CEE 690K ENVIRONMENTAL REACTION KINETICS

Introduction David A. Reckhow

CEE690K Lecture #6 1 Updated: 24 September 2013

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Lecture #6 Estimation of Rates: Practical Methods Brezonik, pp.50-58

Chlorination of Phenol

David A. Reckhow CEE690K Lecture #6

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RCP

k1

k2

k4

k3

k5

k7

k6

k8

How did Morris & Lee get these rate data?

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Reaction scheme for the chlorination of phenoxide ion (adapted from Lee and Morris (1962) and Burttschell et al. (1959)) with rate constants and ratios percentage obtained from Gallard and von Gunten (2002) and Acero et al. (2005b).

From: Deborde & von Gunten, 2008 [Wat. Res. 42(1)13]

Absorbance data?

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Wavelength (nm)

200 220 240 260 280 300 320

Mol

ar A

bsor

ptiv

ity (M

-1cm

-1)

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

Phenol 2-Chlorophenol 4-Chlorophenol

Wavelength (nm)

220 240 260 280 300 320 340 360 380 400 420

Mol

ar A

bsor

ptiv

ity (M

-1cm

-1)

0

50

100

150

200

250

300

350

400

Hypochlorite (OCl-)

Hypochlorous Acid (HOCl)

Kinetic Analysis of Experimental Data 5

Fitting the data to rate equations Integral Methods Already discussed; depends on model Uses all data; but not as robust

Differential Methods Get simple estimates of instantaneous rates and fit these to

a concentration dependent model Quite adaptable

Initial Rate Methods Relatively free from interference from products Not dependent on common assumptions

David A. Reckhow CEE690K Lecture #6

0102030405060708090

0 20 40 60 80Time (min)

Con

cent

ratio

n

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Simple first order

When n=1, we have a simple first-order reaction

This results in an “exponential decay”

Akcdtdc

−=

ktAoA ecc −=

k = −0 032 1. min

productsA k→

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Integral Method: First order

This equation can be linearized

good for assessment of “k” from data

AA kc

dtdc

−=

10

100

0 20 40 60 80Time (min)

Con

cent

ratio

n (lo

g sc

ale)

ktcc AoA −= lnln

k = −0 032 1. min

Slope

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0102030405060708090

0 20 40 60 80Time (min)

Con

cent

ratio

n

Simple Second Order

This results in an especially wide range in rates

More typical to have 2nd order in each of two different reactants

22

1A

A

A

ckdt

dc−=

ν

tckcc

AoAoA

211

+=

min//0015.02 mgLk =

When n=2, we have a simple second-order reaction

productsA k→ 22

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Integral method: Simple Second Order

Again, the equation can be linearized to estimate “k” from data

22

1A

A

A

ckdt

dc−=

ν

0

0.02

0.04

0.06

0.08

0.1

0.120 20 40 60 80

Time (min)1/

Con

cent

ratio

n

tkcc AoA

2211+=

min//0015.02 2 mgLk =

Slope

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Variable Kinetic Order

Any reaction order, except n=1

nnck

dtdc

−=

( )[ ] ( )11111

1−−−+

=nn

on

otckn

cc

( ) tkncc nn

on 111

11 −+= −−

Mixed Second Order

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Two different reactants Initial Concentrations are different; [A]0≠[B]0

The integrated form is:

Which can be expressed as:

productsBA k→+ 2

=≡≡dtAd

dtd

Vrate

A

][11ν

ξ( )( )xBxAk

BAkdtdx

−−=

=

002

2

][][

]][[

tkBAAB

BA 20

0

00 ][][][][ln

][][1

=−

( )0

0002 ][

][log][][43.0][][log

ABtBAk

BA

−−=

][][log

BA

t0

0

][][log

BA

Integral Method: Mixed Second Order

David A. Reckhow CEE690K Lecture #6

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Initial Concentrations are the same; [A]0=[B]0

The integrated form is:

Which can be integrated:

productsBA k→+ 2

( )( )xAxAk

AAkdtdx

−−=

=

002

2

][][

]][[

02 ][

12][

1A

tkA

+=

][1A

t0][

1A

xBxABA −=−== 00 ][][][][

∫ ∫= dtkA

AdA 22][

][ ν tkAA 2

0

2][

1][

1=−

Differential Methods I

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Doesn’t require assumptions on reaction order Simple method, doing it by “eye” Get estimates of instantaneous rates by drawing tangents &

plotting these slopes

[A]

time

Log(

-d[A

]/dt

)

Log [A]

n

nAkdt

Ad ][][=

− ]log[log][log Ankdt

Ad+=

−

0

Log k

Differential Methods II

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Finite difference method Start with the general linear solution

And substituting back, we get:

So the reciprocal of “X” is a linear function of time

nAkdt

Ad ][][=

−

ktnAA nn )1(

][1

][1

10

1 −=− −−

( )1

10

1

][11][

−

−−

+−= nn

AktnA ( )

1

10][

11][][−

−

+−= nn

AktnAA

( )1

10

1

][11][

][

][ −

−−

+−==≡ n

n

AktnkAk

Adt

AdX

( ) 10][

111−+−= nAk

tnX

Differential Methods III

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Finite difference method (cont.) Now we can get “X” from a time-centered finite

difference approximation And, for t=n

11

11 ][][][

−+

+−

−−

≈

nn

nn

n ttAA

dtAd

1/X

time

N-1

0 dt

Ad

AX ][

][1≡

Initial Rate Methods

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Evaluated in very early stages of the reaction where: Only small amounts of products have been formed Reactants have essentially not changed in

concentrations

Avoids many problems of complex reactions where products continue to react

Initial Rate II

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Run multiple reactions at different staring concentrations Measure short-term concentrations of starting materials Estimate initial rate and plot vs starting concentration

Log(

-d[A

]/dt

)

Log [A]0

n

0

][][

=∆

∆

=

ttA

dtAd

]log[log][log Ankdt

Ad+=

−

0

Log k

[A]

time

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