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Analysis Of Data From A Batch Reactor
1
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Objective
Data analysis from indirect measurements
Essens method (learned in p-chem)
Does not usually work
Vant Hoffs method
Accurate but amplifies errors in data
2
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Background: Kinetic Data AfterMeasuring
Indirect method a method where you measuresome other property (i.e. concentration vs time)
and infer a rate equation. 3
0 5 10 15 20 25 30 35
10
100
Time, Hours
Pressu
re,x2,
torr
Figure 3.8 Typical batch datafor reaction (3.7). Data of
Tamaru[1955].
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Objective For Today:Analysis Of Rate Data
Derive basic equations
Essens method
Vant Hoffs method
4
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Derivation Of PerformanceEquation For A Batch Reactor
5
For AB, the moles of Areacted/volume/time will equal the reactionrate, i.e.
A
A
dC =rd
(1)
CA is the concentration of A, is time,and rA is the rate of reaction per unitvolume. Figure 3.11 A batch reactor
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Integration Yields TheFollowing
6
(3.31)
Memorize this equation
0A
fA
C
A
AC
dC=
(-r )
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For A First Order Reaction
rA
= -k1
CA
(3.38)
Substituting equation (3.38) into equation
(3.31) and integrating yields:
7
f
A
0
A
1 C
CLn
k
1 Memorize this equation(3.39)
Derivation
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For An nth Order Reaction:
nAnA Ckr
9
(3.41)
Substituting equation (3.41) into equation (3.31),
integrating, and rearranging yields:
1C
C
Ck1n
11-n
fA
oA
1no
An
(3.42)
Memorize this equation
Derivation
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Plots Of Equations
0% 20% 40% 60% 80% 100%
conversion
0
2
4
6
8
1 0
Time
First Order
Second
Order
11
bl 3
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Table 3.4 Rate Laws For ANumber Of Reactions
12
Rate Laws for a number of reactions
Reaction Rate Law Differential Equation Integral Equation
A Products
A+B Products
rA=kA
A ProductsA+B Products
rA=kA[A]
A Products
A+B Products
rA=kA[A]n
A+B Products rA
=kA
[A][B]
A+2B Products rA=kA[A][B]
A B rA=k1[A]-k2[B]
AA k
d
dX
AAA Xk
ddX
nA
1nnAA
A X)C(kd
dX
)XCC)(X1(kd
dXA
0A
0BAAAA
)XC2C)(X1(kd
dXAA
0BAA
A
A2AAA Xk)X1(k
d
dX
AA
Xk
AA
X11Ln1k
1
X1
1
)C)(1n(
1k
1n
A1n0
A
A
0AA0B
A
0
B0B
0A
ACXC
)X1(CLn
CC(
1k
0AA
0B
A0B
0B
0A
ACX2C
)X1(CLn
CC2(
1k
Ae21
XX
1Ln
1)kk(
Fi i B h D T A R
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Fitting Batch Data To A RateLaw
Steps
Start with a batch reactor and measureconcentrations vs time.
Fit those data to a first order and a
second order rate law and see whichequation fits better.
Whichever rate equation fits best is
assumed to be the correct rate equationfor the reaction.
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Key Challenge: First And Second OrderData Does Not Look That Much Different
14
0 0 . 5 1 1 .
0
0 . 2
0 . 4
0 . 6
0 . 8
1
Time
C
/C
0
A
A
HalfOrder
FirstOrder
Second Orde
ThirdOrder
0 0 . 5 1 1 .
0
0 . 2
0 . 4
0 . 6
0 . 8
1
Time
C
/C0
A
A
HalfO
rder
FirstOrder
SecondOrder
ThirdO
rder
Samek(CA
0)n-1
Vary k to fitdata
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Essens Method
15
f
A
0
A
1 C
CLn
k
1
1C
C
Ck1n
1 1-n
fA
oA
1noAn
First order
nth order
(3.42)
(3.39)
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Essens Method
16
(C
/C)
-1
0A
A
ln(C
/C
)
0A
A
0 0 . 5 1 1 .
0
0 . 2
0 . 4
0 . 6
0 . 8
1
Time
Ha
lfO
rder
Fir
stO
rder
Second
Order
Third
Order
0 0 . 5 1 1 .
0
0 . 2
0 . 4
0 . 6
0 . 8
1
Time
Half
Ord
er
Fir
stOrder
Second
Order
Third
Order
Figure 3.15
E l Th C t ti Of
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Example: The Concentration OfDye As A Function Of Time
17
CA,mmoles
/Lit
, Min CA CA
1 0 0.63 6 0.45 120.91 1 0.59 7 0.43 13
0.83 2 0.56 8 0.42 14
0.77 3 0.53 9 0.40 150.71 4 0.50 10 0.38 16
0.67 5 0.48 11 0.37 17
Table 3.5
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Essen Plot For Example:
18
0 5 10 15
0
0.5
1
1.5
Time, Mins0 5 10 15
0
1
2
3
4
5
6
Time, Mins0 5 10 15
0
0.2
0.4
0.6
0.8
1
Time, Mins
ln(C/
C
)
0A
A
(C
/C
)
-1
2
A
A
(C
/C)
-1
0A
A
0
r2=.984 r2=.999r2=.981
No statistically significant difference between results.
Figure 3.16
E l Sh E M th d
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Example Shows Essens Method
Does Not Distinguish Between Models
In the literature, Essens method is often used.
Useful for impressing your boss since it alwaysfits with good r2 (given good data)
It often gives the incorrect answers.
19
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Vant Hoffs Method
Take batch data as before.
Calculate kone (first order rate constant)ktwo (second order rate).
kone should be constant for a first orderreaction, ktwo should be constant for asecond order reaction. (Use f test tocheck).
20
Equations For kone And
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Equations For kone Andktwo Follow From Before
21
A
0
A
1 C
CLn
k
1
A
0
A1
C
CLn
1k
(3.39) (3.51)
1C
C
C)1n(
1k
1n
A
0A
1n0A
n
(3.52)
1C
C
Ck1n
1 1-n
fA
oA
1noAn
(3.42)
Solve for k1
Solve for kn
Derived previously
Derived previously
Easy Solution: Define A VB
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Easy Solution: Define A VBModule In Microsoft Excel
22
Public Function kone(ca0, ca, tau) As Variant
kone = Log(ca0 / ca) / tau
End Function
Public Function ktwo(ca0, ca, tau) As Variant
ktwo = ((1# / ca) - (1# / ca0)) / tau
End Function
Public Function kthree(ca0, ca, tau) As Variant
kthree = ((1# / ca) ^ 2 - (1# / ca0) ^ 2) / tau
End Function
Microsoft Excel/Visual Basic
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Microsoft Excel/Visual BasicReturn Types
23
As Variant General return type (can be aninteger, real, vector, matrix,logical or text)
As Single Single precision real
As Double Double precision real
As Integer Integer
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Spreadsheet For Vant Hoffs
Method
24
B C D E F1 Ca0= 1
2 Essen's
Method
3 time conc first second third4 ln(Ca0/Ca) (Ca0/Ca)-1 (CA0/CA)^
2-1
5 0 1=kone(ca0,C5,B5) =ktwo(ca0,C5,B5) =kthree(ca0,C5,B5)
6 1 0.91 =kone(ca0,C6,B6) =ktwo(ca0,C6,B6) =kthree(ca0,C6,B6)
7 2 0.83=kone(ca0,C7,B7) =ktwo(ca0,C7,B7) =kthree(ca0,C7,B7)
8 3 0.77=kone(ca0,C8,B8) =ktwo(ca0,C8,B8) =kthree(ca0,C8,B8)
9 4 0.71=kone(ca0,C9,B9) =ktwo(ca0,C9,B9) =kthree(ca0,C9,B9)
T e Numer ca Va ues For
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T e Numer ca Va ues ForVant Hoffs Method
25
B C D E F3 Time conc k1 k2 k3
4 0 1 ln(1/Ca)/t ((Ca0/Ca)-1)/t ((CA0/CA)^2-1)/t/2
5 1 0.91 0.094 0.099 0.104
6 2 0.83 0.093 0.102 0.113
7 3 0.77 0.087 0.1 0.1148 4 0.71 0.086 0.102 0.123
9 5 0.67 0.08 0.099 0.123
10 6 0.63 0.077 0.098 0.127
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Vant Hoff Plot
26
Vant Hoffs method is much more accurate than Essens method.
Essens is more common!
0 5 10 150.05
0.1
0.15
0.2
Time, Mins
RateC
onstant
K3
K2
K1
Oxidation ofRed Dye
Figure 3.18Vant Hoff plot of the data from tables 3.5 and 3.6
scuss on ro em: se an o s
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scuss on ro em: se an o sMethod To Determine The Order For The
Following Data
27
Table 4.1 Buchanans [1871] data for the reaction:
CICH2COOH + H2COOH + HCI at 100 C
Time Hours [CICH2COOH] gms/liter
0
2
3
4
6
10
1319
28
34.5
43
48
4
3.80
3.69
3.60
3.47
3.10
2.912.54
2.26
1.95
1.59
1.39
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Solution:
28
Ca0= 4
Van't Hoff's
time Conc first second third
ln(ca0/Ca) (Ca0/Ca)-1 (CA0/CA)^2-
0 4 =kone(cao,B5
,A5)
=ktwo(cao,B5
,A5)
=kthree(cao
,B5,A5)
2 3.8 =kone(cao,B6
,A6)
=ktwo(cao,B6
,A6)
=kthree(cao
,B6,A6)
3 3.69 =kone(cao,B7
,A7)
=ktwo(cao,B7
,A7)
=kthree(cao
,B7,A7)
4 3.6 =kone(cao,B8
,A8)
=ktwo(cao,B8
,A8)
=kthree(cao
,B8,A8)
6 3.47 =kone(cao,B9
,A9)
=ktwo(cao,B9
,A9)
=kthree(cao
,B9,A9)
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Solution Continued:
29
ca0= 4
time conc first second third
ln(ca0/
Ca)
(Ca0/Ca
)-1
(CA0/CA
)^2-1
0 4 #VALUE! #VALUE! #VALUE!
2 3.8 0.026 0.007 0.003
3 3.69 0.027 0.007 0.004
4 3.6 0.026 0.007 0.004
6 3.47 0.024 0.006 0.003
10 3.1 0.025 0.007 0.004
13 2.91 0.024 0.007 0.004
19 2.54 0.024 0.008 0.005
25 2.26 0.023 0.008 0.005
34.5 1.95 0.021 0.008 0.006
43 1.59 0.021 0.009 0.008
48 1.39 0.022 0.01 0.009
Van't Hoff's
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Vant Hoff Plot
30
0 10 20 30 40 500.02
0.03
0.04
0.05
Time, Mins
RateConstant
K3
K2
K1
Hydration of
Chloracetic Acid
Figure 3.18Vant Hoff plot of the data from tables 3.5 and 3.6
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Discussion Problem 2
Ammonium-dinitramide, (ADN) NH4N(NO2)2, is a oxidant used in solidfuel rockets and plastic explosives. ADN is difficult to process because itcan blow up. Oxley et. Al., J. Phys chem A, 101 (1997) 5646, examinedthe decomposition of ADN to try to understand the kinetics of theexplosion process. At 160 C they obtained the data in Table P3.20.
31
time,
seconds
fraction of
the AND
remaining
time,
seconds
fraction of
the AND
remaining
time,
seconds
fraction of
the AND
remaining
0 1.0 900 0.58 2400 0.24
300 0.84 1200 0.49
600 0.70 1500 0.41
Table P3.20 Oxley's measurements of the decomposition ofdinitramide at 160 C
Discussion Problem 2
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Discussion Problem 2Continued:
a) Is this a direct or indirect measurement of the rate?
b) Use Vant Hoffs Method to fit this data to a rateequation.
c) If you had to process ADN at 160 C, how long couldyou run the process without blowing anything up?Assume that there is an explosion hazard once 5% ofthe ADN has reacted to form unstable intermediates.
32
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Same Spreadsheet As Before To Fit
Data
33
Ca0= =b5
Van't Hoff's
time Conc first second third
0 1 ln(ca0/Ca) (Ca0/Ca)-1 (CA0/CA)^2-1
300 0.84 =kone(cao,B5
,A5)
=ktwo(cao,B5
,A5)
=kthree(cao,
B5,A5)
600 0.7 =kone(cao,B6
,A6)
=ktwo(cao,B6
,A6)
=kthree(cao,
B6,A6)
900 0.58 =kone(cao,B7
,A7)
=ktwo(cao,B7
,A7)
=kthree(cao,
B7,A7)
1200 0.49 =kone(cao,B8
,A8)
=ktwo(cao,B8
,A8)
=kthree(cao,
B8,A8)
1500 0.41 =kone(cao,B9
,A9)
=ktwo(cao,B9
,A9)
=kthree(cao,
B9,A9)
2400 0.24 =kone(cao,B1
0,A10)
=ktwo(cao,B1
0,A10)
=kthree(cao,
B10,A10)
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Solution Cont.
34
Ca0= 1
Van't Hoff's
time Conc first second third
ln(ca0/Ca) (Ca0/Ca)-1 (CA0/CA)^2-1
0 1 #VALUE! #VALUE! #VALUE!
300 0.84 0.000581 0.000635 0.001391
600 0.7 0.000594 0.000714 0.001735
900 0.58 0.000605 0.000805 0.002192
1200 0.49 0.000594 0.000867 0.002637
1500 0.41 0.000594 0.000959 0.003299
2400 0.24 0.000595 0.001319 0.006817
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Solution Cont.
c) from equ 3.39
35
fA
oA
1 C
CLn
k
1
k
k1=0.0006/sec (from spreadsheet)o
AC =1 (given)f
AC =0.95 (what's left if 5% converted)
sec850.95
1Ln
0.0006
1
C
CLn
k
1fA
oA
1
Summary: Two Methods To Fit
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Summary: Two Methods To FitRate Data
Essens Method
Most common method
Plots look the best
Gives great looking results even with incorrect rateequation
Vant Hoffs Method
More accurate than Essen
Rare in literature
Plots noisier Highlights weaknesses in rate equations
36
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Class Question
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