Question

The isomerization reaction, ${\mathrm{CH}}_3\mathrm{NC}\to {\mathrm{CH}}_3\mathrm{CN}$, 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

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

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.

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.$0.20\text{minutes}=0.20\times 60\text{seconds}=12\text{seconds}$.

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:$$[A]=0.10\text{M}\times e^{-(0.46\times 12)}$$$$[A]=0.10\text{M}\times e^{-5.52}$$$$[A]\approx 0.10\text{M}\times 0.0040=0.00040\text{M}$$

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).

Key concepts

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 $({\mathrm{CH}}_3\mathrm{NC}\to {\mathrm{CH}}_3\mathrm{CN})$, 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.