Chemical Kinetics (Pt. 4)

The First-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.

Differential Rate Laws

(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].)

Integrated Rate Laws

Use calculus to integrate the (differential) rate law for each of three common reaction orders.

Now, we have a practical way to determine the order of a reaction.

Determining Reaction Order using Integrated Rate Laws

1) In an experiment, collect concentration data versus time.

2) To determine if the reaction is first order, calculate the ln[A] of each concentration.

3) Plot ln[A] versus time. If it’s a straight line, it’s first order!

First-Order Integrated Rate Law

Using calculus to integrate the differential rate law for a first-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.

First-Order Integrated Rate Law

Rearrange this equation…

This is a linear equation!

Use log rule:

First-Order Integrated Rate Law

A first-order reaction is an exponential decay (in terms of reactant).

The concentration of reactant A decreases exponentially over time.

First-Order PlotsGraphs for a first-order reaction:

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

[ 𝑨 ]𝒕= [𝑨 ]𝟎𝒆−𝒌 𝐭   𝐥𝐧 [ 𝑨 ]𝒕=−𝒌𝐭+𝐥𝐧 [ 𝑨 ]𝟎

Determining Reaction Order using Integrated Rate Laws

Step 1: Collect concentration versus time data.

Step 2: Calculate the natural log for each concentration measured. (ln [A])

Time [A] ln[A]0 0.25 -1.3862960 0.218 -1.5232690 0.204 -1.58964

120 0.19 -1.66073180 0.166 -1.79577

Determining Rxn Order using Integrated Rate Laws

Step 3: Graph ln [A] vs. time

The plot shows a straight line.

The reaction fits 1st order kinetics.

Determining Rxn Order using Integrated Rate Laws

k is the “rate constant”

The slope of the line is k.

k = 0.0023 s1

Half Life for First-Order Reactions

Half-life is defined as the time required for one-half of a reactant to react.

Because [A] at t1/2 is one-half of the original concentration of A,

[A]t = 0.5 [A]0The Half Life of a First Order Reaction from http://2012books.lardbucket.org/books/principles-of-general-chemistry-v1.0m/s18-05-half-lives-and-radioactive-dec.html

Half-Life for a First Order Process

Deriving an expression for the half-life of a first-order process:Let [A]t = 0.5[A]0

Use log rule:

Half-Life for a First Order Process

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

Half-Life for a First Order Process

Cancel negative signs and solve for t1/2

Ln 0.5 is just a number (put it in your calculator!)

Half-Life for a First Order Process

Note that the half life for a first-order process does not depend on the initial concentration [A]0

Example Problemswill be posted separately.

Next up, The Second Order

Integrated Rate Law (Pt 5)