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

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.

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

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

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.

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.

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 = 0.693 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.

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

2nd order Integrated Rate Law Sample 2nd order reaction 2UO2+ + 4H+  U4+ + UO22+ + 2H2O If n = 2, a graph of 1/[UO2+] vs. time should be linear.

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

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

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

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.

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

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

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.

Practice 1: What is the order?

Practice 2: What is the order?