Integrated Rate Equations
Zero and first order equations with half-life expressions. (Chemistry › Chemical Kinetics, NEET UG syllabus.)
What is Integrated Rate Equations?
An equation that expresses the concentration of a reactant or product as a function of time, derived by integrating the differential rate law.
Key formula / rule: Integrated Rate Equation (Zero-Order)
Key points
- Derive integrated rate equations for zero and first-order reactions.
- Calculate the rate constant (k) and half-life (t1/2) for zero and first-order reactions.
- Predict the concentration of a reactant at a given time or the time required for a given concentration change.
- Interpret graphical representations of integrated rate laws to determine reaction order and rate constant.
Common exam trap
Confusing the integrated rate equations for zero and first-order reactions.
Definitions
- Term
Integrated Rate Equation
- Meaning
An equation that expresses the concentration of a reactant or product as a function of time, derived by integrating the differential rate law.
- Term
Rate Constant (k)
- Meaning
A proportionality constant in the rate law that relates the rate of reaction to the concentrations of reactants, specific for a given reaction at a particular temperature.
- Term
Half-life (t1/2)
- Meaning
The time required for the concentration of a reactant to decrease to half of its initial value during a chemical reaction.
Learning objectives
Derive integrated rate equations for zero and first-order reactions.
Calculate the rate constant (k) and half-life (t1/2) for zero and first-order reactions.
Predict the concentration of a reactant at a given time or the time required for a given concentration change.
Interpret graphical representations of integrated rate laws to determine reaction order and rate constant.
Solve numerical problems based on integrated rate equations and half-life.
Formulae
- Name
Integrated Rate Equation (Zero-Order)
- Note
Relates concentration at time t to initial concentration and rate constant.
- Expression
[A]t = [A]0 - kt
- Name
Half-life (Zero-Order)
- Note
Time for concentration to halve; depends on initial concentration.
- Expression
t1/2 = [A]0 / 2k
- Name
Integrated Rate Equation (First-Order, natural log form)
- Note
Relates concentration at time t to initial concentration and rate constant.
- Expression
ln[A]t = ln[A]0 - kt
- Name
Integrated Rate Equation (First-Order, base-10 log form)
- Note
Commonly used for calculations involving base-10 logarithms.
- Expression
k = (2.303/t) log([A]0/[A]t)
- Name
Half-life (First-Order)
- Note
Time for concentration to halve; independent of initial concentration.
- Expression
t1/2 = 0.693 / k
Prerequisites
Basic understanding of differential calculus (integration).
Concept of reaction rate and rate law.
Definition of order of reaction.
Knowledge of logarithms and exponential functions.
Common mistakes
Confusing the integrated rate equations for zero and first-order reactions.
Incorrectly applying natural logarithm (ln) vs base-10 logarithm (log) in calculations.
Errors in calculating half-life, especially for zero-order reactions where it depends on initial concentration.
Misinterpreting the slopes and intercepts of concentration-time graphs.
Forgetting to convert units (e.g., time from minutes to seconds) before calculation.
Keywords
Integrated rate law
zero-order reaction
first-order reaction
half-life
rate constant
concentration-time graphs
chemical kinetics
Practice preview
Which of the following expressions correctly represents the integrated rate law for a zero-order reaction?…
easy
A zero-order reaction A → B has a rate constant of 0.02 mol L⁻¹ s⁻¹. If the initial concentration of A is 0.5 M, what will be the concentration of A after 10 seconds?…
medium
Which of the following statements is INCORRECT regarding the half-life (t₁/₂) of a reaction?…
medium