Glossary

11.2 Integrated rate laws

4 min read•july 30, 2024

Integrated rate laws are crucial tools in chemical kinetics, helping us understand how reactant concentrations change over time. They provide mathematical equations for different reaction orders, allowing us to calculate concentrations, determine reaction rates, and predict half-lives.

By applying these laws, we can analyze experimental data to determine reaction orders and rate constants. This knowledge is essential for understanding reaction mechanisms and predicting the behavior of chemical systems in various applications.

Integrated Rate Laws

Deriving Integrated Rate Laws for Different Reaction Orders

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  • Derive integrated rate law for zero-order reactions
    • Equation: [A]=−kt+[A]0[A] = -kt + [A]₀
      • [A][A] concentration of reactant A at time t
      • kk rate constant
      • [A]0[A]₀ initial concentration of A
  • Derive integrated rate law for first-order reactions
    • Equation: ln[A]=−kt+ln[A]0ln[A] = -kt + ln[A]₀
      • [A][A] concentration of reactant A at time t
      • kk rate constant
      • [A]0[A]₀ initial concentration of A
  • Derive integrated rate law for second-order reactions
    • Equation: 1/[A]=kt+1/[A]01/[A] = kt + 1/[A]₀
      • [A][A] concentration of reactant A at time t
      • kk rate constant
      • [A]0[A]₀ initial concentration of A

Applying Integrated Rate Laws to Calculate Concentrations

  • Calculate concentration of reactant at given time using integrated rate law for specific reaction order
    • Requires knowledge of initial concentration and rate constant
  • : [A]=−kt+[A]0[A] = -kt + [A]₀
    • Example: If k=0.5M/mink = 0.5 M/min and [A]0=2M[A]₀ = 2 M, calculate [A][A] at t=3mint = 3 min
  • : [A]=[A]0e(−kt)[A] = [A]₀e^(-kt)
    • Example: If k=0.2min(−1)k = 0.2 min^(-1) and [A]0=1.5M[A]₀ = 1.5 M, calculate [A][A] at t=5mint = 5 min
  • : [A]=1/(kt+1/[A]0)[A] = 1/(kt + 1/[A]₀)
    • Example: If k=0.1M(−1)min(−1)k = 0.1 M^(-1)min^(-1) and [A]0=4M[A]₀ = 4 M, calculate [A][A] at t=2mint = 2 min

Determining Reaction Order

Analyzing Concentration-Time Data

  • Determine reaction order by analyzing relationship between concentration and time
  • Plot concentration vs. time for zero-order, ln(concentration) vs. time for first-order, or 1/concentration vs. time for second-order
    • Straight line indicates reaction follows specific order
  • Slope of straight line determines
  • Y-intercept corresponds to initial concentration ([A]0[A]₀) for zero-order, ln([A]0)ln([A]₀) for first-order, or 1/[A]01/[A]₀ for second-order

Determining Rate Constant from Appropriate Plot

  • Use slope of straight line obtained from appropriate plot to determine rate constant (k)
    • Zero-order: k=−slopek = -slope
    • First-order: k=−slopek = -slope
    • Second-order: k=slopek = slope
  • Example: Plot concentration vs. time for a reaction and obtain a straight line with slope −0.03M/min-0.03 M/min. Reaction is zero-order with k=0.03M/mink = 0.03 M/min
  • Example: Plot ln(concentration) vs. time for a reaction and obtain a straight line with slope −0.15min(−1)-0.15 min^(-1). Reaction is first-order with k=0.15min(−1)k = 0.15 min^(-1)

Half-Life and Integrated Rate Laws

Relationship between Half-Life and Integrated Rate Laws

  • Half-life (t1/2t_{1/2}) time required for concentration of reactant to decrease to half of initial value
  • Zero-order reaction: t1/2=[A]0/(2k)t_{1/2} = [A]₀/(2k)
    • [A]0[A]₀ initial concentration
    • kk rate constant
    • Half-life depends on initial concentration
  • First-order reaction: t1/2=ln(2)/kt_{1/2} = ln(2)/k
    • kk rate constant
    • Half-life independent of initial concentration
  • Second-order reaction: t1/2=1/(k[A]0)t_{1/2} = 1/(k[A]₀)
    • [A]0[A]₀ initial concentration
    • kk rate constant
    • Half-life depends on initial concentration

Calculating Half-Life for Different Reaction Orders

  • Use half-life equations for zero-order, first-order, and second-order reactions to solve problems
    • Example: For a zero-order reaction with k=0.02M/mink = 0.02 M/min and [A]0=1.6M[A]₀ = 1.6 M, calculate t1/2t_{1/2}
    • Example: For a first-order reaction with k=0.1min(−1)k = 0.1 min^(-1), calculate t1/2t_{1/2}
    • Example: For a second-order reaction with k=0.5M(−1)min(−1)k = 0.5 M^(-1)min^(-1) and [A]0=0.8M[A]₀ = 0.8 M, calculate t1/2t_{1/2}

Solving Integrated Rate Law Problems

Applying Integrated Rate Laws

  • Use appropriate integrated rate law based on given information (initial concentration, rate constant, time) to:
    • Calculate concentration of reactant at specific time
    • Calculate time required to reach certain concentration
  • Example: For a first-order reaction with k=0.2min(−1)k = 0.2 min^(-1) and [A]0=2M[A]₀ = 2 M, calculate [A][A] at t=10mint = 10 min
  • Example: For a second-order reaction with k=0.05M(−1)min(−1)k = 0.05 M^(-1)min^(-1) and [A]0=1.2M[A]₀ = 1.2 M, calculate time required for [A][A] to reach 0.3M0.3 M

Using Half-Life to Solve Problems

  • Use half-life equations for zero-order, first-order, and second-order reactions to solve problems related to:
    • Time required for reactant to reach half of initial concentration
    • Determining rate constant from given half-life
  • Example: For a zero-order reaction with [A]0=2.4M[A]₀ = 2.4 M and t1/2=30mint_{1/2} = 30 min, calculate rate constant kk
  • Example: For a first-order reaction with k=0.08min(−1)k = 0.08 min^(-1), calculate time required for reactant to reach one-fourth of its initial concentration

Analyzing Concentration-Time Data to Determine Reaction Order and Rate Constant

  • Plot appropriate graph (concentration vs. time, ln(concentration) vs. time, or 1/concentration vs. time) to determine reaction order
  • Determine rate constant from slope and y-intercept of resulting straight line
  • Example: Given concentration-time data, plot ln(concentration) vs. time and obtain a straight line with slope −0.12min(−1)-0.12 min^(-1) and y-intercept −0.22-0.22. Determine reaction order and calculate rate constant and initial concentration

Key Terms to Review (18)

Activation energy: Activation energy is the minimum energy required for a chemical reaction to occur. It serves as a barrier that reactants must overcome for the reaction to proceed, influencing reaction rates and mechanisms. Understanding this concept helps in analyzing how changes in temperature, concentration, and the presence of catalysts affect reaction dynamics.
Arrhenius Equation: The Arrhenius equation describes how the rate of a chemical reaction depends on temperature and activation energy. It shows that as the temperature increases, the reaction rate typically increases, highlighting the connection between kinetic energy and molecular collisions. This equation is crucial for understanding reaction kinetics, linking to concepts like ionic conductivity, rate laws, and transition states.
Concentration vs. time graph: A concentration vs. time graph is a visual representation that shows how the concentration of a reactant or product changes over time during a chemical reaction. This type of graph is essential for understanding reaction kinetics, as it allows for the determination of the rate of reaction and helps in deriving integrated rate laws that describe the relationship between concentration and time.
Elementary step: An elementary step is a single reaction event that occurs as part of a larger chemical reaction mechanism. These steps are the building blocks of complex reactions and can involve the formation or breaking of chemical bonds, typically represented as a simple equation showing reactants turning into products. Understanding elementary steps is crucial for analyzing how a reaction proceeds, determining its rate, and forming the overall reaction pathway.
Environmental Degradation Rates: Environmental degradation rates refer to the speed at which natural resources and ecosystems are depleted or deteriorated due to human activity or natural processes. This concept is crucial for understanding how various factors, such as pollution, deforestation, and climate change, impact the environment over time. Monitoring these rates helps in formulating strategies for sustainability and conservation efforts to mitigate damage and restore ecosystems.
Equilibrium Constant: The equilibrium constant, often represented as K, quantifies the ratio of the concentrations of products to reactants at equilibrium for a given chemical reaction at a specific temperature. It provides insight into the extent of a reaction and helps determine whether reactants or products are favored in a chemical process. This concept connects closely to the notions of free energy, chemical potential, and reaction rates, illustrating how changes in conditions can shift equilibria.
First-order reaction: A first-order reaction is a type of chemical reaction where the rate of reaction is directly proportional to the concentration of a single reactant. This means that if the concentration of that reactant doubles, the rate of the reaction also doubles. Understanding this concept helps in analyzing how reaction rates change with concentration and in deriving integrated rate laws, as well as in categorizing reactions based on their molecularity.
Gas Chromatography: Gas chromatography is an analytical technique used to separate and analyze compounds that can be vaporized without decomposition. This method relies on the interaction between the sample and the stationary phase within a column, allowing for the identification and quantification of different components in a mixture. Its precision and speed make it essential in various fields, including chemistry, environmental science, and pharmaceuticals.
Half-life (t1/2): Half-life (t1/2) is the time required for the concentration of a reactant to decrease to half of its initial value during a chemical reaction. This concept is crucial in understanding the rate at which reactants are consumed and products are formed, particularly in first-order reactions, where the half-life remains constant regardless of the concentration of the reactant. It provides insights into reaction kinetics and helps predict how long it will take for a reaction to reach a certain level of completion.
Integrated Rate Equation: The integrated rate equation is a mathematical expression that relates the concentration of reactants or products in a chemical reaction to time. It provides a way to determine the concentration of a species at any given point during a reaction, based on its initial concentration and the rate constant. Understanding these equations is crucial for analyzing how fast reactions occur and how they change over time.
Pharmaceutical drug kinetics: Pharmaceutical drug kinetics is the study of how drugs move through the body over time, focusing on their absorption, distribution, metabolism, and excretion. This field helps in understanding how long a drug stays active in the body and how effectively it reaches its target, which is crucial for determining dosing regimens and ensuring therapeutic efficacy.
Pseudo-first-order reaction: A pseudo-first-order reaction occurs when a reaction that is actually second-order in nature behaves like a first-order reaction because one reactant is present in a large excess compared to the other. This condition simplifies the rate law, allowing for easier analysis of concentration changes over time. In such cases, the concentration of the excess reactant remains relatively constant, making it possible to observe an apparent first-order kinetics in the remaining reactant.
Rate constant (k): The rate constant (k) is a proportionality factor in the rate equation that relates the rate of a chemical reaction to the concentrations of the reactants. It is a key indicator of the reaction's speed and varies depending on factors like temperature and the presence of catalysts. The value of k is crucial for determining half-lives and understanding how reaction order influences kinetics.
Rate Law Constant: The rate law constant, often represented as 'k', is a proportionality constant in the rate law equation that relates the rate of a chemical reaction to the concentration of the reactants. It is specific to a particular reaction at a given temperature and indicates how the reaction rate changes with varying concentrations. Understanding this constant is crucial for analyzing reaction kinetics and predicting how fast a reaction will occur based on reactant concentrations.
Reaction intermediate: A reaction intermediate is a temporary species formed during a chemical reaction that exists between the reactants and products. These intermediates are crucial for understanding the mechanism of a reaction, as they provide insight into the steps involved in transforming reactants into products. Their presence can influence reaction rates, as they often have distinct energy levels and can be involved in rate-determining steps.
Second-order reaction: A second-order reaction is a type of chemical reaction where the rate of reaction is directly proportional to the square of the concentration of one reactant or the product of the concentrations of two different reactants. This means that if you double the concentration of a reactant, the rate will increase by a factor of four. Understanding second-order reactions is essential as they often involve more complex kinetics and provide insights into molecular interactions during reactions.
Spectrophotometry: Spectrophotometry is an analytical technique that measures the amount of light absorbed by a sample at various wavelengths, allowing for the determination of concentration and other properties of the sample. This method relies on the interaction of electromagnetic radiation with matter, which can be utilized to study reaction kinetics and the electronic structure of molecules.
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 even if the concentration of the reactants changes, the rate remains the same, leading to a linear decrease in concentration over time. Zero-order kinetics often occurs in situations where a reactant is present in excess or when a catalyst is involved, making it crucial for understanding reaction rates and mechanisms.
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