A First-order Reaction Is 58% Complete At The End Of 11 Min. What Is The Value Of The Rate Constant?
Understanding reaction kinetics is fundamental in chemistry, especially when dealing with reaction rates and their governing parameters. One of the most common types of reactions studied is the first-order reaction, which has a rate proportional to the concentration of a single reactant. In this article, we explore how to determine the rate constant for a first-order reaction given specific data: that the reaction is 58% complete after 11 minutes. We will walk through the principles, formulas, and calculations step-by-step to find the rate constant (k), ensuring clarity and depth for learners and professionals alike.
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Introduction to First-Order Reactions
What Is a First-Order Reaction?
A first-order reaction is a chemical process where the rate depends linearly on the concentration of a single reactant. The general form of a first-order reaction is:\[ \text{Rate} = k [A] \]
where:
- \( [A] \) is the concentration of the reactant,
- \( k \) is the rate constant with units of inverse time (e.g., min\(^{-1}\)).
Characteristics of First-Order Reactions
- The half-life (\( t_{1/2} \)) is constant and independent of concentration.
- The concentration versus time graph is exponential.
- The integrated rate law takes a simple form, allowing easy calculation of reaction progress and rate constants.
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Understanding the Data: 58% Completion at 11 Minutes
What Does 58% Completion Mean?
The term "58% complete" indicates that 58% of the reactant has reacted, leaving 42% remaining at the specified time. Mathematically, if:\[ [A]_0 \] = initial concentration, then
\[ [A] = (1 - 0.58) \times [A]0 = 0.42 \times [A]0 \]
This information allows us to relate concentration changes over time to the rate constant.
Implications for the Reaction Progress
- At \( t = 11 \) minutes, the concentration of reactant \( [A] \) is 42% of the initial concentration.
- The goal is to find the rate constant \( k \) based on this data.
Mathematical Framework for First-Order Reactions
Integrated Rate Law for First-Order Reactions
The integrated rate law relates concentration and time:\[ \ln \left( \frac{[A]_0}{[A]} \right) = kt \]
where:
- \( [A]_0 \) = initial concentration,
- \( [A] \) = concentration at time \( t \),
- \( k \) = rate constant,
- \( t \) = time elapsed.
Rearranged for Rate Constant \( k \)
Solving for \( k \):
\[ k = \frac{1}{t} \times \ln \left( \frac{[A]_0}{[A]} \right) \]
Since \( [A]_0 \) and \( [A] \) are known or expressed as fractions of the initial concentration, the calculation simplifies.
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Step-by-Step Calculation of the Rate Constant
Step 1: Express Known Quantities
- Time \( t = 11 \) minutes
- Remaining concentration \( [A] = 0.42 \times [A]_0 \)
Step 2: Plug into the Integrated Rate Law
Using the relation:\[ k = \frac{1}{t} \times \ln \left( \frac{[A]_0}{[A]} \right) \]
\[ k = \frac{1}{11} \times \ln \left( \frac{[A]0}{0.42 \times [A]0} \right) \]
The \( [A]_0 \) cancels out:
\[ k = \frac{1}{11} \times \ln \left( \frac{1}{0.42} \right) \]
Step 3: Calculate the Natural Logarithm
Calculate \( \ln (1/0.42) \):\[ \frac{1}{0.42} \approx 2.38095 \]
\[ \ln(2.38095) \approx 0.8675 \]
Step 4: Final Calculation
Now compute \( k \):\[ k = \frac{1}{11} \times 0.8675 \]
\[ k \approx 0.07886 \, \text{min}^{-1} \]
Thus, the rate constant:
\( \boxed{k \approx 0.0789 \, \text{min}^{-1}} \)
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Interpreting the Result and Additional Insights
Significance of the Rate Constant
- The rate constant \( k \) quantifies how quickly the reactant is consumed.
- A higher \( k \) indicates a faster reaction.
- In this case, approximately 7.89% of the reactant is converted per minute on average.
Half-Life for the Reaction
For a first-order reaction, the half-life \( t_{1/2} \) is:\[ t_{1/2} = \frac{\ln 2}{k} \]
\[ t_{1/2} = \frac{0.693}{0.0789} \approx 8.78 \text{ minutes} \]
This aligns with the data, as roughly 58% completion at 11 minutes suggests a half-life close to 8.78 minutes.
Application in Chemical Engineering and Laboratory Settings
Knowing \( k \) allows chemists and engineers to:- Predict how long a reaction will take to reach a desired conversion level.
- Design reactors and processes for optimal efficiency.
- Understand reaction mechanisms and compare different reaction pathways.
Factors Affecting the Rate Constant
Temperature
- Increasing temperature generally increases \( k \) due to the Arrhenius equation.
- Activation energy plays a crucial role in this temperature dependence.
Catalysts
- Catalysts can significantly increase \( k \) by providing alternative pathways.
Pressure and Concentration
- For first-order reactions, the rate constant is independent of concentration but sensitive to other factors like temperature.
Conclusion
Determining the rate constant for a first-order reaction based on reaction completion over time is a fundamental skill in chemical kinetics. Given that a reaction is 58% complete after 11 minutes, we utilized the integrated rate law to calculate the rate constant as approximately 0.0789 min\(^{-1}\). This value not only helps in understanding the speed of the reaction but also aids in designing chemical processes, optimizing reaction conditions, and predicting reaction behavior. Mastery of these calculations reinforces a deeper understanding of reaction dynamics and their practical applications in science and industry.
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References and Further Reading
- Atkins, P., & de Paula, J. (2014). Physical Chemistry. Oxford University Press.
- Laidler, K. J., Meiser, J. H., & Sanctuary, B. C. (1999). Physical Chemistry. Houghton Mifflin.
- Petrucci, R. H., Herring, F. G., Madura, J. D., & Bissonnette, C. (2017). General Chemistry: Principles & Modern Applications. Pearson Education.
- Online resources for chemical kinetics and rate law calculations.
Note: Always ensure units are consistent during calculations. In this case, minutes were used for time, but the principles hold for any unit of time, provided the units are consistent.
