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

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. 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. 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. 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. 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
= −𝑘t
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. First-Order Integrated Rate Law
Rearrange this equation…
ln
A t
A 0
= −𝑘t
ln A t − ln A 0 = −𝑘t
ln A t = −𝑘t + ln A 0
This is a linear equation!
Use log rule:
𝐥𝐧
𝐱
𝐲
= 𝐥𝐧 𝐱 − 𝐥𝐧 𝐲

7. First-Order Integrated Rate Law
A first-order reaction is an exponential
decay (in terms of reactant).
A t = A 0 𝑒−𝑘t
The concentration of reactant A decreases
exponentially over time.

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 𝒆−𝒌𝐭 𝐥𝐧 𝑨 𝒕 = −𝒌𝐭 + 𝐥𝐧 𝑨 𝟎

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. 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. Determining Rxn Order using Integrated Rate Laws
𝐥𝐧 𝑨 𝒕 = −𝒌𝐭 + 𝐥𝐧 𝑨 𝟎
k is the “rate
constant”
The slope of the
line is k.
k = 0.0023 s1

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. 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 = −𝑘t + ln A 0
ln 0.5 A 0 − ln A 0 = −𝑘t
Use log rule: 𝐥𝐧
𝐱
𝐲
= 𝐥𝐧 𝐱 − 𝐥𝐧 𝐲

14. Half-Life for a First Order Process
ln 0.5 A 0 − ln A 0 = −𝑘t
ln
0.5 A 0
A 0
= −𝑘t1
2
ln
0.5 A 0
A 0
= −𝑘t1
2
𝐥𝐧 𝟎. 𝟓 = −𝒌𝐭 𝟏
𝟐
Time is now labeled
for half life with a
subscript (t1/2)

15. Half-Life for a First Order Process
𝐥𝐧 𝟎. 𝟓 = −𝒌𝐭 𝟏
𝟐
−0.693 = −𝑘t1
2
Cancel negative signs and solve for t1/2
𝒕 𝟏
𝟐
=
𝟎. 𝟔𝟗𝟑
𝒌
Ln 0.5 is just a
number (put it in
your calculator!)

16. Half-Life for a First Order Process
𝒕 𝟏
𝟐
=
𝟎. 𝟔𝟗𝟑
𝒌
Note that the half life for a first-order process
does not depend on the initial concentration [A]0

17. Example Problems
will be posted separately.
Next up,
The Second Order Integrated
Rate Law (Pt 5)