Chapter 7: Problem 6
How are the rate constant and the half-life related?
Short Answer
Expert verified
Answer: For a first-order reaction, the half-life (t₁/₂) is inversely proportional to the rate constant (k) and is given by the equation t₁/₂ = ln(2) / k.
Step by step solution
01
Write the first-order rate law
Write down the first-order rate law which describes the rate of change of concentration of a reactant[A] with time. The general form of the first-order rate law is:rate = k * [A]
02
Integrate the rate law
Integrate the rate law to find the concentration of the reactant [A] as a function of time. Divide both sides by [A] and integrate with respect to time:
\[\int_{[A]_0}^{[A]} \frac{1}{A} d[A] = \int_0^t k dt\]
where [A]₀ is the initial concentration of the reactant A and [A] is the concentration of A at time t.
03
Find the concentration as a function of time
Solve the integration from step 2 and obtain the equation that relates the concentration of A at time t to the initial concentration and rate constant:
\[[A] = [A]_0 * e^{-kt}\]
04
Define the half-life
The half-life (t₁/₂) is the time at which the concentration of reactant A has reduced to half its initial value. In terms of the equation derived above:
\[\frac{[A]_0}{2} = [A]_0 * e^{-kt_{1/2}}\]
05
Find the relationship between the rate constant and the half-life
Solve for t₁/₂ in the equation from step 4 to find the relationship between the rate constant and half-life:
\[\frac{1}{2} = e^{-kt_{1/2}}\]
Take the natural logarithm of both sides of the equation:
\[-\ln(2) = -kt_{1/2}\]
So, the relationship between the rate constant and the half-life is given by:
\[t_{1/2} = \frac{\ln(2)}{k}\]
This relationship demonstrates that for a first-order reaction, the half-life of the reaction is inversely proportional to the rate constant, k.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with Vaia!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
First-Order Rate Law
When we talk about chemical reactions, the rate at which they occur can vary greatly. For first-order reactions, there's a unique relationship between the concentration of the reactant and the rate of the reaction.
In simple terms, the first-order rate law states that the rate of reaction is directly proportional to the concentration of one reactant. The mathematical expression for this relationship is rate = k * [A], where 'rate' is the speed of the reaction, 'k' is the rate constant, and [A] is the concentration of the reactant A.
This means that if we were to double the concentration of the reactant A, the rate of the reaction would also double. This is different from second-order reactions or zero-order reactions, where the rate depends on the concentration in a more complex way. For students grappling to understand this concept, imagining the reaction rate as a car's speed and the concentration as the pressure on the accelerator could provide a helpful analogy.
In simple terms, the first-order rate law states that the rate of reaction is directly proportional to the concentration of one reactant. The mathematical expression for this relationship is rate = k * [A], where 'rate' is the speed of the reaction, 'k' is the rate constant, and [A] is the concentration of the reactant A.
This means that if we were to double the concentration of the reactant A, the rate of the reaction would also double. This is different from second-order reactions or zero-order reactions, where the rate depends on the concentration in a more complex way. For students grappling to understand this concept, imagining the reaction rate as a car's speed and the concentration as the pressure on the accelerator could provide a helpful analogy.
Chemical Kinetics
Chemical kinetics is the study of how quickly chemical reactions occur and what factors affect their rates. It's like the 'pace' of a chemical race—how fast can reactants turn into products? Factors such as temperature, concentration, and the presence of catalysts can impact the reaction rate.
Understanding kinetics is crucial for scientists and engineers, who need to control reactions—for example, to make them faster for industrial processes or slower for preservation. When we measure the kinetics of a reaction, we are essentially timing the chemical race and finding out what influences its outcome. Explaining chemical kinetics using real-life scenarios, such as why food spoils and how preservatives work, can make this topic much easier to digest for students.
Understanding kinetics is crucial for scientists and engineers, who need to control reactions—for example, to make them faster for industrial processes or slower for preservation. When we measure the kinetics of a reaction, we are essentially timing the chemical race and finding out what influences its outcome. Explaining chemical kinetics using real-life scenarios, such as why food spoils and how preservatives work, can make this topic much easier to digest for students.
Reaction Rate
The reaction rate is the speed at which the concentrations of substances involved in a chemical reaction change over time. It's the 'heartbeat' of the chemical process, indicating how lively, or sluggish, the reaction is at a given moment. To measure this rate, one could monitor the decrease in concentration of a reactant or the increase in concentration of a product over time.
It's essential to note that reaction rates are not constant over the entire course of a reaction. They can change due to alterations in concentration, temperature, or the introduction of a catalyst. Students can relate to this by thinking of how the intensity of exercise can vary—starting off energetic and slowing down as fatigue sets in. It's a similar concept with reactions; they can start strong but might slow as reactants are consumed.
It's essential to note that reaction rates are not constant over the entire course of a reaction. They can change due to alterations in concentration, temperature, or the introduction of a catalyst. Students can relate to this by thinking of how the intensity of exercise can vary—starting off energetic and slowing down as fatigue sets in. It's a similar concept with reactions; they can start strong but might slow as reactants are consumed.
Integrated Rate Law
While the rate law gives an instantaneous picture of a reaction's pace, the integrated rate law shows how concentrations of reactants change over time. Mathematically, it's the result of taking the rate law and 'integrating' it, which in this context means summing up the changes in concentration over time.
For first-order reactions, the integrated rate law can be written as
\([A] = [A]_0 * e^{-kt}\), where \( [A] \) represents the concentration of the reactant at time t, \( [A]_0 \) is the initial concentration, 'k' is the rate constant, and 'e' is the base of natural logarithms. What this tells us is that the concentration of the reactant decreases exponentially over time. Students can picture this as a smartphone losing charge; it doesn't happen linearly but instead slows down as the battery drains.
For first-order reactions, the integrated rate law can be written as
\([A] = [A]_0 * e^{-kt}\), where \( [A] \) represents the concentration of the reactant at time t, \( [A]_0 \) is the initial concentration, 'k' is the rate constant, and 'e' is the base of natural logarithms. What this tells us is that the concentration of the reactant decreases exponentially over time. Students can picture this as a smartphone losing charge; it doesn't happen linearly but instead slows down as the battery drains.
