Decomposition Reaction Time Calculation: Step-by-Step

Let's dive into calculating the time it takes for a reactant to decompose to a certain percentage of its original concentration. This is a common type of problem in chemical kinetics, and we'll break it down step-by-step so you can tackle similar problems with confidence. Specifically, we're looking at a decomposition reaction with a rate constant (k) of 0.0012 yr⁻¹. Our goal is to find out how long it takes for the reactant concentration to reach 12.5% of its initial value. Understanding the order of the reaction is crucial because it dictates which equation we use. In this scenario, we'll assume it's a first-order reaction. For first-order reactions, the rate of the reaction is directly proportional to the concentration of the reactant. This means as the reactant gets used up, the reaction slows down proportionally. The integrated rate law for a first-order reaction is given by: ln([A]t/[A]0) = -kt, where [A]t is the concentration of the reactant at time t, [A]0 is the initial concentration of the reactant, k is the rate constant, and t is the time. This equation is the key to solving our problem. Remember, the rate constant, k, is a measure of how quickly a reaction proceeds. A larger k means a faster reaction. Now, let's look at what we're given. We know that the final concentration [A]t is 12.5% of the initial concentration [A]0. Mathematically, this means [A]t = 0.125[A]0. We're also given the rate constant k = 0.0012 yr⁻¹. What we need to find is t, the time it takes for the reaction to reach this point. By understanding these fundamental principles and applying the appropriate equations, we can accurately predict reaction times and reactant concentrations. This is essential in many fields, including pharmaceuticals, environmental science, and materials science, where controlling reaction rates is crucial.

Applying the First-Order Integrated Rate Law

Alright, guys, let's plug in the values and solve for t. The first-order integrated rate law is ln([A]t/[A]0) = -kt. We know [A]t = 0.125[A]0, and k = 0.0012 yr⁻¹. Substituting these values into the equation, we get: ln(0.125[A]0/[A]0) = -0.0012 * t. Notice that [A]0 appears in both the numerator and denominator, so it cancels out: ln(0.125) = -0.0012 * t. Now we need to isolate t. To do this, we divide both sides of the equation by -0.0012: t = ln(0.125) / -0.0012. Using a calculator, we find that ln(0.125) ≈ -2.079. Therefore, t ≈ -2.079 / -0.0012 ≈ 1732.5 years. So, it takes approximately 1732.5 years for the reactant concentration to reach 12.5% of its original value. It's a good idea to always double-check your units and make sure your answer makes sense in the context of the problem. In this case, the rate constant is in years⁻¹, so our time is correctly calculated in years. Remember that understanding the math is just one part of the puzzle. It's equally important to understand the underlying chemical principles. Knowing when to use a particular equation and why it applies to a given situation is what sets apart a good problem-solver from a great one. With practice, you'll become more comfortable recognizing different types of kinetic problems and applying the appropriate techniques. Keep up the great work! Always remember to critically assess the results to ensure they align with the chemical reality and context. This step solidifies the comprehension and enhances problem-solving abilities.

Understanding the Implications

So, what does this result actually mean? A reaction that takes over 1700 years to reach 12.5% completion is incredibly slow! This highlights the importance of the rate constant. A small rate constant, like 0.0012 yr⁻¹, indicates a very slow reaction. This kind of slow decomposition is relevant in several real-world scenarios. For example, the decay of certain radioactive isotopes can have extremely long half-lives, meaning they decompose very slowly. This is why radioactive waste disposal is such a long-term challenge. Similarly, the degradation of some persistent organic pollutants (POPs) in the environment can take decades or even centuries, leading to long-term environmental contamination. On the other hand, in industrial processes, such slow reactions might be undesirable. Chemists and engineers often use catalysts to speed up reactions and make them economically viable. A catalyst lowers the activation energy of a reaction, allowing it to proceed faster without being consumed itself. Understanding the factors that influence reaction rates, such as temperature, pressure, and the presence of catalysts, is crucial for controlling and optimizing chemical processes. Furthermore, consider how changes in temperature would affect the reaction rate. Generally, increasing the temperature increases the rate constant, leading to a faster reaction. The Arrhenius equation quantifies this relationship: k = A * exp(-Ea/RT), where A is the pre-exponential factor, Ea is the activation energy, R is the gas constant, and T is the absolute temperature. Understanding this equation allows us to predict how reaction rates will change under different temperature conditions, which is vital in various applications, from cooking to chemical manufacturing. The slow rate observed in this calculation underscores the importance of these considerations when dealing with chemical reactions.

Practice Problems and Further Exploration

To solidify your understanding, let's try a few practice problems. What if the rate constant was 0.0024 yr⁻¹? How long would it take for the reactant to reach 12.5% of its original value then? Remember, the key is to use the same first-order integrated rate law and simply substitute the new value for k. Or, what if we wanted to know how long it takes for the reactant to reach 50% of its original value (the half-life)? Again, just plug in 0.5[A]0 for [A]t and solve for t. You'll find that the half-life of a first-order reaction is independent of the initial concentration, which is a unique characteristic of first-order kinetics. Also, explore what happens if the reaction is second order. The integrated rate law is different for second-order reactions, and the half-life does depend on the initial concentration. You can research the integrated rate laws for different reaction orders and practice applying them to various problems. Understanding the differences between reaction orders and how to identify them experimentally is a crucial skill in chemical kinetics. Consider exploring resources like textbooks, online courses, and chemistry websites to deepen your knowledge. Many universities also offer open-access lecture notes and problem sets that can be extremely helpful. Additionally, try searching for real-world examples of first-order reactions to see how these principles apply in various fields. From radioactive decay to enzyme kinetics, first-order reactions are ubiquitous in nature and industry. By actively engaging with the material and practicing problem-solving, you'll develop a strong foundation in chemical kinetics and be well-equipped to tackle more complex problems in the future. Keep experimenting and exploring, and don't be afraid to ask questions!