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💏intro to chemistry review

Ln[A]t

Written by the Fiveable Content Team • Last updated September 2025
Written by the Fiveable Content Team • Last updated September 2025

Definition

ln[A]t, or the natural logarithm of the concentration of a reactant A at time t, is a term used in the context of integrated rate laws. It represents the mathematical expression that describes the relationship between the concentration of a reactant and the time elapsed during a chemical reaction.

5 Must Know Facts For Your Next Test

  1. The term ln[A]t is used in the integrated rate law equations for first-order reactions, where the reaction rate is proportional to the concentration of the reactant.
  2. For a first-order reaction, the integrated rate law is expressed as ln[A]t = -kt + ln[A]0, where [A]0 is the initial concentration of the reactant and k is the rate constant.
  3. The plot of ln[A]t versus time (t) for a first-order reaction is a straight line with a slope of -k, the rate constant.
  4. The half-life of a first-order reaction is independent of the initial concentration and is given by the equation t1/2 = ln(2)/k.
  5. The value of ln[A]t can be used to determine the reaction rate constant (k) and the half-life of a first-order reaction.

Review Questions

  • Explain the relationship between ln[A]t and the integrated rate law for a first-order reaction.
    • For a first-order reaction, the integrated rate law is expressed as ln[A]t = -kt + ln[A]0, where [A]0 is the initial concentration of the reactant and k is the rate constant. The term ln[A]t represents the natural logarithm of the concentration of the reactant A at time t. This linear relationship between ln[A]t and time (t) allows for the determination of the rate constant (k) from the slope of the plot, as well as the half-life of the reaction using the equation t1/2 = ln(2)/k.
  • Describe how the value of ln[A]t can be used to determine the reaction rate constant (k) and the half-life of a first-order reaction.
    • The value of ln[A]t can be used to determine the reaction rate constant (k) and the half-life of a first-order reaction. For a first-order reaction, the integrated rate law is expressed as ln[A]t = -kt + ln[A]0. By plotting ln[A]t versus time (t), the slope of the resulting straight line will be equal to -k, the rate constant. Additionally, the half-life of a first-order reaction is given by the equation t1/2 = ln(2)/k, where the rate constant (k) can be determined from the slope of the ln[A]t versus time plot.
  • Analyze how the term ln[A]t is used to differentiate between first-order and other-order reactions in the context of integrated rate laws.
    • The term ln[A]t is a key distinguishing feature between first-order and other-order reactions in the context of integrated rate laws. For a first-order reaction, the integrated rate law is expressed as ln[A]t = -kt + ln[A]0, where the plot of ln[A]t versus time (t) results in a straight line with a slope of -k, the rate constant. This linear relationship is unique to first-order reactions. In contrast, the integrated rate laws for zero-order and second-order reactions do not involve the natural logarithm of the reactant concentration, and their plots would have different functional forms. Therefore, the presence and behavior of the ln[A]t term is crucial in identifying the reaction order and applying the appropriate integrated rate law.