Glossary

4.1 Zero-order integrated rate law

2 min read•july 22, 2024

Zero-order reactions are unique in chemical kinetics. They occur when the stays constant, regardless of reactant concentration. This behavior is often seen in enzyme-catalyzed reactions and surface reactions.

The zero-order , [A] = [A]₀ - kt, helps predict reactant concentrations over time. It's characterized by a linear concentration vs. time plot with a negative slope. Understanding this law is crucial for analyzing and predicting behavior.

Zero-Order Integrated Rate Law

Derivation of zero-order rate law

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  • Starts with differential rate law for zero-order reaction d[A]dt=−k\frac{d[A]}{dt} = -k
    • [A][A] represents concentration of reactant A
    • tt represents time
    • kk represents zero-order
  • Rearrange differential rate law to isolate d[A]d[A] d[A]=−k⋅dtd[A] = -k \cdot dt
  • Integrate both sides of equation ∫[A]0[A]d[A]=−k∫0tdt\int_{[A]_0}^{[A]} d[A] = -k \int_{0}^{t} dt
    • [A]0[A]_0 represents initial concentration of reactant A
  • Solve integrals [A]−[A]0=−kt[A] - [A]_0 = -kt
  • Rearrange to obtain zero-order integrated rate law [A]=[A]0−kt[A] = [A]_0 - kt

Application of zero-order rate law

  • Zero-order integrated rate law [A]=[A]0−kt[A] = [A]_0 - kt determines reactant concentration at any time tt
  • To find concentration of reactant A at any time tt
    1. Substitute initial concentration [A]0[A]_0, rate constant kk, and desired time tt into equation
    2. Solve for [A][A]
  • If [A]0=1.0 M[A]_0 = 1.0 \text{ M}, k=0.2 M/sk = 0.2 \text{ M/s}, and t=3 st = 3 \text{ s}, then [A]=1.0 M−(0.2 M/s)(3 s)=0.4 M[A] = 1.0 \text{ M} - (0.2 \text{ M/s})(3 \text{ s}) = 0.4 \text{ M}

Characteristics of zero-order reactions

  • Linear concentration vs. time plot with negative slope
    • Slope of line equals negative of zero-order rate constant, −k-k
  • y-intercept of line equals initial concentration of reactant, [A]0[A]_0
  • Reactant concentration decreases linearly with time (glucose, hydrogen peroxide)
  • Rate of reaction constant and independent of reactant concentration (enzyme-catalyzed reactions, surface reactions)

Rate constant calculation for zero-order reactions

  • Zero-order integrated rate law rearranged to solve for rate constant kk k=−[A]−[A]0tk = -\frac{[A] - [A]_0}{t}
  • To calculate kk using experimental data
    1. Determine initial concentration [A]0[A]_0 and concentration [A][A] at specific time tt
    2. Substitute values into rearranged equation and solve for kk
  • If [A]0=1.0 M[A]_0 = 1.0 \text{ M}, [A]=0.4 M[A] = 0.4 \text{ M}, and t=3 st = 3 \text{ s}, then k=−0.4 M−1.0 M3 s=0.2 M/sk = -\frac{0.4 \text{ M} - 1.0 \text{ M}}{3 \text{ s}} = 0.2 \text{ M/s}

Key Terms to Review (14)

Arrhenius: The Arrhenius concept, formulated by Svante Arrhenius in the late 19th century, describes how the rate of a chemical reaction depends on temperature and activation energy. This idea forms the basis for understanding reaction rates and is crucial in both zero-order kinetics and reaction orders, as it highlights the temperature dependency of these processes and how they can influence reaction rates.
Concentration vs. Time Graph: A concentration vs. time graph is a visual representation that shows how the concentration of a reactant or product in a chemical reaction changes over time. This type of graph is particularly useful for analyzing reaction kinetics, as it helps to identify the order of the reaction and to determine rate constants through integrated rate laws, especially in zero-order reactions where the concentration decreases linearly with time.
Constant Temperature: Constant temperature refers to a condition where the temperature of a system remains unchanged throughout a process. In the context of chemical reactions, maintaining constant temperature is crucial as it influences reaction rates and equilibrium positions, particularly in zero-order reactions where the rate is independent of concentration.
Enzyme inhibition: Enzyme inhibition refers to the process by which a molecule, known as an inhibitor, decreases or completely stops the activity of an enzyme. This can affect reaction rates and pathways, impacting overall metabolic processes. Understanding enzyme inhibition is crucial because it helps illustrate how enzymes function and how their activity can be modulated, which relates closely to the kinetics of reactions, particularly in zero-order reactions where the rate is constant and independent of substrate concentration.
Independence of Concentration: Independence of concentration refers to the characteristic of certain reactions where the rate of reaction does not depend on the concentration of reactants. In the context of zero-order reactions, this means that regardless of the changes in the concentration of the reactants, the rate remains constant and is determined solely by a different factor, often a catalyst or temperature. This unique behavior highlights the distinctive nature of zero-order kinetics, where the concentration does not influence the speed at which products are formed.
Integrated Rate Law: The integrated rate law is a mathematical equation that relates the concentration of a reactant to time for a chemical reaction. This law allows chemists to determine how the concentration of a reactant decreases over time and helps in understanding the relationship between reaction rate and concentration, which is crucial for identifying reaction orders and analyzing the kinetics of reactions.
Rate Constant: The rate constant is a proportionality factor in the rate law that quantifies the speed of a chemical reaction at a given temperature. It connects the concentration of reactants to the reaction rate, showing how quickly the reaction proceeds. The value of the rate constant is influenced by factors such as temperature, activation energy, and the presence of catalysts, making it a key element in understanding reaction kinetics and dynamics.
Reaction Mechanisms: Reaction mechanisms describe the step-by-step sequence of elementary reactions that occur during a chemical transformation. Understanding these mechanisms is crucial because they provide insight into how reactants convert into products, highlighting the molecular interactions and changes that occur throughout the reaction process.
Reaction rate: The reaction rate is a measure of how quickly reactants are converted into products in a chemical reaction, typically expressed as the change in concentration of a reactant or product per unit time. Understanding this concept is essential for analyzing how factors like temperature, concentration, and catalysts influence the speed of reactions and the mechanisms involved in these processes.
Saturation Concentration: Saturation concentration refers to the maximum concentration of a solute that can dissolve in a solvent at a given temperature and pressure. This concept is crucial in understanding the behavior of chemical reactions, especially in zero-order kinetics, where the rate of reaction is constant and independent of reactant concentration until saturation is reached.
Slope of graph: The slope of a graph represents the rate of change of one variable in relation to another, typically illustrated as a straight line in a Cartesian coordinate system. In the context of chemical kinetics, it is crucial for interpreting the relationship between reactant concentration and time, particularly in zero-order reactions, where the slope indicates how quickly reactants are consumed over time.
Substrate concentration: Substrate concentration refers to the amount of substrate present in a reaction mixture that can be converted into product by an enzyme. It plays a critical role in determining the rate of enzyme-catalyzed reactions and is fundamental in understanding how enzymes function, including their kinetics, inhibition, and reactions under various conditions.
Zero-order kinetics: Zero-order kinetics refers to a reaction rate that is independent of the concentration of the reactants. In this type of reaction, the rate remains constant over time, meaning that the same amount of reactant is consumed in each unit of time regardless of how much reactant is present. This concept is particularly important in fields like pharmacology and environmental science, as it helps predict how substances behave under certain conditions.
Zero-order reaction: A zero-order reaction is a type of chemical reaction where the rate of the reaction is constant and independent of the concentration of the reactants. This means that no matter how much reactant is present, the rate at which the reaction occurs remains unchanged, leading to a linear relationship between concentration and time. This concept is essential in understanding integrated rate laws, differential rate laws, and the fundamental principles of chemical kinetics.
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