1. Integrated Rate Laws 1 1

2. How are “integrated rate laws” different from “rate laws?”
A rate law gives the speed of reaction at a given point in time, and how it is affected by the concentration of the reactants.
An integrated rate law gives the change in concentration from time zero up to a given point in time. It describes how the rate changes over time. 2 2

3. Integrated Rate Law
The following formulae are given on the AP data sheet: They can be derived using calculus:
First order reactions
Second order reactions
Chem FAQ: How do I calculate reaction rate using a rate law? 3

4. Use of the Integrated Rate of 1st Order Rxns
Example Dinitrogen pentoxide is not very stable. In the gas phase or dissolved in a nonaqueous solvent, like CCl4, it decomposes by a 1st order rxn into dinitrogen tetroxide and molecular oxygen.
2 N2O  N2O O2
The rate law is Rate = k[N2O5]
At 45oC, the rate constant for the rxn is 6.22 x 10-4s-1. If the initial concentration of N2O5 at 45oC is M, what will its concentration be after exactly one hour?
Ans M 4

5. Half-life (t½) of a Reactant
"Half-life" of a reactant is the time it takes for ½ of the reactant to react.
It is NOT half the time it takes for all of the reactant to react.
For example, t½ of the radioisotope I-131 is 8.0 days. Starting with 20 g of I-131, after 8.0 days, there would be 10 g left.
After a total of 16.0 days (two half-lives), there would be……
5 g left

6. Application of Half-life
I-131 has a half-life of 8 days.
After 4 half-lives (32 days), it is down to 1/16 of its original concentration.

7. If we start with 20.0 g of I-131, after 40 days, how much of I-131 is still there? (Half-life of I-131 = 8 days.) What percent of I-131 remains after 24 days? What fraction of I-131 remains after 48 days?

8. Example: A patient is given a certain amt of I-131 as part of a diagnostic procedure for a thyroid disorder. What fraction of the initial I-131 would be present in a patient after 25 days if none of it were eliminated through natural body processes?
t1/2 = 8.02 days What to do if # days is not an exact multiple of the half-life?
We solve this problem using the integrated rate law for 1st order rxn?

9. Half-life is a measure of the speed of reaction: The shorter the half-life, the faster is the reaction.
By definition, after one half-life,
[A]t = ½ [A]o
For a 1st order rxn,
Note that half-life is not affected by concentration.
How do you know that?
There is no concentration
term in the equation.

10. Half-life = 8.02 days, after 25 days?

11. The half-life of I-132 is 2.295h. What percentage remains after 24 hours? (Previous problem was for I-131.)
% is based on 100
Ans % I-132 remains.

12. Integrated Rate Law of 2nd Order Reactions
They are of several types:
Rate=k[A]2,
Rate=k[A]1[B]1 and
Rate=k[A]2[B]0, etc…
2nd order means the powers must add up to 2.
The integrated equation is of the form
ChemFAQ: How does concentration vary with time in a second order reaction?

13. Example:Nitrosyl chloride, NOCl, decomposes slowly to NO and Cl2.
2NOCl NO Cl2
The rate law shows that the rate is second order in NOCl. Rate = k[NOCl]2
The rate constant k equals L mol-1 s-1 at a certain temp. If the initial conc of NOCl in a closed reaction vessel is M, what will the concentration be after 35 minutes?
Ans M

14. Half-life of 2nd Order Reactions
Derived from
Note that unlike 1st order reactions, half-life of 2nd order rxns depend on the initial conc of the reactant.
2HI(g)  H2(g) + I2 (g)
has the rate law, Rate = k[HI]2 with k = L mol-1s-1 at 508oC. What is the half-life of this rxn at this temp when the initial HI concentration is 0.10 M?
Ans. 1.3x102s

15. Using a graph to find K for 2nd Order Reactions
How can we determine rate constant graphically? Which are the variables?
Rearrange
ChemFAQ: How does concentration vary with time in a second order reaction?
y = mx + b
What do we plot on the x-axis? y-axis? What does the slope represent?

16. Using a graph to find K for 1st Order Reactions
Slope is calculated to be
- 6.0x10-4 s-1.
So, k = 6.0x10-4 s-1
Rate = k[N2O5] Units are correct!
What is the slope equal to?
slope = - k
So, k = - slope
Note: (-) does not mean k is negative but it has the opposite sign of the slope.

17. Determining the order of a reaction by seeing which plot gives a striaght line.
Second order:
Zero order:[A] vs time
First order
ln [A] vs time:

18. Summary of Integrated Rate Laws
1st Order Rxn
2nd Order Rxn