Integrated Rate Law Equations

Integrated rate laws describe how the concentration of reactants changes over time in chemical reactions. Understanding zero, first, and second-order reactions helps predict reaction behavior, calculate half-lives, and analyze data, making it essential in the study of chemical kinetics.

1. Zero-order integrated rate law

   • The rate of reaction is constant and independent of the concentration of reactants.
   • The integrated rate law is expressed as [A] = [A]₀ - kt, where [A]₀ is the initial concentration, k is the rate constant, and t is time.
   • The reaction proceeds at a steady rate until the reactant is depleted.

2. First-order integrated rate law

   • The rate of reaction is directly proportional to the concentration of one reactant.
   • The integrated rate law is given by ln[A] = ln[A]₀ - kt, which can also be expressed as [A] = [A]₀e^(-kt).
   • This order is commonly observed in radioactive decay and many organic reactions.

3. Second-order integrated rate law

   • The rate of reaction is proportional to the square of the concentration of one reactant or the product of the concentrations of two reactants.
   • The integrated rate law is 1/[A] = 1/[A]₀ + kt.
   • This order is often seen in reactions involving bimolecular collisions.

4. Half-life equations for each order

   • Zero-order: t₁/₂ = [A]₀ / (2k), half-life decreases as concentration decreases.
   • First-order: t₁/₂ = 0.693 / k, half-life is constant and independent of concentration.
   • Second-order: t₁/₂ = 1 / (k[A]₀), half-life increases as concentration decreases.

5. Graphical representations of each order

   • Zero-order: A plot of [A] vs. time is linear with a slope of -k.
   • First-order: A plot of ln[A] vs. time is linear with a slope of -k.
   • Second-order: A plot of 1/[A] vs. time is linear with a slope of k.

6. Units of rate constants for each order

   • Zero-order: Units are M/s (molarity per second).
   • First-order: Units are s⁻¹ (inverse seconds).
   • Second-order: Units are M⁻¹s⁻¹ (inverse molarity per second).

7. Relationship between concentration and time for each order

   • Zero-order: Concentration decreases linearly over time.
   • First-order: Concentration decreases exponentially over time.
   • Second-order: Concentration decreases in a hyperbolic manner over time.

8. Methods for determining reaction order from experimental data

   • Method of initial rates: Compare initial rates at varying concentrations.
   • Integrated rate laws: Analyze concentration vs. time data to determine linearity.
   • Half-life analysis: Examine how half-life changes with concentration.

9. Integrated rate law for parallel reactions

   • For parallel reactions, the rate law can be expressed as r = k₁[A] + k₂[B], where k₁ and k₂ are the rate constants for each pathway.
   • Each reaction pathway can be analyzed separately to determine individual rate laws.

10. Integrated rate law for consecutive reactions

    • For consecutive reactions, the rate law can be complex, often requiring differential equations to solve.
    • The integrated rate law for a simple two-step reaction A → B → C can be expressed using rate constants k₁ and k₂, with the concentration of intermediates needing special consideration.