The equation $$[a]t = [a]0 e^{-kt}$$ describes the concentration of a reactant at time $$t$$ during a first-order reaction. This mathematical expression connects the initial concentration, the decay constant $$k$$, and the time elapsed, illustrating how the concentration decreases exponentially over time as the reaction progresses.
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This equation applies specifically to first-order reactions, where the rate of reaction is proportional to the concentration of one reactant.
The term $$[a]0$$ represents the initial concentration of the reactant before any reaction has occurred.
As time increases, the value of $$[a]t$$ approaches zero, indicating that the reactant is consumed over time.
The decay constant $$k$$ is crucial as it dictates how fast or slow a reaction proceeds; larger values of $$k$$ result in faster reactions.
In first-order kinetics, the plot of $$ ext{ln}([a]t/[a]0)$$ versus time is linear, with a slope equal to -k.
Review Questions
How does the equation $$[a]t = [a]0 e^{-kt}$$ illustrate the behavior of first-order reactions over time?
The equation shows that in first-order reactions, the concentration of a reactant decreases exponentially as time progresses. The initial concentration $$[a]0$$ indicates what we start with, while the term $$e^{-kt}$$ shows how this concentration drops based on the decay constant $$k$$ and time $$t$$. This means that as time goes on, fewer molecules are available for reaction, which aligns with the fundamental concept of first-order kinetics.
Discuss how the decay constant $$k$$ influences the rate of reaction in the context of this equation.
The decay constant $$k$$ directly impacts how quickly a reactant's concentration decreases over time. A larger value of $$k$$ means that the reaction occurs more rapidly, leading to a quicker drop in concentration. Conversely, a smaller value results in a slower rate, allowing more time for reactants to remain in solution. Therefore, understanding and calculating $$k$$ is essential for predicting how long a reaction will take and how much reactant will be left at any given moment.
Evaluate the significance of half-life in relation to the equation $$[a]t = [a]0 e^{-kt}$$ and its application in practical scenarios.
Half-life is a crucial concept linked to the equation because it provides an easy way to measure and understand reaction kinetics. For first-order reactions, half-life remains constant regardless of initial concentration, making it easier to predict how long it takes for half of a reactant to be consumed. This is especially useful in fields like pharmacology or environmental science, where knowing how quickly substances degrade can inform safety guidelines and treatment plans. Thus, half-life offers practical insights derived from this exponential decay model.
A reaction whose rate depends linearly on the concentration of one reactant, leading to an exponential decrease in concentration over time.
Rate Constant (k): A specific constant that quantifies the speed of a reaction; it is unique to each reaction and affects how quickly reactants are converted to products.
Half-Life: The time required for the concentration of a reactant to decrease to half of its initial value, which remains constant for first-order reactions.