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Chem 2 - Chemical Kinetics IV: The First-Order Integrated Rate Law

  1. 1. Chemical Kinetics (Pt. 4) The First-Order Integrated Rate Law By Shawn P. Shields, Ph.D. This work is licensed by Shawn P. Shields-Maxwell under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
  2. 2. Differential Rate Laws (Differential) Rate Laws for 3 common reaction orders: First Order: Rate = k [A]1 Second Order: Rate = k [A]2 Zero Order: Rate = k [A]0 (No dependence of reaction rate on [A].)
  3. 3. Integrated Rate Laws Use calculus to integrate the (differential) rate law for each of three common reaction orders. Now, we have a practical way to determine the order of a reaction.
  4. 4. Determining Reaction Order using Integrated Rate Laws 1) In an experiment, collect concentration data versus time. 2) To determine if the reaction is first order, calculate the ln[A] of each concentration. 3) Plot ln[A] versus time. If it’s a straight line, it’s first order! 
  5. 5. First-Order Integrated Rate Law Using calculus to integrate the differential rate law for a first-order process gives us ln A t A 0 = βˆ’π‘˜t Where, [A]0 is the initial concentration of A, and [A]t is the concentration of A at some time, t, during the course of the reaction.
  6. 6. First-Order Integrated Rate Law Rearrange this equation… ln A t A 0 = βˆ’π‘˜t ln A t βˆ’ ln A 0 = βˆ’π‘˜t ln A t = βˆ’π‘˜t + ln A 0 This is a linear equation! Use log rule: π₯𝐧 𝐱 𝐲 = π₯𝐧 𝐱 βˆ’ π₯𝐧 𝐲
  7. 7. First-Order Integrated Rate Law A first-order reaction is an exponential decay (in terms of reactant). A t = A 0 π‘’βˆ’π‘˜t The concentration of reactant A decreases exponentially over time.
  8. 8. First-Order Plots Graphs for a first-order reaction: Graphs for a First Order Reaction from http://2012books.lardbucket.org/books/principles-of-general-chemistry- v1.0m/s18-03-methods-of-determining-reactio.html 𝑨 𝒕 = 𝑨 𝟎 π’†βˆ’π’Œπ­ π₯𝐧 𝑨 𝒕 = βˆ’π’Œπ­ + π₯𝐧 𝑨 𝟎
  9. 9. Determining Reaction Order using Integrated Rate Laws Step 1: Collect concentration versus time data. Step 2: Calculate the natural log for each concentration measured. (ln [A]) Time [A] ln[A] 0 0.25 -1.38629 60 0.218 -1.52326 90 0.204 -1.58964 120 0.19 -1.66073 180 0.166 -1.79577
  10. 10. Determining Rxn Order using Integrated Rate Laws Step 3: Graph ln [A] vs. time The plot shows a straight line. The reaction fits 1st order kinetics.
  11. 11. Determining Rxn Order using Integrated Rate Laws π₯𝐧 𝑨 𝒕 = βˆ’π’Œπ­ + π₯𝐧 𝑨 𝟎 k is the β€œrate constant” The slope of the line is ο€­k. k = 0.0023 sο€­1
  12. 12. Half Life for First-Order Reactions Half-life is defined as the time required for one-half of a reactant to react. Because [A] at t1/2 is one-half of the original concentration of A, [A]t = 0.5 [A]0 The Half Life of a First Order Reaction from http://2012books.lardbucket.org/books/principles-of-general-chemistry- v1.0m/s18-05-half-lives-and-radioactive-dec.html
  13. 13. Half-Life for a First Order Process Deriving an expression for the half-life of a first-order process: Let [A]t = 0.5[A]0 ln 0.5 A 0 = βˆ’π‘˜t + ln A 0 ln 0.5 A 0 βˆ’ ln A 0 = βˆ’π‘˜t Use log rule: π₯𝐧 𝐱 𝐲 = π₯𝐧 𝐱 βˆ’ π₯𝐧 𝐲
  14. 14. Half-Life for a First Order Process ln 0.5 A 0 βˆ’ ln A 0 = βˆ’π‘˜t ln 0.5 A 0 A 0 = βˆ’π‘˜t1 2 ln 0.5 A 0 A 0 = βˆ’π‘˜t1 2 π₯𝐧 𝟎. πŸ“ = βˆ’π’Œπ­ 𝟏 𝟐 Time is now labeled for half life with a subscript (t1/2)
  15. 15. Half-Life for a First Order Process π₯𝐧 𝟎. πŸ“ = βˆ’π’Œπ­ 𝟏 𝟐 βˆ’0.693 = βˆ’π‘˜t1 2 Cancel negative signs and solve for t1/2 𝒕 𝟏 𝟐 = 𝟎. πŸ”πŸ—πŸ‘ π’Œ Ln 0.5 is just a number (put it in your calculator!)
  16. 16. Half-Life for a First Order Process 𝒕 𝟏 𝟐 = 𝟎. πŸ”πŸ—πŸ‘ π’Œ Note that the half life for a first-order process does not depend on the initial concentration [A]0
  17. 17. Example Problems will be posted separately. Next up, The Second Order Integrated Rate Law (Pt 5)

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Chem 2 - Chemical Kinetics IV: The First-Order Integrated Rate Law

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