Chemical Kinetics (Pt. 5)

The Second-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.

Recall: Differential Rate Laws

RECALL: (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].)

Determining Reaction Order with Integrated Rate Laws

1) Collect concentration data versus time.

2) To determine if the reaction is second order, calculate the value of for each concentration.

3) Plot versus time. If it’s a straight line, it’s second order!

Second-Order Integrated Rate Law

Using calculus to integrate the differential rate law for a second-order process gives us

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.

Second-Order Integrated Rate Law

Plotting versus time will yield a line if the process is second order.

Y = mx + b

Second-Order Plots

Graphs for a Second Order Reaction from http://2012books.lardbucket.org/books/principles-of-general-chemistry-v1.0m/s18-03-methods-of-determining-reactio.html

1[A ]t

=𝑘 t+ 1[A ]0plot of [A ] vs time

Determining Reaction Order using Integrated Rate Laws

Step 1: Collect concentration versus time data.

Step 2: Calculate the reciprocal for each concentration measured. (1/[A])

Time [A] 1/[A]0 0.0400 25.00010 0.0303 33.00320 0.0244 40.98430 0.0204 49.02040 0.0175 57.143

Determining Rxn Order using Integrated Rate Laws

Step 3: Graph 1/[A] vs. time

The plot shows a straight line.

The reaction fits 2nd order kinetics.

Plot for a Second Order Reaction

k is the “rate constant”

The slope of the line is k.

k = 0.803 M1 s1

Half-Life for a Second-Order Process

Deriving the half life for a second order process:Let [A]t = 0.5[A]0

Subtract from both sides…

Half-Life for a Second-Order Process

Subtract from both sides…

Time is now labeled for half life with a subscript (t1/2)

Half-Life for a Second-Order Process

Note that the half life for a second-order process DOES depend on the initial

concentration [A]0

Example Problemswill be posted separately.

Next up, Microscopic Aspects of Kinetics- Chemical Reaction Mechanisms

(Pt 6)