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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.
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].)
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.
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!
First-Order Integrated Rate Law
Using calculus to integrate the differential rate law for a first-order process gives us
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.
First-Order Integrated Rate Law
Rearrange this equation…
This is a linear equation!
Use log rule:
First-Order Integrated Rate Law
A first-order reaction is an exponential decay (in terms of reactant).
The concentration of reactant A decreases exponentially over time.
First-Order PlotsGraphs 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 ]𝟎𝒆−𝒌 𝐭 𝐥𝐧 [ 𝑨 ]𝒕=−𝒌𝐭+𝐥𝐧 [ 𝑨 ]𝟎
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 -1.3862960 0.218 -1.5232690 0.204 -1.58964
120 0.19 -1.66073180 0.166 -1.79577
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.
Determining Rxn Order using Integrated Rate Laws
k is the “rate constant”
The slope of the line is k.
k = 0.0023 s1
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]0The 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
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
Use log rule:
Half-Life for a First Order Process
Time is now labeled for half life with a subscript (t1/2)
Half-Life for a First Order Process
Cancel negative signs and solve for t1/2
Ln 0.5 is just a number (put it in your calculator!)
Half-Life for a First Order Process
Note that the half life for a first-order process does not depend on the initial concentration [A]0
Example Problemswill be posted separately.
Next up, The Second Order
Integrated Rate Law (Pt 5)