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 disappear.
It is NOT half the time it takes for all of the reactant to disappear.
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. Integrated Rate Law of 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. Determination of the Rate Constant for the Integrated Rate Law
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. Summary of Integrated Rate Laws
1st Order Rxn
2nd Order Rxn