Calculating The Ratio Of Rate Constants In First-Order Reactions A Comprehensive Guide
In chemical kinetics, understanding the rates of reactions is crucial for predicting and controlling chemical processes. First-order reactions are a fundamental class of reactions where the reaction rate is directly proportional to the concentration of a single reactant. Analyzing these reactions often involves determining rate constants, which quantify how quickly a reaction proceeds. This article delves into calculating the ratio of rate constants in two simultaneous first-order reactions, providing a detailed explanation and step-by-step solution. Specifically, we will address the scenario where two reactants, A and B, react with a reagent to form a product, and we aim to find the ratio of their rate constants (K₁/K₂) given the extents of their reactions at a particular time. Let's explore the intricacies of chemical kinetics and understand how to derive the ratio of rate constants in such reactions. This article aims to provide a comprehensive understanding of the principles and calculations involved, making it easier for students and professionals alike to grasp the concepts of chemical kinetics. By breaking down the problem into manageable steps, we will illustrate how the integrated rate laws for first-order reactions can be applied to solve real-world problems. Furthermore, we will discuss the significance of these calculations in various fields, including chemical engineering, pharmaceuticals, and environmental science. Understanding the kinetics of chemical reactions is essential for optimizing processes, predicting reaction outcomes, and ensuring the efficient use of resources. This knowledge is particularly valuable in industries where chemical reactions are central to the production process. By mastering the techniques presented in this article, readers will be better equipped to analyze and solve complex kinetic problems, ultimately enhancing their understanding of chemical reactions and their applications. This article will guide you through the necessary steps and provide a clear methodology for calculating the ratio of rate constants, ensuring a solid grasp of the underlying principles and practical applications.
Theoretical Background
To tackle this problem, we first need to revisit the basics of first-order kinetics. For a first-order reaction, the rate of the reaction is given by:
Rate = -d[A]/dt = K[A]
Where:
- [A] is the concentration of reactant A at time t
- K is the rate constant
- -d[A]/dt is the rate of decrease of [A] with respect to time
Integrating this rate equation gives us the integrated rate law for a first-order reaction:
ln([A]t/[A]₀) = -Kt
Where:
- [A]t is the concentration of A at time t
- [A]₀ is the initial concentration of A
This integrated rate law is the foundation for solving problems involving first-order reactions. It allows us to relate the concentration of a reactant at any given time to its initial concentration and the rate constant. The rate constant, K, is a crucial parameter that reflects the speed of the reaction. A larger K value indicates a faster reaction, while a smaller K value indicates a slower reaction. In the context of the given problem, we have two first-order reactions, each with its own rate constant. By comparing the extents of the reactions at a specific time, we can determine the ratio of these rate constants. Understanding the integrated rate law and its applications is essential for anyone studying chemical kinetics. It provides a powerful tool for analyzing experimental data and predicting the behavior of chemical reactions. This knowledge is particularly useful in fields such as chemical engineering, where optimizing reaction conditions is crucial for efficient production. By mastering the principles of first-order kinetics, readers can gain a deeper appreciation for the dynamic nature of chemical reactions and their role in various scientific and industrial processes. The integrated rate law also plays a vital role in determining the half-life of a reaction, which is the time required for half of the reactant to be consumed. This concept is particularly important in areas such as nuclear chemistry and drug metabolism, where understanding the decay rates of substances is critical. By connecting the integrated rate law to real-world applications, we can better appreciate its significance and utility in various scientific disciplines. This article will further illustrate how this fundamental equation can be used to solve complex problems and provide valuable insights into the behavior of chemical reactions.
Problem Statement
We are given two first-order reactions:
- A + Reagent → Product
- B + Reagent → Product
We are told that when 94% of A has reacted, 50% of B has reacted. Our goal is to calculate the ratio of the rate constants, K₁/K₂. Understanding the problem statement is the first step towards finding a solution. We need to identify the key pieces of information provided and determine how they relate to each other. In this case, we know the percentage of reactants A and B that have reacted at a particular time, and we are asked to find the ratio of their rate constants. This requires us to apply the principles of first-order kinetics and utilize the integrated rate law. The problem highlights the importance of understanding the stoichiometric relationships in chemical reactions. Although both reactions involve a reagent and a product, the rates at which A and B react are different, as indicated by their respective rate constants. By comparing the extents of the reactions, we can gain insights into their relative speeds and determine the ratio of their rate constants. This type of problem is commonly encountered in chemical kinetics and serves as a good example of how theoretical concepts can be applied to solve practical problems. The ability to analyze and solve such problems is essential for anyone working in the field of chemistry or related disciplines. This article will provide a detailed step-by-step solution, making the process clear and understandable. By breaking down the problem into smaller, more manageable parts, we will ensure that the underlying concepts are fully grasped. This approach will not only help in solving this specific problem but also in tackling other similar problems in chemical kinetics.
Step-by-Step Solution
1. Define the Integrated Rate Laws
For reaction A:
ln([A]t/[A]₀) = -K₁t ...(1)
For reaction B:
ln([B]t/[B]₀) = -K₂t ...(2)
These equations are the cornerstones of our solution. They describe how the concentrations of reactants A and B change over time in first-order reactions. The integrated rate laws are derived from the differential rate laws, which express the rate of reaction in terms of the concentrations of reactants. By integrating these differential equations, we obtain the integrated rate laws, which provide a direct relationship between the concentration of a reactant and time. Understanding the derivation of these equations is crucial for appreciating their significance and limitations. The integrated rate laws are applicable only to reactions that follow a specific order, such as first-order reactions. For reactions of other orders, different integrated rate laws must be used. In the context of this problem, we are dealing with two first-order reactions, so the integrated rate laws (1) and (2) are perfectly suited for our analysis. These equations allow us to relate the initial and final concentrations of reactants A and B to their respective rate constants and the time elapsed. By manipulating these equations and using the information provided in the problem statement, we can ultimately determine the ratio of the rate constants, K₁/K₂. The integrated rate laws are also essential for determining the half-life of a reaction, which is the time required for half of the reactant to be consumed. This concept is particularly important in various fields, including nuclear chemistry and drug metabolism, where understanding the decay rates of substances is critical. By mastering the application of integrated rate laws, readers can gain a deeper understanding of chemical kinetics and its practical implications.
2. Express Concentrations in Terms of Reaction Percentage
When 94% of A has reacted, the remaining concentration of A is 6% of its initial concentration. Therefore:
[A]t = 0.06[A]₀
Similarly, when 50% of B has reacted, the remaining concentration of B is 50% of its initial concentration. Thus:
[B]t = 0.5[B]₀
Converting percentages to concentrations is a crucial step in solving this problem. The integrated rate laws relate the concentrations of reactants at different times, so we need to express the given percentages in terms of these concentrations. When 94% of A has reacted, it means that only 6% of A remains. This remaining amount is expressed as 0.06 times the initial concentration of A. Similarly, when 50% of B has reacted, the remaining amount is 0.5 times the initial concentration of B. These conversions are straightforward but essential for the subsequent calculations. It is important to note that the percentages given in the problem statement refer to the extent of the reaction, not the absolute concentrations of the reactants. The initial concentrations of A and B may be different, but the percentages provide a relative measure of how much each reactant has been consumed at a particular time. By expressing these percentages in terms of concentrations, we can directly apply the integrated rate laws and relate the extents of the reactions to the rate constants. This step highlights the importance of carefully interpreting the information provided in the problem statement and translating it into a form that can be used in the relevant equations. The ability to perform these conversions accurately is a fundamental skill in chemical kinetics and is essential for solving a wide range of problems. By mastering this step, readers can gain confidence in their ability to tackle more complex kinetic problems.
3. Substitute Concentrations into Integrated Rate Laws
Substitute [A]t in equation (1):
ln(0.06[A]₀/[A]₀) = -K₁t
ln(0.06) = -K₁t ...(3)
Substitute [B]t in equation (2):
ln(0.5[B]₀/[B]₀) = -K₂t
ln(0.5) = -K₂t ...(4)
Substituting the concentrations into the integrated rate laws is a key step in relating the extents of the reactions to the rate constants. By replacing [A]t and [B]t with their respective expressions in terms of the initial concentrations, we can simplify the equations and isolate the rate constants. In equation (3), we substitute 0.06[A]₀ for [A]t, which represents the concentration of A remaining after 94% has reacted. The [A]₀ terms cancel out, leaving us with ln(0.06) = -K₁t. Similarly, in equation (4), we substitute 0.5[B]₀ for [B]t, which represents the concentration of B remaining after 50% has reacted. The [B]₀ terms cancel out, resulting in ln(0.5) = -K₂t. These simplified equations now directly relate the rate constants K₁ and K₂ to the time t and the natural logarithms of the concentration ratios. The natural logarithms arise from the integration of the first-order rate equations and are a characteristic feature of first-order kinetics. It is important to remember that the natural logarithm of a number less than 1 is negative, which is consistent with the negative sign on the right-hand side of the equations. These substitutions demonstrate the power of the integrated rate laws in simplifying complex kinetic problems. By expressing the concentrations in terms of the initial concentrations and using the appropriate substitutions, we can transform the problem into a more manageable form. This step is essential for anyone seeking to understand and apply the principles of chemical kinetics.
4. Divide Equation (3) by Equation (4)
Divide equation (3) by equation (4) to eliminate time t:
ln(0.06) / ln(0.5) = (-K₁t) / (-K₂t)
ln(0.06) / ln(0.5) = K₁ / K₂
Eliminating time (t) by dividing the two equations is a crucial step in solving for the ratio of the rate constants. Since both reactions occur simultaneously, the time elapsed for both reactions is the same. By dividing equation (3) by equation (4), we effectively cancel out the time variable, leaving us with an expression that directly relates the ratio of the rate constants to the ratio of the natural logarithms. This step highlights the elegance of using mathematical manipulations to simplify complex problems. By carefully considering the relationships between the variables and using appropriate algebraic techniques, we can isolate the desired quantity and arrive at a solution. The division step also underscores the importance of understanding the underlying assumptions of the problem. In this case, the assumption that both reactions occur simultaneously and therefore have the same time elapsed is critical for the elimination of the time variable. If the reactions occurred at different times, this step would not be valid. The resulting equation, ln(0.06) / ln(0.5) = K₁ / K₂, provides a direct means of calculating the ratio of the rate constants. All that remains is to evaluate the natural logarithms and perform the division. This step demonstrates the power of combining theoretical concepts with mathematical techniques to solve practical problems in chemical kinetics. By mastering these techniques, readers can gain confidence in their ability to analyze and interpret experimental data and make predictions about the behavior of chemical reactions.
5. Calculate the Ratio
Now, calculate the natural logarithms:
ln(0.06) ≈ -2.813
ln(0.5) ≈ -0.693
Therefore,
K₁ / K₂ ≈ -2.813 / -0.693
K₁ / K₂ ≈ 4.06
Calculating the natural logarithms and performing the division yields the final answer for the ratio of the rate constants. The natural logarithm of 0.06 is approximately -2.813, and the natural logarithm of 0.5 is approximately -0.693. These values are obtained using a calculator or a table of logarithms. Dividing -2.813 by -0.693 gives us a ratio of approximately 4.06. This result indicates that the rate constant for reaction A (K₁) is about 4.06 times larger than the rate constant for reaction B (K₂). In other words, reaction A proceeds about 4.06 times faster than reaction B under the given conditions. This calculation demonstrates the practical application of the integrated rate laws and the power of chemical kinetics in quantifying reaction rates. By determining the ratio of the rate constants, we can gain insights into the relative speeds of different reactions and make predictions about their behavior. This information is valuable in various fields, including chemical engineering, pharmaceuticals, and environmental science. For example, in chemical engineering, understanding reaction rates is crucial for optimizing industrial processes and maximizing product yields. In pharmaceuticals, it is essential for determining the shelf life of drugs and predicting their metabolism in the body. By mastering the techniques presented in this article, readers can develop a deeper appreciation for the quantitative aspects of chemical reactions and their applications in various scientific and industrial settings.
Final Answer
The ratio of K₁ / K₂ is approximately 4.06.
Therefore, the correct answer is (1) 4.06.
This final answer confirms our step-by-step solution and provides the numerical value for the ratio of the rate constants. The ratio K₁ / K₂ ≈ 4.06 indicates that reaction A proceeds significantly faster than reaction B under the given conditions. This result is consistent with the observation that 94% of A has reacted while only 50% of B has reacted at the same time. The larger rate constant for reaction A implies that it has a higher affinity for the reagent or a lower activation energy compared to reaction B. This information can be valuable in understanding the underlying mechanisms of the reactions and in predicting their behavior under different conditions. The process of arriving at this final answer has involved a series of logical steps, starting from the definition of first-order kinetics and the integrated rate laws, to the conversion of percentages to concentrations, the substitution of concentrations into the rate laws, the elimination of time, and the final calculation. Each step has been carefully explained to ensure a clear understanding of the underlying principles and techniques. By following this step-by-step approach, readers can gain confidence in their ability to solve similar problems in chemical kinetics. The final answer also serves as a validation of the methodology used. By comparing the calculated ratio with the information provided in the problem statement, we can ensure that the result is consistent with the observations. This self-checking mechanism is an important aspect of problem-solving and helps to avoid errors. In conclusion, the ratio K₁ / K₂ ≈ 4.06 represents a quantitative measure of the relative rates of two first-order reactions and provides valuable insights into their kinetics.
In this article, we have demonstrated a detailed method for calculating the ratio of rate constants in two simultaneous first-order reactions. By applying the integrated rate laws and performing the necessary mathematical manipulations, we were able to determine that K₁ / K₂ is approximately 4.06. This exercise highlights the importance of understanding first-order kinetics and the ability to apply these principles to solve real-world problems. Understanding the principles and methods discussed in this article is crucial for anyone studying or working in the field of chemistry. Chemical kinetics is a fundamental aspect of chemistry, and the ability to analyze reaction rates is essential for a wide range of applications. Whether you are a student learning the basics or a professional working in a chemical industry, the knowledge and skills gained from this article will be invaluable. The step-by-step approach presented here provides a clear and logical framework for solving similar problems. By breaking down the problem into smaller, more manageable parts, we have made the process more accessible and easier to understand. This approach can be applied to other types of kinetic problems as well, making it a valuable tool for problem-solving in general. The importance of understanding the assumptions and limitations of the equations used has also been emphasized. The integrated rate laws are applicable only to reactions that follow a specific order, such as first-order reactions. It is crucial to verify that the assumptions are valid before applying the equations. In addition to the specific problem solved in this article, the general principles of chemical kinetics have been discussed. The concepts of rate constants, integrated rate laws, and reaction rates are fundamental to understanding chemical reactions and their behavior. By mastering these concepts, readers can gain a deeper appreciation for the dynamic nature of chemical processes and their role in various scientific and industrial applications. This article serves as a comprehensive guide to calculating the ratio of rate constants in first-order reactions and provides a solid foundation for further exploration of chemical kinetics.