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Chemical Kinetics
Rate Laws, Arrhenius Equation - Experiments

Peter Keusch, University of Regensburg




German version





einstein "Everyone has Problems -

but Chemists have Solutions"




Zero Order Reaction

A reaction is of zero order when the rate of reaction is independent of the concentration of materials. The rate of reaction is a constant. When the limiting reactant is completely consumed, the reaction stops abruptly.

The zero order rate law for the general reaction


is written as

This means that the rate of the reaction never changes; it's always equal to the value of the rate constant.

Rearranging of equation  (1)  gives

which on integration of both sides

leads to

When  t  =  0,   the concentration of  A  is  [ A ] o.  The constant of integration must be  [ A ] o.

Now the integrated form of zero-order kinetics can be written as follows



Plotting  [ A ]  versus  t  will give a straight line with slope  -k.



First Order Reaction

For a general unimolecular reaction



where  A  is a reactant and  P  is a product, the reaction rate expression for a first order reaction is


This means that the reaction rate is proportional to the concentration of a single reactant raised to the first power.


Separation of the variables in equation  (1)

and integration of equation  (2)

yields

The constant of integration  C  can be evaluated by using boundary conditions. When  t   =  0,   [ A ]  =  [ A ] o.   [ A ] o  is the initial concentration of  A.

Substituting the values above into equation  (4)  one obtains:
Therefore the constant of integration is

Substituting equation  (6)  into equation  (5)  leads to the integrated rate law:



Plotting   ln [ A ]   or   ln [ A ] / [ A ] o  against time creates a straight line with slope  -k.  The plot should be linear up to a conversion of about 90 %.

Equation  (7)  can also be written as:
This means that the concentration of  A  decreases exponentially as a function of time.

The rate constant  k  can also be determined from the half-life  t 1 / 2.  Half-life is the time it takes for the concentration to fall from  [ A ] o  to  [ A ] o / 2  ==>  [ A ]  =  1 / 2 [ A ] o.

Equation  (7)  is rewritten as:

Thus, half-life of a first order reaction is independent of the initial concentration of the reactant.



Pseudo First Order Reaction

A  and  B  react to produce  P:



If the initial concentration of the reactant  A  is much larger than the concentration of  B,   the concentration of  A  will not change appreciably during the course of the reaction The concentration of the reactant in excess will remain almost constant. Thus the rate's dependence on  B  can be isolated and the rate law can be written


Equation  (1)  represents the differential form of the rate law. Integration of this equation and evaluation of the integration constant  C  produces the corresponding integrated law.

Substituting  [ B ]  =  c  into equation  (1)  yields


Integrating equation  (2)  gives:
The constant of integration  C  can be evaluated by using boundary conditions. At  t  =  0,   the concentration of  B  is  co.

Therefore
Accordingly, equation  (3)  can be rewritten as follows:

If the decrease in concentration of  B  is followed by photometric measurement the  Beer' Law  must be taken into account.

Combining equation  (5)  and  Beer' Law

A  =  Absorbance,  e = Molar absorbtivity with units of  L · mol -1 · cm -1
c  =  Concentration of the absorbing species in solution, expressed in  mol · L -1,  d  =  Path length through the sample
Io =  Intensity of the initial light beam,  I  =  Intensity of the transmitted light


gives the relationship between  k'  and  lnA:



According to equation  (7),  a plot of lnA versus time should lead to a straight line whose slope is the pseudo-first order rate constant  k'.  The value of  k'  can then be divided by the known, constant concentration of the excess compound to obtain the true constant second order  k:


The  pseudo-first order rate constant k'  can be also determined from the  half-life t 1 / 2.



Measuring a second order reaction rate (see below) can be problematic: the concentrations of the two reactants must be followed simultaneously, which is more difficult; or measure one of them and calculate the other as a difference, which is less precise. A common solution for that problem is the pseudo first order approximation. As a general rule, a minimum of a 20-fold stoichiometric excess is necessary. A 50-fold or 100-fold stoichiometric excess is preferable.



Second Order Reaction

The rate of a second order reaction is proportional to either the concentration of a reactant squared, or the product of concentrations of two reactants.

For the general case of a reaction between  A  and  B,  such that



the rate of reaction will be given by

1. Initial concentrations of the two reactants are equal:

Equation  (1)  can be written as:
Separating the variables and integrating


gives

Provided that  [ A ]  =  [ A ] o  at  t  =  0, the constant of integration  C  becomes equal to  1 / [ A ] o.

Thus the second order integrated rate equation is


A plot of  1 / [ A ]   vs   t   produces a straight line with slope   k  and intercept  1 / [ A ]o.  The plot should be linear up to a conversion of about 50 %.


2. Starting concentrations of the two reactants are different:

If  [ A ] o  and  [ B ] o  are different the variable  x  is used.

Noting that   d[ A ] = - dx,  the general rate equation  (1)  can be expressed by


where  [ A ] o  -  x  =  [ A ],   [ B ] o  -  x  =  [ B ]  and   x   is the decrease in the concentration of  A  and  B respectively.

In order to integrate the left-hand side of equation  (6),   it is necessary to perform a partial fraction expansion. The fraction on the left hand-side of equation below is splitted into a sum of simple fractions on the right hand-side.


c'  and  c'' are two constants. Putting them back over a common denominator



leads to

Provided that  - x (c' + c'') = 0

The equations above allow to determine the factors  c'  and  c'':



Using c', c'' notation equation  (6)  becomes the form


whose integration

results in

where  C  is the constant of integration.

Using the condition that  x  =  0,  when  t  =  0,  the value of  C  can be found:



and equation  (15)  becomes

If  [ A ] o  >  [ B ] o,  then a plot of


against  t  will have a positive slope, equal  ([ A ] o  -  [ B ] o) · k.

Because equivalent amounts of  A  and  B  are reacting,  [ A ]  can be expressed in terms of  [ B ]:

If the experimental method yields reactant concentrations rather than x, the equivalent form of equation  (17)  is




The rate constant can be determined using equation  (20),  provided that the initial concentration of  A  is twice the initial concentration of  B  (see Kinetic equations - Reaction Second Order - Download PDF file):


where  xo  =  [ B ] o and  x  =  [ B ] .

Summary

Reaction Order
Differential Rate Law
Integrated Rate Law
Linear Plot
Slope of
Linear Plot
Units of
Rate Constant
0
- d [A]  /  dt  =  k
[ A ]  =  [ A ] o - kt
[ A ]  versus  t
- k
mol · L -1 · s -1
1
- d [ A ]  /  dt  =  k [ A ]
[ A ]  =  [ A ] o · e - kt
ln [ A ]  versus  t
- k
s -1
2
- d [ A ]  / dt  =  k [A] 2
  1 /  [ A ]  =  1  /  [ A ] o  +  kt  
1 /  [ A ]  versus  t
k
L · mol -1 · s -1








Arrhenius Equation

Svante Arrhenius 		It is a well-known fact that raising the temperature increases the reaction rate. Quantitatively this relationship between the rate a reaction proceeds and its temperature is determined by the Arrhenius Equation:

E a = activation energy
R  =  8.314 [ J · mol -1 · K -1 ]
T  =  absolute temperature in degrees Kelvin
A  =   pre-exponential or frequency factor
A  =  p · Z, where Z is the collision rate and p is a steric factor.
Z turns out to be only weakly dependant on temperature.
Thus the frequency factor is a constant, specific for each reaction.



Effective collisions 		The Arrhenius equation is based on the collision theory which supposes that particles must collide with both the correct orientation and with sufficient kinetic energy if the reactants are to be converted into products.

The two animations are taken from Effects of temperature, concentration, catalysts, inhibitors on reaction rates

Ineffektive collisions


The Arrhenius equation is often written in the logarithmic form:



Determination of  E a A plot of lnk versus 1 / T produces a straight line with the familiar form y  =  - mx + b, where

x  =  1 / T
y  =  lnk
m  =  - E a / R
b  =  lnA

The activation energy E a can be determined from the slope m of this line:  E a  =  - m · R

The value of the activation energy  E a  is rounded to one decimal place. The value of lnA shall be expressed with an accuracy of two decimal places.

An accurate determination of the activation energy requires at least three runs completed at different reaction temperatures. The temperature intervals should be at least 5 °C.


"Two-Point Form" of the Arrhenius Equation

The activation energy can also be found algebraically by substituting two rate constants (k1, k2) and the two corresponding reaction temperatures (T1, T2) into the Arrhenius Equation  (2).




Substracting equation  (4)  from equation  (3)  results in



Rerrangement of equation  (5)  and solving for  E a  yields



References:

How to Determine the Rate Law from Experimental Data
Eyring Equation




Experiments


Conductivity Measurement

Hydrolysis of Methyl Formate
Objectives: Test for a First Order Behaviour, Determination of Rate Constants, Temperature Effect on Rate		 Experiment Description

Alkaline Hydrolysis of Esters
Objectives: Test for a Second Order Behaviour, Determination of Rate Constants and Activation Parameters		 Experiment Description

Alkaline Hydrolysis of Ethyl Acetate
Objectives: Test for a Second Order Behaviour, Determination of Rate Constants and Activation Parameters		 Experiment Description

Hydrolysis of t-Butyl Chloride
Objectives: Test for a First Order Behaviour, Determination of Rate Constants and Activation Parameters		 Experiment Description

Hydrolysis of t-Butyl Halides.
Objectives: Effect of Solvent or Leaving Group on Rate, Determination of Rate Constants		 Experiment Description

Hydrolysis of Benzoyl Chloride
Objectives: Test for a First Order Behaviour, Determination of Rate Constants and Activation Parameters		 Experiment Description

Enzymatic Hydrolysis of Urea
Objectives: Determination of the Temperature Optimum, the Michaelis Constant KM and the Maximal Velocity Vmax, Competitive Inhibition		 Experiment Description

Photometric Measurement

Acid catalyzed Iodination of Acetone
Objectives: Determination of Rate Constants, Test for a Pseudo Zero Order Reaction		 Experiment Description

Acid Catalyzed Iodination of Acetone
Objectives: Test for a Pseudo First Order Behaviour, Determination of Rate Constants and Activation Parameters		 Experiment Description

Bromination of reactive Aromatics
Objectives: Test for a Pseudo First Order Behaviour, Determination of Rate Constants and Activation Parameters		 Experiment Description

Fading of Phenolphtalein in Alkaline Solution.
Objectives: Test for a Pseudo First Order Behaviour, Determination of the Half-Life and the Rate Constant		 Experiment Description

Reaction of Methyl Orange with Tin(II) Chloride
Objectives: Test for a Pseudo First Order Behaviour, Determination of Rate Constants and Activation Parameters		 Experiment Description

Fading of Triphenylmethane Dyes
Objectives: Test for a Pseudo First Order Behaviour, Determination of Rate Constants and Activation Parameters		 Experiment Description

Volume Measurement

Catalyzed Decomposition of Hydrogen Peroxide
Objectives: Test for a First Order Behaviour, Determination of Rate Constants and Activation Parameters		 Experiment Description

Enzymatic Decomposition of Hydrogen Peroxide
Objectives: Dependance of Reaction Rate on Temperature and on Concentration of Enzyme or Substrate, First Order Reaction 		Experiment Description

Temperature Measurement

Decomposition of Hydrogen Peroxide catalyzed by Potassium Iodide
Objective: Dependance of the Reaction Rate upon the Concentration of the Catalyst		 Experiment Description

Decomposition of Hydrogen Peroxide catalyzed by Potassium Dichromate
Objective: Dependance of the Reaction Rate upon the Concentration of the Catalyst		 Experiment Description

Addition of Bisulfite to Ketones
Objectives: Nucleophilic Addition to Carbonyl Group of Ketones, Effect of Alkyl Groups Electron-Donating Effect, Steric Hindrance 		Experiment Description




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