This equation represents the relationship between the concentration of a reactant (A) at time t, the initial concentration of the reactant, and the rate constant (k) in the context of a first-order reaction. It is a fundamental expression in the study of reaction kinetics and rate laws.
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The equation 1/[A]_t = 1/[A]_0 + kt is derived from the integrated rate law for a first-order reaction.
The term [A]_t represents the concentration of the reactant A at time t, while [A]_0 represents the initial concentration of A.
The rate constant k is a measure of the intrinsic reactivity of the chemical species and the frequency of successful collisions.
A plot of 1/[A]_t versus t will yield a straight line with a slope of k, allowing for the determination of the rate constant.
The equation can be used to predict the concentration of the reactant A at any given time during the course of the reaction.
Review Questions
Explain the significance of the equation 1/[A]_t = 1/[A]_0 + kt in the context of first-order reactions.
The equation 1/[A]_t = 1/[A]_0 + kt is a crucial expression in the study of first-order reactions. It describes the relationship between the concentration of the reactant A at any given time (t), the initial concentration of A ([A]_0), and the rate constant (k) of the reaction. This equation allows for the determination of the rate constant from experimental data, as well as the prediction of the reactant concentration at any point during the reaction. Understanding this equation is essential for analyzing the kinetics and rate laws of first-order chemical processes.
Describe how the plot of 1/[A]_t versus t can be used to determine the rate constant (k) for a first-order reaction.
For a first-order reaction, the plot of 1/[A]_t versus t will yield a straight line with a slope equal to the rate constant (k). This is because the equation 1/[A]_t = 1/[A]_0 + kt can be rearranged to the linear form y = mx + b, where y = 1/[A]_t, x = t, m = k, and b = 1/[A]_0. By plotting the experimental data and fitting a linear regression line, the slope of the line will directly give the value of the rate constant k. This graphical method provides a simple and effective way to determine the rate constant for a first-order reaction from the given concentration data.
Analyze how the equation 1/[A]_t = 1/[A]_0 + kt can be used to predict the concentration of a reactant A at any time t during a first-order reaction.
The equation 1/[A]_t = 1/[A]_0 + kt can be rearranged to solve for the concentration of the reactant A at any time t during a first-order reaction. Specifically, the equation can be manipulated to obtain the expression [A]_t = [A]_0 / (1 + k[A]_0 t), which allows for the calculation of the reactant concentration at a given time t, based on the known initial concentration [A]_0 and the rate constant k. This predictive capability of the equation is crucial in understanding and modeling the progress of first-order chemical reactions, as it enables the forecasting of reactant concentrations at any point in time during the reaction process.
Related terms
First-Order Reaction: A reaction where the rate of the reaction is directly proportional to the concentration of a single reactant.
Rate Constant (k): A proportionality constant that represents the speed or rate at which a chemical reaction occurs.
Reaction Kinetics: The study of the rates of chemical reactions and the factors that affect those rates.