Collection and Analysis of Rate
Data
Dr. AKM Shafiqul Islam
Types of Chemical Reactions

Two types of reaction for rate data


The Batch reactor, which is used primarily for
homogenous reaction
The Differential reactor, which is used for
solid-fluid heterogeneous reactions
Batch Reaction


In Batch reaction experiments, concentration, pressure,
and or volume are usually measured and recorded at the
different times during the course of reaction.
Measurements on the differential reactor are made
during steady-state operation. The product
concentrations are monitored for different sets of feed
conditions
Rate law
A



B
rA = the rate of formation of species A per unit volume
-rA = the rate of a disappearance of species A per unit
volume
rB = the rate of formation of species B per unit volume
Example, A  B


If B is being created at a rate of 0.2 moles per decimeter
cubed per second, i.e., the rate of formation of B is,
rB = 0.2 mole/dm3/s
Then A is disappearing at the same rate:
-rA = 0.2 mole/dm3/s
the rate of formation of A is
rA = -0.2 mole/dm3/s
Power Law Model & Elementary Rate
Laws



The dependence of reaction rate, -rA on the
concentration of the species present fn(Cj)
The order of a reaction refers to the powers to which the
concentrations are raised in the kinetic rate law
Here a order with respect to reactant A and b order with
respect to reactant B
Steps in Analyzing Rate Data
1.
Postulate a rate law
A.
Power law models for homogenous reactions
a
a
b
 rA  kCA ,  rA  kCA CB
A.
Langmuir-Hinshelwood models or
heterogeneous reactions
kPA
 rA 
,
1  K A PA
kPA PB
 rA 
2
(1  K A PA  PB )
Select reactor type and corresponding
mole balance
2.
A.
B.
If batch reactor use mole balance on Reactant A
dC A
 rA  
TE5-1.1
dt
If different PBR use mole balance on product P (AP)
FP
 rA 
 C P vo / W
W
TE5-1.2
3.
Process your data in terms of measured
variable (e.g., NA, CA or PA)
If possible write your mole balance in terms of the
measured variable
4.
Look for simplifications
For example, if one of the reactant is excess,
assume its concentration is constant. If the gas
phase mole fraction of reactant is small, set e0
5.
For a batch reactor, calculate –rA as function of
concentration CA to determine reaction order
A.
Differential analysis
Combine the mole balance and power law model
a
 rA  kCA
dC A

 kCaA
dt
(TE5-1.3)
(TE5-1.4)
taking the natural log
 dC 
ln   A   ln  rA   ln k  a ln C A
 dt 
1)
Find  dC A
(TE5-1.5)
from CA versus t data by
dt
a)
b)
c)
2)
Graphical method
Finite differential method
Polynominal
Plot  dC A vs dt and find reaction order a which
dt
3)
Find k
B.
Integral method
For  rA  kCaA the combined mole balance and rate
law is
dC A

 kCaA
(TE5-1.4)
dt
1)
2)
Guess a and integrate equation
Nonlinear regression (Polymath)
Integrate equation (TES5-1.4) to obtain
1  C A10a   C1Aa 
t 
for a  1
k

(TE5-1.6)
For differential PBR calculate  rA as a function
of C A and PA
6.
A.
B.
Calculate  rA  v0C P as a function of reactant concentration C
A
W
Choose model, e.g.,
 rA 
C.
7.
kPA
,
1  K A PA
Use nonlinear regression to find the best model and model
parameters
Analyze the rate model for “goodness of fit”.
Batch Reactor Data



Batch reactors are used primarily to determine rate law
parameters for homogeneous reactions
Determination is achieved by measuring concentration as
a function of time
Use differential, integral and nonlinear regression
method of data analysis to determine reaction order, a
and specific reaction rate constant, k

if reaction is irreversible, it is possible to determine
reaction order a and the specific rate constant by either
nonlinear regression or numerically differentiating
concentration vs time data

Example, for the decomposition reaction
A  Products
 rA  k AC aA

The differential method may be used


It is possible to determine the relationship between
and the concentration of other reactants
 rA
For irreversible reaction
A + B  Products
with the rate law
 rA  k AC aA CBb
Where a and b are both unknown, the reaction could
first be run in an excess of B so that CB remains
essentially unchanged during the course of the reaction
 rA  k C aA
Where
b
b

k  k AC B  k AC B 0
After determining a, the reaction is carried out in excess
of A, for which the rate law is approximated as
 rA  k CBb
Where
a
a


k  k AC A  k AC A 0

Once a and b are determined, kA can be calculated from
the measurement of –rA at known concetration of A and B
 rA
k A  a b  (dm 3 /mol )a  b 1/ s
C A CB

Both a and b can be determined by using method of
excess, coupled with a differential analysis of data for
batch system
Differential Method of Analysis
Consider a reaction carried out isothermally in a
constant-volume batch reactor and concentration
recorded as a function of time. By combining the mole
dC A

 k AC aA
dt
After taking natural log on both side of the equation
 dC A 
ln  
  ln k A  a ln C A
 dt 
The reaction order can be found from a ln-ln plot of
 dC A  vs ln C
ln  

A

dt 
Three Ways to Determine (-dCA/dt) from ConcentrationTime Data



Graphical differentiation
Numerical differentiation
Differentiation of a Polynomial fit to the data

Graphical Method
The graphical method involves plotting –CA/n as a
function of t and then using equal-area differentiation to
obtain –dCA/dt.

Numerical differentiation
Numerical differentiation formulas can be used when the
data points in the independent variables are equally
spread, such as t1 – t0 = t2 – t1 = t
time (s)
0
t1
t2
t3
concentration (mol/dm3)
CAo
CA1
CA2
CA3

The three-point differentiation formulas
Initial point:
Interior points:
Last point:
 3C A0  4C A1  C A 2
 dC A 

 
2t
 dt t0
1
 dC A 


C Ai 1  C Ai 1 


 dt t1 2t
1
 dC 
C A4  C A2 
e.g.,  A  
 dt t3 2t
1
 dC A 
C A3  4C A4  3C A5 

 
dt
2

t

 t5

Polynomial Fit
Another technique to differentiate the data is to first fit the
concentration-time data to the nth-order polynomial:
CA = ao + a1t + a2t2 + a3t3 +a4t4
Many software package program that will calculate best
values for the constants ai only by entering concentrationtime data
After determining the constants, ai, the differentiate
equation with respect to time can get
The concentration and the time rate of change of
concentration are both known at any time t