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12.4 Integrated Rate Laws

3 min read•june 25, 2024

are essential tools in chemistry, linking reactant concentration to time in chemical reactions. They enable us to predict reaction progress and kinetics without constant monitoring, providing a mathematical relationship between concentration and time.

These laws come in different forms for zero-, first-, and second-order reactions, each with unique characteristics. Understanding and how to determine reaction order are crucial skills in applying integrated rate laws to real-world chemical processes.

Integrated Rate Laws

Purpose of integrated rate laws

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  • Relate reactant concentration to time in a chemical reaction enables determination of concentration at any given time or time required to reach a specific concentration
  • Derived by integrating the provides a mathematical relationship between concentration and time
  • Allows prediction of reaction progress and kinetics without continuous monitoring of reactant concentration (spectroscopy or titration)

Calculations with integrated rate laws

  • Zero-order reactions:
    • Integrated rate law: [A]t=−kt+[A]0[A]_t = -kt + [A]_0 concentration decreases linearly with time
    • units: concentration/time () indicates the change in concentration per unit time
    • Graphical representation: [A] vs. t is linear with a slope of −[k](https://www.fiveableKeyTerm:K)-[k](https://www.fiveableKeyTerm:K) straight line with negative slope
  • First-order reactions:
    • Integrated rate law: ln⁡[A]t=−kt+ln⁡[A]0\ln[A]_t = -kt + \ln[A]_0 natural logarithm of concentration decreases linearly with time
    • units: 1/time (s−1^{-1}) indicates the fraction of reactant consumed per unit time
    • Graphical representation: ln[A] vs. t is linear with a slope of −k-k straight line with negative slope
  • Second-order reactions:
    • Integrated rate law: 1[A]t=kt+1[A]0\frac{1}{[A]_t} = kt + \frac{1}{[A]_0} reciprocal of concentration increases linearly with time
    • Rate constant units: 1/(concentration × time) (M−1^{-1}s−1^{-1}) indicates the change in reciprocal concentration per unit time
    • Graphical representation: 1/[A] vs. t is linear with a slope of kk straight line with positive slope

Half-life in chemical reactions

  • Time required for reactant concentration to decrease to half of its initial value represents the speed of the reaction
  • Related to rate constant and reaction order allows calculation of from rate constant and vice versa
    1. Zero-order: t1/2=[A]02kt_{1/2} = \frac{[A]_0}{2k} half-life increases with initial concentration
    2. First-order: t1/2=ln⁡2kt_{1/2} = \frac{\ln 2}{k} half-life is constant and independent of initial concentration
    3. Second-order: t1/2=1k[A]0t_{1/2} = \frac{1}{k[A]_0} half-life decreases with increasing initial concentration
  • For first-order reactions, half-life is constant and independent of initial concentration simplifies calculations and comparisons
  • For zero-order and second-order reactions, half-life depends on initial concentration requires recalculation for different starting concentrations

Determination of reaction order

  • Plot concentration-time data using integrated rate law equations for different reaction orders:
    • Zero-order: [A] vs. t
    • First-order: ln[A] vs. t
    • Second-order: 1/[A] vs. t
  • The plot that yields a straight line indicates the correct reaction order allows visual determination of reaction order
    • Slope of the line is related to rate constant (k) enables calculation of rate constant from graph
  • Compare half-lives at different initial concentrations an alternative method to determine reaction order
    • If half-life is constant, the reaction is first-order
    • If half-life is inversely proportional to initial concentration, the reaction is second-order
    • If half-life is directly proportional to initial concentration, the reaction is zero-order
  • The describes how the rate of a reaction depends on the concentration of reactants

Reaction Kinetics and Rate Laws

  • Differential rate law expresses the rate of reaction as a function of reactant concentrations
  • Integrated rate law relates concentration to time and is derived from the differential rate law
  • Rate constant (k) is a proportionality factor that relates reaction rate to reactant concentrations
  • study how fast chemical reactions occur and the factors affecting reaction rates

Key Terms to Review (24)

[A]0: [A]0 represents the initial concentration of a reactant, A, in a chemical reaction. It is a key parameter used in the integrated rate law equations, which describe the relationship between the concentration of a reactant and the reaction time.
[A]t: [A]t is a key term in the context of Integrated Rate Laws, which describe the relationship between the concentration of reactants and the time elapsed during a chemical reaction. This term is crucial in understanding how the concentration of a reactant changes over time under different reaction conditions.
1/[A]0: 1/[A]0 is a term that appears in the integrated rate law equations, which describe the relationship between the concentration of a reactant and time in a chemical reaction. It represents the initial concentration of the reactant, and is a crucial parameter in understanding the kinetics of a reaction.
1/[A]t: 1/[A]t is a key term in the context of Integrated Rate Laws, which describe the relationship between the concentration of a reactant and the time elapsed during a chemical reaction. It represents the inverse of the concentration of the reactant (A) at a given time (t), and is a crucial parameter in understanding the kinetics of a reaction.
Differential Rate Law: The differential rate law, also known as the rate equation, describes the instantaneous rate of a chemical reaction as a function of the concentrations of the reactants. It is a fundamental concept in chemical kinetics that helps determine the order of a reaction and the rate constant.
First-Order Reaction: A first-order reaction is a chemical reaction where the rate of the reaction is directly proportional to the concentration of a single reactant. The reaction rate is independent of the concentrations of other reactants or products involved in the reaction.
Half-life: Half-life is the time required for half of the radioactive nuclei in a sample to decay. It is a characteristic property of each radioactive isotope.
Half-life: Half-life is the time it takes for a radioactive substance to decay to half of its original amount. It is a fundamental concept in nuclear chemistry that describes the exponential decay of radioactive isotopes and is crucial for understanding the behavior of radioactive materials.
Half-life of a reaction (t1/2): The half-life of a reaction ($t_{1/2}$) is the time required for the concentration of a reactant to decrease to half of its initial value. It provides insight into the rate at which a reaction proceeds.
Integrated Rate Laws: Integrated rate laws are mathematical expressions that describe the change in the concentration of a reactant or product over time in a chemical reaction. They provide a way to quantify the kinetics of a reaction and determine the order of the reaction based on experimental data.
K: K is a variable used to represent various constants and parameters in the context of chemical processes and principles. It is a versatile term that appears in multiple areas of chemistry, including the study of gas behavior, reaction kinetics, chemical equilibrium, and thermodynamics.
Ln 2: ln 2, also known as the natural logarithm of 2, is a mathematical constant that has numerous applications in various fields, including chemistry. It is an important value that arises in the context of integrated rate laws, a topic covered in section 12.4 of the chemistry curriculum.
Ln[A]0: ln[A]0 is a mathematical expression used in the context of integrated rate laws, which describe the relationship between the concentration of a reactant and time in a chemical reaction. It represents the natural logarithm of the initial concentration of the reactant, A, at time t = 0.
Ln[A]t: 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.
M-1s-1: M-1s-1 is a unit used to express the rate constant in chemical kinetics, specifically in the context of integrated rate laws. It represents the change in concentration of a reactant or product per unit of time, where the concentration is measured in moles per liter (M) and the time is measured in seconds (s).
M/s: M/s is a unit of velocity, specifically meters per second, which is commonly used to measure the rate of change in a chemical reaction. It is an important concept in the context of Integrated Rate Laws, as it provides a quantitative way to describe the speed at which a reaction is progressing over time.
Order of Reaction: The order of a reaction refers to the relationship between the rate of the reaction and the concentrations of the reactants. It is a measure of how the rate of a chemical reaction changes with changes in the concentrations of the reactants involved.
Rate constant: The rate constant, often denoted as $k$, is a proportionality factor in the rate equation that relates the reaction rate to the concentration of reactants. Its value is specific to a particular reaction and changes with temperature.
Rate Constant: The rate constant is a measure of the speed or rate at which a chemical reaction occurs. It is a fundamental parameter that describes the intrinsic reactivity of the reactants and the reaction mechanism, and it is an essential component in understanding and predicting the kinetics of chemical processes.
Reaction Kinetics: Reaction kinetics is the study of the rate at which chemical reactions occur and the factors that influence those rates. It provides a quantitative description of the progress of a chemical reaction over time, allowing for the prediction and control of reaction rates in various applications.
S-1: s-1 is a unit of measurement that represents the rate of change or the speed at which a reaction is occurring. It is commonly used in the context of integrated rate laws, which describe the relationship between the concentration of reactants and the time elapsed during a chemical reaction.
Second-Order Reaction: A second-order reaction is a chemical reaction where the rate of the reaction is proportional to the square of the concentration of one of the reactants or to the product of the concentrations of two reactants. This type of reaction is important in understanding the kinetics and mechanisms of chemical processes.
T1/2: t1/2, or half-life, is the time it takes for the concentration or amount of a substance to decrease to half of its initial value. It is a fundamental concept in the study of chemical kinetics and is particularly relevant in the context of 12.4 Integrated Rate Laws.
Zero-Order Reaction: A zero-order reaction is a chemical reaction where the rate of the reaction is independent of the concentration of the reactants. In other words, the reaction rate remains constant regardless of how much of the reactants are present.
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