ChE 452 Lecture 08

Analysis Of Data From A Batch Reactor

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Objective

Data analysis from indirect measurements Essen’s method (learned in p-chem)

Does not usually work Van’t Hoff’s method

Accurate but amplifies errors in data

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Background: Kinetic Data After Measuring

Indirect method – a method where you measure some other property (i.e. concentration vs time) and infer a rate equation. 3

0 5 10 15 20 25 30 3510

100

Time, Hours

Pre

ssur

e, x

2, to

rr

Figure 3.8 Typical batch data for reaction (3.7). Data of Tamaru[1955].

Objective For Today: Analysis Of Rate Data

Derive basic equations Essen’s method Van’t Hoff’s method

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Derivation Of Performance Equation For A Batch

Reactor

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For A B, the moles of A reacted/volume/time will equal the reaction rate, i.e.

A

A

dC=r

dτ(1)

CA is the concentration of A, is time, and rA is the rate of reaction per unit volume.

Figure 3.11 A batch reactor

Integration Yields The Following

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(3.31)

Memorize this equation

0A

fA

C

A

AC

dC=τ

(-r )

For A First Order Reaction

rA = -k1CA

(3.38)

Substituting equation (3.38) into equation (3.31) and integrating yields:

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fA

0A

1 C

CLn

k

1 Memorize this equation(3.39)

Derivation

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

1noAn

(3.42)

Memorize this equation

Derivation

Plots Of Equations

0% 20% 40% 60% 80% 100%

conversion

0

2

4

6

8

10

Tim

e

First Order

SecondOrder

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Table 3.4 Rate Laws For A Number Of Reactions

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Rate Laws for a number of reactions

Reaction Rate Law Differential Equation Integral Equation

A ProductsA+B Products

rA=kA

A ProductsA+B Products

rA=kA[A]

A ProductsA+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

ddX

AAA Xk

ddX

nA

1nnAA

A X)C(kd

dX

)XCC)(X1(kd

dXA

0A

0BAAA

A

)XC2C)(X1(kd

dXAA

0BAA

A

A2AAA Xk)X1(k

d

dX

A

AX

k

A

A X11

Ln1

k

1X1

1

)C)(1n(

1k

1n

A1n0

AA

0AA

0B

A0B

0B

0A

ACXC

)X1(CLn

CC(

1k

0AA

0B

A0B

0B

0A

ACX2C

)X1(CLn

CC2(

1k

Ae

21 XX

1Ln

1)kk(

Fitting Batch Data To A Rate Law

Steps• Start with a batch reactor and

measure concentrations vs time.• Fit those data to a first order and a

second order rate law and see which equation fits better.

• Whichever rate equation fits best is assumed to be the correct rate equation for the reaction.

13

Key Challenge: First And Second Order Data Does Not Look That Much

Different

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0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

Time

C /

C0

AA

Second Order

0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

Time

C /

C0

AA

Same k(CA

0)n-1

Vary k to fit data

Essen’s Method

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fA

0A

1 C

CLn

k

1

1C

C

Ck1n

11-n

fA

oA

1noAn

First order

nth order

(3.42)

(3.39)

Essen’s Method

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(C /

C )

- 1

0 AA

ln(C

/C

)0 A

A

0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

TimeH

alf O

rder

Firs

t Ord

er

0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

Time

Hal

f Ord

er

Figure 3.15

Example: The Concentration Of Dye As A Function Of

Time

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CA,mmoles /Lit

, Min CA CA

1 0 0.63 6 0.45 120.91 1 0.59 7 0.43 130.83 2 0.56 8 0.42 140.77 3 0.53 9 0.40 150.71 4 0.50 10 0.38 160.67 5 0.48 11 0.37 17Table 3.5

Essen Plot For Example:

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0 5 10 150

0.5

1

1.5

Time, Mins0 5 10 150

1

2

3

4

5

6

Time, Mins0 5 10 150

0.2

0.4

0.6

0.8

1

Time, Mins

ln(C

/C

)0 A

A

(C /

C )

- 1

2A

A

(C /

C )

- 1

0 AA

0

r2=.984 r2=.999r2=.981

No statistically significant difference between results.Figure 3.16

Example Shows Essen’s MethodDoes Not Distinguish Between

Models

In the literature, Essen’s method is often used.

Useful for impressing your boss since it always fits with good r2 (given good data)

It often gives the incorrect answers.

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Van’t Hoff’s Method

• Take batch data as before.• Calculate kone (first order rate

constant) ktwo (second order rate).• kone should be constant for a first

order reaction, ktwo should be constant for a second order reaction. (Use f test to check).

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Equations For kone And ktwo Follow From Before

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A

0A

1 C

CLn

k

1

A

0A

1 C

CLn

1k

(3.39) (3.51)

1C

C

C)1n(

1k

1n

A

0A

1n0A

n

(3.52)

1C

C

Ck1n

11-n

fA

oA

1noAn

(3.42)

Solve for k1

Solve for kn

Derived previously

Derived previously

Easy Solution: Define A VB Module In Microsoft

Excel

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Public Function kone(ca0, ca, tau) As Variantkone = Log(ca0 / ca) / tauEnd Function Public Function ktwo(ca0, ca, tau) As Variantktwo = ((1# / ca) - (1# / ca0)) / tauEnd Function Public Function kthree(ca0, ca, tau) As Variantkthree = ((1# / ca) ^ 2 - (1# / ca0) ^ 2) / tauEnd Function

Microsoft Excel/Visual Basic Return Types

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As Variant General return type (can be an integer, real, vector, matrix, logical or text)

As Single Single precision real

As Double Double precision real

As Integer Integer

The Formulas In The Spreadsheet For Van’t

Hoff’s Method

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B C D E F1 Ca0= 12 Essen's

Method3 time conc first second third4 ln(Ca0/Ca) (Ca0/Ca)-1 (CA0/CA)^

2-15 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)

The Numerical Values For

Van’t Hoff’s Method

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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 0.1046 2 0.83 0.093 0.102 0.1137 3 0.77 0.087 0.1 0.1148 4 0.71 0.086 0.102 0.1239 5 0.67 0.08 0.099 0.123

10 6 0.63 0.077 0.098 0.127

Van’t Hoff Plot

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Van’t Hoff’s method is much more accurate than Essens’ method.Essen’s is more common!

0 5 10 150.05

0.1

0.15

0.2

Time, Mins

Rat

e C

onst

ant

K3

K2

K1

Oxidation ofRed Dye

Figure 3.18 Van’t Hoff plot of the data from tables 3.5 and 3.6

Discussion Problem: Use Van’t Hoff’s Method To Determine The Order For

The Following Data

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Table 4.1 Buchanan’s [1871] data for the reaction:

CICH2COOH + H2COOH + HCI at 100º C

Time Hours [CICH2COOH] gms/liter

02346

10131928

34.54348

43.803.693.603.473.102.912.542.261.951.591.39

Solution:

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Ca0= 4Van't Hoff's

time Conc first second thirdln(ca0/Ca) (Ca0/Ca)-1 (CA0/CA)̂ 2-

10 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)

Solution Continued:

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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 0.0033 3.69 0.027 0.007 0.0044 3.6 0.026 0.007 0.0046 3.47 0.024 0.006 0.00310 3.1 0.025 0.007 0.00413 2.91 0.024 0.007 0.00419 2.54 0.024 0.008 0.00525 2.26 0.023 0.008 0.005

34.5 1.95 0.021 0.008 0.00643 1.59 0.021 0.009 0.00848 1.39 0.022 0.01 0.009

Van't Hoff's

Van’t Hoff Plot

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0 10 20 30 40 500.02

0.03

0.04

0.05

Time, Mins

Rat

e C

onst

ant

K3

K2

K1

Hydration ofChloracetic Acid

Figure 3.18 Van’t Hoff plot of the data from tables 3.5 and 3.6

Discussion Problem 2

Ammonium-dinitramide, (ADN) NH4N(NO2)2, is a oxidant used in solid fuel rockets and plastic explosives. ADN is difficult to process because it can blow up. Oxley et. Al., J. Phys chem A, 101 (1997) 5646, examined the decomposition of ADN to try to understand the kinetics of the explosion process. At 160º C they obtained the data in Table P3.20.

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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 of dinitramide at 160 C

Discussion Problem 2 Continued:

 a) Is this a direct or indirect measurement of the

rate? b) Use Van’t Hoff’s Method to fit this data to a rate

equation. c) If you had to process ADN at 160° C, how long

could you run the process without blowing anything up? Assume that there is an explosion hazard once 5% of the ADN has reacted to form unstable intermediates.

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This Is An Indirect Measurement! Use Same Spreadsheet As Before

To Fit Data

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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,B10,A10)

=ktwo(cao,B10,A10)

=kthree(cao,B10,A10)

Solution Cont.

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  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

Solution Cont.

c) from equ 3.39

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fA

oA

1 C

CLn

k

1

kk1=0.0006/sec (from spreadsheet)

oAC =1 (given) fAC =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 Rate Data

• Essen’s Method• Most common method• Plots look the best• Gives great looking results even with incorrect

rate equation • Van’t Hoff’s Method

• More accurate than Essen• Rare in literature• Plots noisier• Highlights weaknesses in rate equations

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Class Question

What did you learn new today?

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