chemical equation for the overall change the governs the reaction rate. Reaction rate is experimentally quantity and not necessary depends on stoichiometric coefficients

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