Integrated Rate Law Problems

Rate Law Expression for Ordered Reactions

• Zero Order Reaction: Rate = k
  • Rate is independent of reactant concentration.
• First Order Reaction: Rate = k[A]^1
  • Rate is proportional to reactant concentration.
  • Doubling concentration doubles the rate.
• Second Order Reaction: Rate = k[A]^2
  • Rate changes with the square of concentration changes.
  • Doubling concentration quadruples the rate.

Units of the Rate Constant (k)

• Zero Order: M^1 T^-1
• First Order: T^-1
• Second Order: M^-1 T^-1
• General Formula: M^(1-n) T^-1
  • n = order of reaction

Half-Life (t₁/₂)

• Zero Order: (Initial Concentration) / 2k
• First Order: ln(2) / k
  • Independent of initial concentration.
• Second Order: 1 / (k * Initial Concentration)*

Integrated Rate Law Expressions

• Zero Order: [A] = -kt + [A]₀
  • Plot: [A] vs. time (linear)
• First Order: ln[A] = -kt + ln[A]₀
  • Plot: ln[A] vs. time (linear)
• Second Order: 1/[A] = kt + 1/[A]₀
  • Plot: 1/[A] vs. time (linear)

Key Concepts

• Order of Reaction: Determines how concentration affects rate.
• Slope of Graph: Indicates rate constant:
  • Zero Order: Negative slope
  • First Order: Negative slope
  • Second Order: Positive slope

Example Problems and Solutions

Problem 1

• Determine units of k for third-order reaction:
  • Units: M^-2 T^-1 (Example Choice: C)

Problem 2

• Straight-line plot for first-order reaction:
  • ln[A] vs. time with slope negative k (Example Choice: D)

Problem 3

• Analyze statements about half-life:
  • True: Half-life of first-order is independent of concentration.
  • False (Correct): Half-life of zero-order is not constant (Example Correct Choice: D)

Problem 4

• Zero-order reaction concentration calculations:
  • Use equation: [A] = -kt + [A]₀

Problem 5

• Calculate initial concentration from given data:
  • Utilize units of k to identify reaction order.

Problem 6

• Determine rate constant from concentration vs. time data:
  • Slope of straight line = -k

Problem 7

• Calculate half-life for first-order reaction:
  • Use formula: ln(2) / k

Problem 8

• Identify plot for first order: ln[A] vs. time

Problem 9

• Rate law expression determination from reaction order:
  • Overall order = 5
  • Determine units of k accordingly

Problem 10

• Determine overall order from given units:
  • Liters^3 moles^-3 seconds^-1 indicates fourth-order (Example Correct Choice: E)