1. Integrated Rate Laws
Describe how the concentration of reactants changes over TIME
Compare this to the Differential Rate laws that describe how the concentration of the reactants affects the initial RATE

2. Integrated Rate Laws
The overall reaction order of a reaction changes the form of the integrated rate law.
We will mainly look at integrated rate laws for reactions with one reactant.
We will have integrated rate laws for ZERO, FIRST, and SECOND order reactions.

3. Uses of Integrated Rate Laws
Can be used to find
Concentration of the reactant at any time
Time necessary for a given amount of reactant to react
Time necessary for the concentration of the reactant to reach a certain level

4. 1st Order Integrated Rate Law
If n = 1, then -D[A]/Dt = k[A]
Integrate the equation:
ln [A]t = ln [A]o - kt
Can also be written as
ln ( [A]t ) = -kt
( [A]o )
t = time [A]t = concentration at time
k = rate constant [A]o = initial concentration

5. 1st Order Integrated Rate Law
Sample First Order Reaction
2N2O5  4NO2 + O2(g)
If n = 1, a graph of ln [N2O5] vs. time should be linear.

6. 1st order Integrated Rate Law
We can determine that a reaction is first order by graphing the natural log of concentration v. time
ln [ ]
time
This will be a straight line for a first order reaction.

7. 1st order Integrated Rate Law
Can be modified to find the half-life of a reaction
Half-life = time for half of the sample to react
Half-life equation takes the form
t1/2 = t1/2 = half-life
k k = rate constant
The half-life of a first order reaction does not depend on the initial concentration of the reactant.

8. 2nd order Integrated Rate Law
If n = 2, then -D[A]/Dt = k[A]2
Integrate above equation to get:
1 = kt + 1
[A]t [A]o

9. 2nd order Integrated Rate Law
Sample 2nd order reaction
2UO2+ + 4H+  U4+ + UO H2O
If n = 2, a graph of 1/[UO2+] vs. time should be linear.

10. 2nd order Integrated Rate Law
We can determine that a reaction is second order by graphing one over the concentration v. time 1
[A]
time
Doing this will give a straight line for second order reactions

11. 2nd order Integrated Rate Law
To determine the half-life, use the equation
t1/2 = 1
k [A]o
Half-life DOES depend on the concentration
Each successive half-life is longer

12. Zero order Integrated Law
If n = 0, then -D[A]/Dt = k
Integrate above expression to get:
[A]t = -kt + [A]o

13. Zero order Integrated Law
Sample Zero order reaction:
Mg + 2HCl  MgCl2 + H2
If D[H2]/Dt = k, a graph of [H2] vs. time should be linear.

14. Zero order Integrated Law
We can determine that a reaction is zero order by graphing concentration v. time
[ ]
time

15. Zero order Integrated law
To find the half-life use the equation
t1/2 = [A]o 2k
Half-life depends on concentration
Each successive half-life will be shorter

16. So, How do we determine rate laws?
Summary:
n = 0 -D[A]/Dt = k [A]t = [A]o - kt
n = 1 -D[A]/Dt = k[A] ln [A]t = ln [A]o - kt
n = 2 -D[A]/Dt = k[A]2 1/[A]t = 1/[A]o + kt
Graph [A] vs. t, ln [A] vs. t, and 1/[A] vs. t and see which graph is linear.

17. Practice 1: What is the order?

18. Practice 2: What is the order?