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Kinetics and Mechanism of Reactions of T ransition Metal Com plexes
Ralph G.WiMns
Copyright 2002 Wiley-VCH Verlag GmbH & Co. KGaA
ISBNs: 3-527-28253-X (Hardback); 3-527-60082-5 (Electronic)
R.G.Wilkins
Kinetics and Mechanism of Reactions
ofTransitionMetal Complexes
VCH
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Kinetics and Mechanism of Reactions of T ransition Metal Com plexes
Ralph G.WiMns
Copyright 2002 Wiley-VCH Verlag GmbH & Co. KGaA
ISBNs: 3-527-28253-X (Hardback); 3-527-60082-5 (Electronic)
VCH Verlagsgesellschaft mbH, D-6940 Weinheim (Federal Republic of Germany), 1991
Distribution:
VC H, P. O. Box
101161,
D-6940 Weinheim (Federal Republic of Germany)
Switzerland: VCH, P.O. Box, CH-4020 Basel (Switzerland)
United Kingdom and Ireland: VCH (UK ) Ltd., 8 Wellington C ourt, Cambridge CB1 1H Z (England)
USA and Canada: VCH, Suite 909, 220 East 23rd Street, New York, NY 10010-4606 (USA)
ISBN 3-527-28389-7 (V CH , W einheim ) ISBN 1-56081-198-6 (V CH , New York)
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Kinetics and Mechanism of Reactions of T ransition Metal Com plexes
Ralph G.WiMns
Copyright 2002 Wiley-VCH Verlag GmbH & Co. KGaA
ISBNs: 3-527-28253-X (Hardback); 3-527-60082-5 (Electronic)
Ralph G.Wilkins
Kinetics and
Mechanism of Reactions
of Transition
Metal Complexes
2nd Thoroughly Revised Edition
VCH
Weinheim
New York
Basel
Cambridge
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Kinetics and Mechanism of Reactions of T ransition Metal Com plexes
Ralph G.WiMns
Copyright 2002 Wiley-VCH Verlag GmbH & Co. KGaA
ISBNs: 3-527-28253-X (Hardback); 3-527-60082-5 (Electronic)
Prof.
Ralph G. Wilkins
University of Warwick
Dept. of Chemistry
Coventry CV4 7AL
Great Britain
This book was carefully produced. Nevertheless, author and publisher do not warrant the information
contained th erein to be free of error s. Rea ders are advised to keep in mind that statem ents, data , illustra-
tions,
procedural details or other items may inadvertently be inaccurate.
1st edition 1974
2nd edition 1991
Published jointly by
VCH Verlagsgesellschaft mbH, Weinheim (Federal Republic of Germany)
VCH Publishers, Inc., New York, NY (USA)
Editorial Director: Karin von der Saal
Production Manager: Elke Littmann
Library of Congress Card No. applied for.
A CIP catalogue record for this book is available
from the British Library.
Deutsche Bibliothek Cataloguing-in-Publication Data:
Wilkins, Ralph G.:
Kinetics and mechanism of reactions of transition metal
complexes / Ralph G. W ilkins. -2 . , thoroughly rev. ed. -
Weinheim ; New York ; Basel ; Cambridge : VCH, 1991
ISBN 3-527-28389-7 (W einheim ...) bros ch.
ISBN 3-527-28253-X (Weinheim ...) Gb.
ISBN 1-56081-198-6 (New York) bros ch.
ISBN 1-56081-125-0 (New York) Gb .
VCH Verlagsgesellschaft mbH, D-6940 Weinheim (Federal Republic of Germany), 1991
Printed on acid-free and low chlorine paper.
All rights reserved (including those of translation into other lang uages). No part of this book may be repro duced
in any form by photoprinting, microfilm, or any other means - nor transmitted or translated into a machine
language without written permission from the publishers. Registered names, trademarks,etc.used in this book ,
even when not specifically marked as such, are not to be considered unprotected by law.
Composition: Filmsatz Unger und Sommer GmbH, D-6940 Weinheim. Printing and bookbinding: Konrad
Triltsch, Graphischer Betrieb, D-8700 Wurzburg. Cover design: TWI, Herbert J. Weisbrod, D-6943 Birkenau.
Printed in the Federal Republic of Germany
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Kinetics and Mechanism of Reactions of T ransition Metal Com plexes
Ralph G.WiMns
Copyright 2002 Wiley-VCH Verlag GmbH & Co. KGaA
ISBNs: 3-527-28253-X (Hardback); 3-527-60082-5 (Electronic)
Preface
Seventeen years is a long time between editions of a book. In order to add some of the vast
amount of new material which has been published in that time, I have needed to abridge the
older edition and in so doing apologise to oldtimers (myself included ) whose work may have
been removed or modified. Nevertheless, the approach used is unchanged. In the first three
chapters I have dealt with the acquisition of experimental data and discussed use for building
up the rate law and in the deduction of mechanism. In the second part of the book, the
mechanistic behavior of transition metal complexes of the Werner type is detailed, using exten-
sively the principles and concepts developed in the first part.
There are noticeable changes from the first edition. The past decade or so has seen a marked
increase in the use of photolytic and nmr methods, pressure effects and so on, to discover the
intima te details of mecha nism. These developments have been incorpo rated. So too, the grow-
ing interest in the biological aspects of inorganic chemistry has not been ignored. On the other
hand, photochemical behavior and organometallic compounds are used for illustrative pur-
poses only. I have succumbed to the use of SI units for energy, recognizing that much of the
past (and a sizeable amount of the present) literature do not employ these. I shall never forget
the factor 4.184 which converts old to new. I have increased the number of problems threefold
and provided a section containing hints on their solution.
I would like to thank Dave Pennington for his careful examination of Chapter 5, Steve
Davies for producing the structures, Ellen Foley and Merlin Callaway for valient typing and
the staff of VCH, especially Michael Weller and Karin von der Saal, for their kind help in
the production of the book. Many copyright holders gave permission to reproduce material.
All these people have my sincere thanks. At the end of course, errors that remain are my
responsibility alone.
My especial thanks go to my wife, Pat. She critically read the whole manuscript and gave
much encouragement and understanding from the beginning. The book would never have ap-
peared without her support and I dedicate it to her, with much love.
Ralph G. Wilkins, June 1991
Wappenbury, England
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Kinetics and Mechanism of Reactions of Transition Metal Complexes
Ralph G.WiMns
Copyright 2002 Wiley-VCH Verlag GmbH & Co. KGaA
ISBNs: 3-527-28253-X (Hardback); 3-527-60082-5 (Electronic)
Contents
Ligands and Complexes XI I I
1 The Determination of the Rate Law 1
1.1 The Rate of a Reaction and the Rate Law 1
1.2 The Rate Law Directly from Rate M easurem ents 2
1.2.1 Initial-R ate M etho d 3
1.2.2 Cri tique of the Initial-R ate M etho d 4
1.2.3 Steady-State Ap pro ach 5
1.3 Integrate d Form s of the Rate Expre ssion 5
1.4 M onoph asic Unidirectional Reactions 7
1.4.1 Zero-O rder Dep enden ce 7
1.4.2 First-O rder Dep enden ce 8
1.4.3 Second -Ord er Dep enden ce 10
1.4.4 Conversion of Pseu do to Real Rate Co nstan ts 11
1.5 M ono pha sic Reversible Reactions 13
1.5.1 Conversion of Reversible to Un idirectio nal Reactions 15
1.6 M ultiph asic Un idirectio nal Reactions 16
1.6.1 Co ncur rent Reactions 17
1.6.2 Consecutive Reactions with no Elem ents of Reversibility 18
1.6.3 Two-Step Rea ction s with an Elem ent of Reversibility 23
1.6.4 Reaction Schemes Asso ciated with (1.98) 24
1.6.5 Two-Step Re action s with Total Reversibility 27
1.7 Reca pitulatio n 31
1.8 Rel axatio n Kinetics 32
1.8.1 Single-Step Re action s 32
1.8.2 M ultistep Reactions 33
1.9 Exc hang e Kinetics 38
1.10 The Inclusio n of [ H
+
] Terms in the Rate Law 41
1.10.1
One M onop rotic Reactant, One Acid-Base Equilibrium 41
1.10.2 Two Acid-Base Equ ilibria 43
1.10.3
The Effect of Hig h Acid Co ncen tratio n 47
1.11 Kinetics and The rm ody nam ics 49
1.12 Con cludin g Rem arks 50
References 50
Problems 56
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VIII Contents
2 The Deduction of Mechan ism
65
2.1 The Rate Law and Mechan ism 65
2.1.1 First-O rder Reactions 67
2.1.2 Second-Order Reactions 68
2.1.3 Third-O rder Reactions 69
2.1.4 Even Higher-Order Reactions 72
2.1.5 Negative-O rder Reactions 72
2.1.6 Fractional-O rder Reactions 73
2.1.7 The Inclusion of [ H
+
] Terms in the Rate Law 75
2.2 Furth er Checks of Mechanism 80
2.2.1 The Detection and Study of Intermediates 80
2.2.2 The Determ ination of Bond Cleavage 84
2 .3 Ac t iva tion Parameters , Thermodynamic Func tions and Mechanism . . . . 87
2.3.1 The Effect of Tem perature on the Rate of a Reactio n 87
2.3.2 The Variation of
a
or AH- with Temperature 89
2.3.3 The Effect of Pressure on the Rate of a Reaction 90
2.3.4 The Variation of AV - with Pressure 91
2.3.5 Activation Parameters and Con centration Units 91
2.3.6 Reactio n Profiles 91
2.4 Free Energy of Activation and Mechan ism 93
2.5 Linear Free-Energy Relationships AG " and
AC
96
2.5.1 Ha m me tt Relationship 99
2.5.2 Taft Relatio nship 101
2.5.3 Bren sted Relatio nship 101
2.5.4 Swain-Sco tt Relatio nship 103
2.6 Enthalpy of Activation and Mechanism 104
2.7 Entrop y of Activation and Mechanism 105
2.7.1 AS* and the Char ge of the Reacta nts 105
2.8 Volume of Activa tion and M echanism 106
2.8.1 A // - and AS " Values - The Isokinetic Relationship 108
2.8.2 AS* and AK* Values 109
2.8.3 Use of All Param eters 110
2.9 M edium Effects on the Rate 110
2.9.1 The Effect of Electrolytes 110
2.9.2 The Effect of Electolytes - M edium or M echa nistic ? 115
2.9.3 The Solvent Effect and Me chanism 116
References 118
Problems 123
3 The Experimental Determination of the Rate of Reaction 131
3.1 Essential Prelim inaries 131
3.1.1 Reacta nt Species in Solutio n 131
3.1.2 Stoich iom etry of Reaction 133
3.1.3 The Natu re of the Produ cts 133
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Contents IX
3.1.4 The Influence of Impurities 134
3.1.5 The Control of Experimental Conditions 134
3.2 The Methods of Initiating Reaction and their Time Ranges 135
3.3 Flow Methods 136
3.3.1 Continuous Flow 137
3.3.2 Quenched Flow 138
3.3.3 Stopped Flow 139
3.4 Relaxation Methods 140
3.4.1 Temperature Jump 141
3.4.2 Pressure Jump 141
3.4.3 Electric-Field Jump 143
3.4.4 Ultrasonic Absorption 144
3.5 Large Perturbations 145
3.5.1 Flash or Laser Photolysis 145
3.5.2 Pulse Radiolysis 148
3.5.3 Comparison of Large Perturbation Methods 151
3.6 Competition Methods 151
3.7 Accessible Rate Constants Using Rapid Reaction Methods 151
3.8 The Methods of Monitoring the Progress of a Reaction 153
3.9 Spectrophotometry 154
3.9.1 Ultraviolet and Visible Regions 154
3.9.2 Infrared Region 158
3.9.3 Fluorescence 159
3.9.4 Polarimetry 160
3.9.5 Nmr Region 161
3.9.6 Nmr Line Broadening 163
3.9.7 Epr Region 169
3.9.8 Epr Line Broadening 170
3.10 Non-Spectrophotometric Methods 171
3.10.1 [H
+
] Changes 171
3.10.2 Cationic and Anionic Probes 173
3.10.3 Conductivity 173
3.10.4 Thermal Changes 174
3.10.5 Pressure Changes 174
3.10.6 Electrochemical Methods 174
3.11 Batch Methods 175
3.12 Competition Methods 176
3.12.1 Stern-Volmer Relationship 177
3.12.2 Isotope Fractionation 178
3.13 The Study of Transients 178
3.13.1 Spectral Properties of Transients 179
3.13.2 Thermodynamic Properties of Transients 180
3.13.3 Chemical Reactivity of Transients 180
References 181
Problems 192
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X Contents
4 Substitution Reactions
199
4.1 The Chara cteristics of Subs titution Reactions 199
4.1.1 Solvated M etal Ions 200
4.1.2 Representation of Substitution Mechanism s 201
4.2 Substitution in Octahe dral Complexes 201
4.2.1 Solvent Exchan ge with M etal Ions 202
4.2.2 The Interchange of Different Unid entate Ligands 205
4.2.3 Ou ter Sphere Com plexes 206
4.2.4 Char acteristics of the Various M echanism s 207
4.2.5 The Limiting First-Order Rate Cons tant 208
4.2.6 Second-Order Rate Consta nts 210
4.2.7 Sum ma ry 211
4.3 Accelerated Substitution of Unid entate Ligands 212
4.3.1 H
+
-Assiste d Removal 212
4.3.2 Me tal Ion-Assisted Removal 213
4.3.3 Ligand-A ssisted Removal 214
4.3.4 Base-Assisted Removal 215
4.3.5 The Quest for Five Co ord inate Interm ediates 217
4.4 Replacement Reactions Involving Mu ltidentate Ligands 219
4.4.1 The Form ation of Chelates 219
4.4 .2 Effect of [H
+
] on the Rates of Sub stitutio n in Chelate Com plexes 221
4.4.3 M etal Ion-Assisted Dech elation 222
4.4.4 Ligand-Assisted Dech elation 223
4.5 Replacem ent Reactions Involving Macrocycles 224
4.5.1 Azam acrocycles 226
4.5.2 Crown-E thers and Cryp tands 227
4.5.3 Porphyrins 229
4.6 Substitution in Square-P lanar Complexes 232
4.6.1 The Kinetics of Replacement Involving Uniden tate Ligands 232
4.6.2 Activation Parameters 235
4.7 Ligan d Effects on the Rate 236
4.7.1 Effect of Ente ring Ligan d 236
4.7.2 Effect of Leaving Gr oup 237
4.7.3 Effect of Ligan ds Alre ady Present 237
4.7.4 Effect of Solvent 238
4.7.5 Reaction Pathways 238
4.7.6 Chelation in Square-P lanar Complexes 240
4.8 Sub stitutio n in Tetrahed ral Com plexes 242
4.9 Substitution in Five-Coordinate Complexes 243
4.10 Sub stitutio n in Organ ized Surfa ctant Systems 244
4.11 Substitution in Metalloproteins 245
References 246
Problems 252
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Contents
XI
5 Oxidation-Reduction Reactions 257
5.1 Genera l Charac teristics 257
5.2 Classification of Redox Reactions 258
5.3 Characterization of Mechanism 259
5.4 Ou ter Sphere Reactio ns 262
5.4.1 The Applications of the Marcus Expression 269
5.5 Inne r Sphere Redox Reactions 269
5.6 The Bridging Ligan d in Inner-S phere Redox Reactions 270
5.7 Some Oth er Features of Redox Reactions 272
5.7.1 Mixed Outer- and Inner-S phere Reactions 275
5.7.2 Two-Electron Transfer 276
5.7.3 Redox Catalyzed Sub stitutio n 276
5.7.4 Stereosele ctivity 277
5.8 Intra mo lecula r Electro n Transfer 279
5.8.1 Between Two M etal Centers 279
5.8.2 M etal Com plexes with Reducible Ligan ds 282
5.8.3 Indu ced Electron Transfer 284
5.9 Electr on Transfer in Protein s 285
5.10 Th e Futu re 288
References 288
Problems 292
6 The Modification of Ligand Reactivity by Complex Formation
299
6.1 The Metal as a Collecting Point Reactant 299
6.1.1 Neig hborin g Gr oup Effects 300
6.1.2 Temp late Chem istry 301
6.1.3 Collecting Reacta nt Mo lecules 303
6.2 Pr om otion of Reaction within the M etal-Bound Ligand 305
6.3 Hydrolysis of Co ord inate d Ligands 308
6.3.1 Carboxylate Esters : - C O
2
R - -CO
2
H 308
6 .3 .2 Amides and Pep t ides : - C O N H R -> - C O
2
H 311
6 .3 .3 N i t ri le s : - C N -> -C O N H
2
313
6.3.4 Phosphate Esters : RO PO
3
2
- -> PO
4
3
~ 314
6.3.5 Other Group s 317
6.4 Th e Acidity of Co ordin ated Ligan ds 317
6.4.1 Coo rdinated Water 318
6.4.2 Coordinated Am mon ia and Amines 320
6.4.3 Other Coord inated Ligands 320
6.5 Electrop hilic Sub stitutio n in M etal Complexes 322
6.6 M asking Effects 322
6.7 Disturban ce of Reaction Stoichiometry 323
6.8 M olecula r Strain Alte rations 323
6.9 Fun ctio n of the Ligan d 324
6.10 Con clusions 325
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X II Contents
References 325
Problems 328
7 Isomerism and Stereochemical Change 337
7.1 Con form ational Isomerism 334
7.2 Configuration al Isomerism 335
7.2.1 Plan ar, Tetrahedral 336
7.2.2 Planar, Square Pyram idal 336
7.2.3 Planar, Octahed ral 337
7.2.4 Tetrahed ral, Octa hedr al 338
7.3 Spin Equ ilibria in Octa hedra l Com plexes 339
7.4 Linka ge Isom erism 340
7.4.1 Rea rrang em ent Studies 341
7.5 Geom etrical and Optical Isomerism 343
7.6 Octah edral Complexes 343
7.6.1 Complexes of the Type M (AA )
3
343
7.6.2 Comp lexes of the type M (A A,)
3
348
7.6.3 Com plexes of the Type M (L,) (L
2
) 351
7.6.4 Complexes of the Type M (AA )
2
X
2
and M(AA)
2
XY 351
7.6.5 Isom eric Form s As Biological Prob es 354
7.7 Four-C oordinated Complexes 355
7.7.1 Op tical Isom erism in Tetrahed ral Com plexes 355
7.7.2 Geom etrical Isomerism in Square Plana r Complexes 356
7.8 Five-, Seven- and Eig ht-C oor dina ted Com plexes 359
7.9 Inversion and Proto n Exchange at Asymm etric Nitrogen 360
References 365
Problems 368
8 A Survey of the Transition Elemen ts 373
References 424
Problems 433
Problems Hints to Solutions 443
Index 455
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Kinetics and Mechanism of Reactions of T ransition Metal Com plexes
RalphG.WiMns
Copyright 2002 Wiley-VCH Verlag GmbH & Co. KGaA
ISBNs:3-527-28253-X(Hardback);3-527-60082-5(Electronic)
Ligands and Complexes
Ligand Abbreviations
N H
2
(C H
2
) ,NH
2
NH
2
(C H
2
)
2
NH(CH
2
)2NH2
en (x=2) ;
tn (x=3)
dien
NH
2
(C H
2
)
2
NH ( CH
2
)
2
NH ( CH
2
)
2
N H
2
trien
N[(CH
2
)
X
NH
2
]
3
tren (x=2); trpn (x= 3)
N H ( C H
2
C O
2
) |- ida
2
-
N(CH
2
CO
2
)1- nta
3
-
[ - O C O C H
2
]
2
N - X - N [ C H
2
C O O - ]
2
edta
4
- (X = (CH
2
)
2
);
pdta
4
- ( X=CH ( CH
3
)C H
2
);
cydta
4
" (X = l,2-cyclohexyl)
imid
py (R =
2,2'-bpy
4,4'-bpy
mbpy
+
tpy
pz
//A\
N N
bp z
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XIV
Ligands and Complexes
NH,-C(L
phen
acacH
d t c
2
-
HC
u
[14]ane-l,4,8,ll-N
4
[14]aneN
4
or cyclam
H,C
H,C
HoC
CH
3
C H ,
CH,
rac-(5,12)-Me
6
[14]aneN
4
tet b
H,C
C H ,
M e
2
pyo[14]trieneN
4
[18]crown-6
/
\ H
3
C
N \ O
/
-O
P \ N
0 .
cryptand 2
O
2
O
2
O
(the numbers indicate
the oxygens on each
strand of the
macrocyclic ligand)
2+
[18] di en eN
4
)
2
Some commonly used macrocycles.
The num ber in brackets indicates the size of the ring. The terms
ane and ene denote saturated and unsaturated rings respectively. The number of Hgating atoms is in-
dicated by a subscript. A trivial name is often used (G. A. Melson in Coordination Chemistry of
Macrocyclic Compounds, Ed. G. A. Melson, Plenum, New York, 1979, Chapter 1; Comprehensive Coor-
dination Chemistry, Ed. G. Wilkinson, Pergamon, Oxford, 1987. Several chapters in Vol. 2 (Ligands)).
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Ligands and Complexes
X
XV
R = H, tpypH
2
R = CH
3
, tmpypH
2
X = SOf , tppsH
2
X = N(CH
3
)
3
+
, tapH
2
Some Useful Water-Soluble Porphyrins
Co ( se p )
2 + / 3
Co(sar)
2
Co ( a z a m e sa r )
2 + / 3
N 0
2
Co(d inosar )
2+ / 3
+
Some Caged Cobalt Complexes The synthesis of these m acrocycles is based on b oth the metal template
effect as well as metal-ion activation of the imine moiety to nucleophilic attack, which initiates encapsula-
tion (Chap. 6). A large variety of groups can replace NO
2
in Co(dinosar)
3 +
yielding complexes with a
wide range of pro pertie s. Aga in, trivial abb reviatio ns are em ployed e.g. sep replaces 1,3,6,8,10,13,16,19-
octaazabicyclo[6.6.6]eicosane. A. M. Sargeson, Pure Appl. Chem. 56, 1603 (1984).
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Kinetics and Mechanism of Reactions of T ransition Metal Com plexes
Ralph G.WiMns
Copyright 2002 Wiley-VCH Verlag GmbH & Co. KGaA
ISBNs: 3-527-28253-X (Hardback); 3-527-60082-5 (Electronic)
Chapter 1
The Determination of the Rate Law
The single most important factor that determines the rate of a reaction is concentration
primarily the concentrations of the reactants, but sometimes of other species that may not
even appear in the reaction equation. The relation between the rate of a reaction and the con-
centration of chemical species is termed the rate law; it is the cornerstone of reaction
mechan isms. Th e rate law alone allows much insight into the mechanism . T his is usually sup-
plemented by an examination of other factors which can also be revealing. (For these, see
Chap. 2)
1.1 The Rate of a Reaction and the Rate Law
The rate, or velocity, of a reaction is usually defined as the change with time t of the concen-
tration (denoted by square brackets) of one of the reactants or of one of the products of the
reaction; that is ,
rate = V = -G?[reactant]Aft = n x fi?[pro duct] /
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2 1 TheDetermination of the Rate Law
order
in A. These are the most frequently encountered orders of reaction. The overall order
of the reaction is
(a + b).
The rate may be independent of the concentration of A, even if
it participates in the overall stoichiometry. In this case,
a =
0, and the reaction is
zero-order
in A. The concentration of A does not then feature in the rate law.
The formation of a number of chromium(III) complexes Cr(H
2
O )
5
X
2 +
from their consti-
tuent ions Cr(H
2
O ) |
+
a n d X " ,
Cr(H
2
O )^
+
+ X~ - C r (H
2
O )
5
X
2 +
+ H
2
O (1.3)
where X " represents a unidentate ligand, obeys the two-term rate law (the coordinated water
being usually omitted from the formula)
V = d[CvX
2+
]/dt = / t , [ C r
3 +
] [ X - ] + A :
2
[C r
3 +
] [ X - ] [ H
+
] -
1
(1.4)
over a wide range of concentration of reactants and acid.
2
Symbols such as
a
and
b
instead
of k
x
and k
2
may be used in (1.4), since these quantitites are often composite values, made up
of rate and equilibrium constants. To maintain both sides of (1.4) dimensionally equivalent,
V = M s"
1
= k
]
X M
2
= k
2
X M (1.5)
k
l
and
k
2
must obviously be expressed in units of M^
1
s "
1
and s^
1
respectively. This simple
application of dimension theory is often useful in checking the correctness of a complex rate
law (see 1.121 and Probs. 1, Chap. 1 and Chap. 2).
If we consider specifically the formation of CrBr
2 +
, the rate law
d[CrBr
+
]/dt = 3.0 x 10"
8
[C r
3 +
] [Br~] + 3.6 x 10~
9
[C r
3 +
] [Br"] [H
+
] "
1
(1.6)
holds at 25.0
C ,
and an ionic strength of 1.0 M.
2
The majority of kinetic determinations are
carried out at a constant ionic strength by the addition of an "unreactive" electrolyte which
is at a much higher concentration than that of the reactants. At high acid concentrations
( > 1 M), the first term in (1.6) is larger than the second, and the reaction rate is virtually acid-
independent. At lower acid concentrations (~10"
2
M), the second term dominates and the
rate of reaction is now inversely proportional to [H
+
]. This emphasizes the importance of
studying the rate of a reaction over as wide a range of concentrations of species as possible
so as to obtain an extensive rate law.
1.2 Th e Ra te Law Directly from R ate M easurem ents
If the rate of a reaction can be measured at a time for which the concentrations of the reac-
tants are known, and if this determination can be repeated using different concentrations of
reactants, it is clear that the rate law (1.2) can be deduced directly. It is not often obtained
in this manner, however, despite some distinct advantages inherent in the method.
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1.2 The Rate Law Directly from Rate Measurements
3
1.2.1 Initial-Rate Method
The rate of reaction is measured at the commencement of the reaction, when the concentra-
tions of the reactants are accurately known, indeed predetermined.
The decomposition of H
2
O
2
(1.7)
2 H
2
O
2
2 H
2
O + O
2
(1.7)
is catalyzed by many metal complexes and enzymes.
3
The rate law for catalysis by an
iron(III) macrocycle complex 1 has been deduced by measuring the initial rates
V
o
of oxygen
produc t ion .
3
A selected number from many data are shown in Table 1.1.
CH
Table 1.1.
Initial Rate Data for the Catalyzed Decomposition of H
2
O
2
by an Fe(III) Macrocycle at25 C
3
Run
1
2
3
4
5
6
7
8
9
10
11
12
[H
2
O
2
]
0
M
0.18
0.18
0.18
0.12
0.18
0.36
0.18
0.18
0.18
0.18
0.18
0.18
Fe(III)
m M
0.41
0.74
1.36
0.66
0.66
0.66
0.66
0.66
0.66
0.66
0.66
0.66
[ H
+
]
1 0
5
M
2.45
2.45
2.45
2.45
2.45
2.45
1.0
2.45
11.5
2.45
2.45
2.45
[OAc]
T
M
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.02
0.05
0.17
10
3
x
Initial Rate
1 0
3
M s~ '
54
85
160
37
78
306
191
86
24
233
86
26
1 0
3
/ t
a
S "
1
5.0
4.4
4.5
5.0
4.5
4.5
4. 9
4.8
6.5
5.2
4.8
4. 9
a
Obtained from k = F
0
[H
+
]
[ O A c ]
T
[Fe( I I I ) ] - ' [H
2
O
2
]n
2
, a rearrangement of (1.8). Note the units
oik.
It is apparent from these data that within experimental error the initial rate is proportional
to the initial concentration of iron(III) complex, (Runs 1-3) to the square of the initial H
2
O
2
concentra tion [H
2
O
2
]
0
(4-6) and inversely dependent on both H
+
(7-9) and total acetate
[OAc]
T
(10-12) concentrations. A rate law involving the initial rate V
o
,
V
o
= +d[O
2
]/dt = * [ F e ( I I I ) ] [ H . O ^ t H+r ' t O A c l T
1
(1.8)
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4
1 TheDetermination of theRateLaw
applies therefore
and
values
of
k calculated
on
this basis
are
reasonably c onstant. They
are
shownin
the
last colum nofTable 1.1. Raw initial rate data
are
also includ edinpapers dealing
withtheCr(VI)-I~ reaction,
4
and thereaction of pyridoxal phosphate with glutamatein the
presence of copper ions .
5
SeeProb.2.
Initial reaction rates areoften estimated from the steepest tangents to absorbance/t ime
traces. These initial gradients (absorbance units/sec)
are
easily conv erted into
M s "
1
unitsby
using theknown molar absorptivities of products andreactants. Alternatively, absorbance
(>)/Time (T) plots can be fit to a function (1.9)leading to (1.10)and(1.11)
F(>) = Cl + C2T + C3T
2
+ CAT
3
. . . (1.9)
d[D]/dt
= C2 +
2C3T
+ 3C4T
2
(1.10)
A t r = 0 ,
d[D]/dt
=C2 (1.11)
as illustrated in
the
studyof
the
reaction ofH
2
O
2
with 1,3-dihydroxybenzene in
the
presence
of Cu(II) ions ,
6 ( a )
and in the
disproportionation
of
Mn(VI) .
6 ( b )
1.2.2 Critique
of
the Initial-Rate M ethod
The merits anddifficulties in the use of this method aresummarized:
(a )
The
initial-rate method isuseful in
the
studyof reactions complicated by side reactions
or subsequent steps.
The
initial step of hydrolytic polymerization ofCr(III) is dimerization
of the monomer.Assoonas thedimerisformed however,itreacts withthemo n o me r orwith
dimerto form trimerortetram er, respectively. Initia l rates (usingapHstat, Sec. 3.10.1 (a))are
almost essential
in the
m easurement
of
dimer formation.
7
The reverse reaction in(1.12)
Cr(H
2
O)j
+
+ NCS~ C r ( H
2
O )
5
N C S
2 +
+ H
2
O (1.12)
is unimportant during the early stages ofthe forward reaction when theproduc ts havenot
accumulated.
A
small initial loss
of
free th iocyan ate
ion can be
accurately mo nitored
by
using
the sensitive Fe(III) colorimetric method. The reverse reaction
can be
similarly studiedfor
the
initial rate of thiocyanate ionappearance by starting with Cr(H
2
O )
5
N C S
2 +
ions .
8
(b) Complicated kinetic expressions andmanipulations maybethus avoided. Theinitial-
rate method
is
extensively employed
for the
study
of
enzyme kinetics
and in
enzyme assay
(Prob.
3),
simplifying
the
kinetics
(see
Selected B ibliography ). O bviously
the
method cannot
cope with
the
situationof induction periodsorinitial burst behavior. Fu rtherm oreacomplex
rate law, givingawealth of mechanistic information, may nowbedisguised(for anexample
seeRef. 9).Despite thistheinitial-rate approa chcan beuseful for distinguishing betweentwo
similar reaction schemes e.g.(1.13)and (1.14) (Sec.
1.6.4(d)).
A + B ^ C -> D (1.13)
C ^ A + B -> D (1.14)
in whichAplusByieldsDeither viaC ordirectly.Theresolutio n otherwise requires veryac-
curate relaxation data.
1 0
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1.3 Integrated Forms of the Rate Expression 5
(c) Two obvious disadvantages
of
the method are that many individual runs must
be
carried
out
to
build
up a
rate law
and a
sensitive monitoring technique
is
required
in
order
to
obtain
accurate concentra tion/t ime datafor thefirst fewpercentage of the reaction. While nothing
can be done about the first point, theadvent of very sensitive computer-linked monitoring
devices
is
ameliorating
the
sensitivity problem.
1.2.3 Steady-State Approach
In thesteady-state appro ach to determining therate law, solutions contain ing r eactan tsare
pumped separately
at a
constant flow rate into
a
vessel ("reac tor") , t he contents
of
which
are
vigorously stirred. After
a
while, products
and
some reactan ts w ill flow from
the
reactor
at
the same total rate
of
inflow and
a
steady state will be attaine d,
in
which the reaction will take
placein thereactor witha
constant concentration of reactants,
andtherefore a
constant rate.
Thisisthe basisof the stirred-flow reactor,orcapacity-flow m eth od ." A lthough the method
has been little used,
it has the
advantage
of a
simplified kinetic treatment even
for
complex
systems.
1.3
Integrated Forms of the Rate Expression
Considera reaction tha t proceedstocompletioninwhich the co ncentration ofonly one reac-
tant,
A,
changes appreciably during the reaction. This may arise because (1) there
is
only
one
reactant
A
involved
for
example,
in a
stereochemical change; (2) all the other possible reac-
tants
are in
much larger
(^
tenfold) concentration than
A; or (3) the
concentration
of one
of the other reactants maybeheld co nstant bybuffering, or be constantly replenished, as it
wouldbe if itwere actingin a catalytic role. Attention needsto befocused therefore onlyon
the change
of
concentration
of A as
the reaction proceeds, a nd we can,
for
the presen t, forget
about
the
other reactants.
Now,
-d[A]/dt
=
k[A]
a
(1.15)
The manner
in
which [A] varies with time determines the order
of
the reaction with respect
to
A.
Since
it is
usually much easier
to
measure
a
concentration than
a
rate,
the
form (1.15)
is in tegra ted .
n
Thethree situations
a =
0, 1, and 2account for theoverwhelming numb er
of kinetic systems weshall encounter, with
a =
1 by far themost c om mon behavior.
a =
0,
zero-order
in A:
-d[A]/dt = k (1.16)
[A], = [A]
o
-
kt (1.17)
a =
1,
first order
in A:
-d[A]/dt
= k[A]
(1.18)
lo g[A],
=
log[A]
0
-
kt/23 (1.19)
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6
The Determination of the Rate Law
or now increasingly common,
In
[A],
= ln [A]
0
- kt
a =
2, second-order in A:
-d[A]/dt = k[A]
2
1 1
[ A ] ,
[ A ]
o
+ kt
(1.19a)
(1.20)
(1.21)
The differential (rate) forms are (1.16), (1.18) and (1.20), and the corresponding integrated
forms are (1.17), (1.19) (or (1.19a)) and (1.21). The designations [A]
o
and [A], represent t he
concentrations of A at zero time and time /. Linear plots of [A],, In
[A],
or
[A]
r
~'
vs time
therefore indicate zero-, first, or second order dependence on the concentration of A. The im-
portant characteristics of these order reactions are shown in Fig. 1.1. Notwithstanding the ap-
pea ran ce of the p lots in 1.1 (b) and 1.1 (c), it is not always easy to differe ntiate betwe en first-
and second-order kinetics.
13
'
14
Sometimes a second-order plot of kinetic data might be
mistaken for successive first-order reactions (Sec.
1.6.2)
with similar rate constants.
1 5
3A
0
Ao
2
Ao
4
Ao
3A
0
4
A,
4
Ao
3A
0
4
A
o
2
A
o
4
\
I
-\
; \
i
- \
\
c)
T
i i
b)
a)
T 2T
Time
Fig. 1.1 The characteristics of (a) zero- (b) first- and (c)
second-order reactions. In (a) the concentration of A
decreases linearly with time until it is all consumed at time
T. The value of the zero-order rate constant is given by
A
o
/T. In (b) the loss of A is exponential with time. The
plot of In [A], vs time is linear, the slope of which is
k,
th e
first-order rate constant. It obviously does not matter at
which point on curve (b) the first reading is taken. In (c)
the loss of A is hyperbolic with time. The plot of [A]," ' vs
time is linear with a slope equal to
k,
the second-order rate
constant.
In the unlikely event that a in (1.15) is a non-integer, the appropriate function must be plot-
ted. In general, for (1.15), a = 1
1 2
(see Prob. 8)
[Air
^-r = ( -
(1.22)
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1.4 Mono phasic Unidirectional Reactions
7
It should be emphasized that in the case that A is in deficiency over other reactants, B etc.,
then
k,
relating to loss of A, is a pseudo rate constant and the effect of the other reactants
on the rate must still be assessed separately (Sec. 1.4.4).
1.4 Monophasic Unidirectional Reactions
We shall first consider some straightforward kinetics, in which the loss of A, in the treatment
referred to above, is monophasic and the reaction is unidirectional, that is, it leads to
loss of A.
1.4.1 Zero-Order Dependence
It is impossible to conceive of a reaction rate as being independent of the concentration of
all the species involved in the reaction. The rate might, however, very easily be independent
of the concentration of one of the reactants. If this species, say A, is used in deficiency, then
a pseudo zero-order reaction results. The rate
d[A]/dt
will not vary as [A] decreases, and
will not depend on the initial concentration of A.
In the substitution reaction
Ni(POEt
3
)
4
+ C
6
H
n
N C N i ( P O E t
3
)
3
(C
6
H
n
NC) + POEt
3
(1.23)
the loss of C
6
H
n
NC has been followed in the presence of excess Ni complex.
16
The linear
plot of absorbance, which is proportional to isonitrile concentration, vs time indicates a reac-
tion zero-order in isonitrile (Fig. 1.2).
100 200
Time (min)
Fig. 1.2 Zero-order kinetic plots for reaction (1.23). Con-
centrations of Ni(POEt
3
)
4
are 95 mM(I), 48 mM(II) and
47 mM(III). Those of C
6
H
n
NC are 6.5 mM(I),
5.0 mM(II) and 9.7 mM(III).
1 6
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8
/
TheDetermination
of
the Rate Law
The slope
of the
zero-order plot (when absorb ance
is
converted into concentration)
is
k,M s~'. T he value
of
k
is
found
to be
proportional
to the
concentration
of
N i (P OE t
3
)
4
,
which
is
used
in
excess (Fig.1.2),
17
-d[Ni(POEt,)
4
]/dt
=
- l / 2 r f [ C
6
H
n
N C ] / < f t
= k =
/fc, [N i(P O Et
3
)
4
] (1.24)
The reaction
is
theref ore overall first-order, w ith
a
first-order rate constant Ar^s"
1
)-
The
zero-order situation
is not
often en counter ed (Prob . 4).
A
number
of
examples are compiled
in Ref. 18
and
on e
is
shown
in
Fig. 8.3.
1.4.2 First-Order Dependence
First-order reactions are extremely comm on and form the bulk of reported kinetic studies. The
rateoflossof the reactant Adecreasesasthe concentration of Adecreases. The differential
form (1.18) leads
to a
number
of
equivalent integrated expressions,
in
addit ion
to
(1.19)
and
(1.19
a) :
[ A ] , = [ A ]
o
exp(-*O (1 .25)
ln ( [A]
0
/ [A] , ) = *f (1.26)
-dln[A],/dt = k (1.27)
Important quantities characteristic of a first-order reaction aret
i/2
, thehalf-life of the
reaction, which is the valueoft when [A]
(
= [A ]
0
/2 , and T, the relaxation time,orme an life-
time, defined ask~
l
.
k
=0.693/t
U2
=1/T
(1.28)
The latter isinvariably used in the relaxation orphotochemical approach torate m easure-
ment (Sec. 1.8), and is the time taken for A to fall to1/e(1/2.718)of its initial value. Half-lives
or relaxation times are constants over
the
complete reaction
for
first-order
or
pseu do first-
order reactions. The loss
of
reactant
A
with time may
be
described
by a
single exponential
bu t
yet
may hide two
or
more concurrent first-order an d/o r pseudo first-order reactions.
The changeinthe absorbanceat450 nm when cis-Ni([14]aneN
4
)( H
2
O)2
+
(for structure see
"Ligands andCom plexes") isplunged into 1.0 MHC1O
4
isfirst-order (rate constan t = k).
It
is
comp atible with concurrent isomerization
and
hydrolysis (1.29)
trans-Ni([l 4]aneN
4
)(H
2
O)
2
H
"
CM-Ni([14]aneN
4
)(H
2
O)f
+
^ " (1.29)
hyd
N i
2 +
+ protonated [14]aneN
4
4
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1.4 MonophasicUnidirectional Reactions
fast
k-1
H ,0
H ,0
k
2
H,O
N i
2+
+
protonated
ligand
For this system
-d[2]/dt
= k[2] = (k
is
k
m
)[2]
(1.30)
Estimation of the amountsof 3 and 4produced from the final absorbance change
1 9
allows
determination of bo th k
isom
andk
hyd
.
zo
Seealso Ref. 21 .
(1.31)
Concurrent first-order changes invariably arise from thedecayof anexcited state *A which
can undergoanumberof first-order changes.Inthe presenceof anexcessofadded B, pseudo
first-order transformations
can
also occur:
A
hv
photochemical reaction products
A +h\' (luminescence, includes fluorescence
and phosphorescence)
A
+
heat (radiationless deactivation)
A and/or products (quenching process)
(1.32)
The analysis
of the
immediate products
or the
isolation
of one of the
steps
is
essential
in
order
to
resolve
the
complexity.
22
The reaction
of a
mixture
of
species
A and Aj
which interconvert rapidly compared with
the reaction under study, can also lead to a single first-order process. Inorder toresolvethe
kinetic data, information on the A ^ A equilibrium is essential. When therelative am ounts
ofA and A[ arepH -con trolle d however (Sec.
1.10.1)
or when theproducts of reaction of A
and
Aj
differ
and do not
interconvert readily, resolution
is
also
in
principle possible.
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10
/
The
Determination
of the
Rate
Law
1.4.3
Second-Order Dependence
Second-order kinetics playan important rolein thereactions of complex ions.Twoidentical
reactantsmay be
2 A > products
involved. Eqn (1.21) is modified slightly to (1.34).By convention, the
= 2kt
1
[A], [A]
o
(1.33)
(1.34)
second-order rate constant for (1.33)isdesignated
2k,
becausetwoidentical molecules disap-
pear
for
each encounter leading
to
react ion.
23
Reaction (1.33)
is a key
step
in the
process
of
dimerizat ion or disproport ionation of A and is often observed whena transient radicalA is
produced
by
photolysis
or
pulse radiolysis. Flash photolysis
of
hexane solutions
of CO and
M
2
(CO)
10
(M = Mn or Re)produces M(CO )j radicals .At the termination of
M
2
(CO)
1
2 M ( C O ) ;
(1.35)
the flash the radicals recombinein a second-order manner toregenerate M
2
(CO)
10
witha rate
la w
24
'
25
-d[M(COY
5
]/dt =
(1.36)
iOO 800
Time
(ps)
Fig.
1.3
Second-order kinetic plot
for
decay
of
Re(CO)j
obtained
by
flash photolysis
of 120
\iM Re
2
(CO)
10
with
10
mM CO in
isooctane.
(CO
slows further reactions
of
Re(CO)j)
At X = 535 nm, the
mola r absorbanc e coeffi-
cient,
e
M
of
Re(CO);
is 1.0 x
l O ' M - ' c m "
1
. Since
therefore from Beer-Lambert
law (D,-D
t
)
=
[A]e
m
l, where
D's are
optical absorbances
and
1
is the
path length
(10
cm), the
slope
of Fig. 1.3 is
2Ar/s
m
l
and
2k therefore
equals
(5.3 x
1 0
5
) ( l0
3
) ( l 0 )
= 5.3 x
l O ' M - ' s "
1
.
Ref. 25.
Reproduced with permission
of the
Journa l
of the
American Chemical Society.
1982,
American Chemical
Society.
The appropiate absorbance function
vs
time
is
linear
(Fig. 1.3).
25
The
values
of k are
9.5 x 1 0
8
M - ' s^
1
(M = Mn) and 3.7 x l O ' M "
1
s"
1
(M = Re) .
24
'
2 5
The second-order decay
of Cr(V)affords ano ther example, of many, of the type (1.33).
26
Second-order reactions between two dissimilar moleculesA and B are invariably studied
under pseudo first-order conditions (Sec. 1.4.4) because thisis by far the simpler procedure.
If however this condition cannot
be
used because,
for
example, both reactants absorb heavily
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1.4 MonophasicUnidirectional Reactions 11
or low concentrations of both reactants must be used because of high rates, then for the reac-
tion
A + B C /t, (1.37)
-d[A]/dt
= k,
[A] [B] (1.38)
Integration results in
1 ln [A]
0
[B] ,
[B]
o
- [A]
o
[B]
0
[A],
=
k
x
t
(1.39)
with subscripts t and 0 representing time t and 0, respectively. Applications of this equation
are given in Refs. 27-29 and include the treatment of flow traces for the rapid second-order
reaction between Eu(II) and Fe(III).
2 7
A simplified form of (1.39) equivalent to (1.21) arises
when the starting concentrations of A and B are equal. This condition [A]
o
= [B]
o
must be
set up experimentally with care
3 0
although in rare cases equal concentrations of A and B
may be imposed by the conditions of their generation.
31
(See Prob. 9.)
1.4.4 Conve rsion of Pseud o to Rea l Ra te Con stants
Having obtained the exponent
a
in (1.15) by monitoring the concentration of A in deficiency
we may now separately vary the concentration of the other reactants, say B and C, still keeping
them however in excess of the concentration of A. The variation of the pseudo rate-constant
k with [B] and [C] will give the order of reaction b and c with respect to these species,
32
leading to the expression
k =
ArJ B]
6
[C ]
c
(1.40)
and therefore the full rate law
-d[A]/dt =
1
[A] [B]
6
[C ]
C
(1.41)
We can use this approach also to examine the effects on the rate, of reactants that may not
be directly involved in the stoichiometry (for example, H
+
) or even of products. It is the most
popular method for determining the rate law, and only rarely cannot be used.
Considering just one other reactant B, we generally find a limited number of observed varia-
tions of k{ = V/[A\") with [B]*. These are shown as (a) to (c) in Fig. 1.4. Nonlinear plots of
k
vs
[B ]
b
signify complex multistep behavior (Sec.1.6.3).
(a) The rate and value of k may be independent of the concentrations of other reactants
(Fig. 1.4
(a)).
This situation may occur in a dimerization, rearrangement or conformational
change involving A. It may also arise in the solvolysis of A since the reaction order with
respect to solvent is indeterminable, because its concentration cannot be changed.
(b) The value of k may vary linearly with [B]
6
, and have a zero intercept for the ap-
propriate plot (Fig. 1.4 (b)). This conforms to a single-term rate law
- d[A]/dt = k [A ]
a
= k-i
[
A]
"
[B] * (1.42)
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12 1 TheDeterminationof the Rate Law
(a )
M = k
IAI
* . * .
3
[A ]
a
[B ]
i)
a
m
b
4+
H,CN
Fig. 1.4 Common variations of
V/[A]
with [B]*.
N C H ,
In the oxidation of Fe(II)P
4 +
(5) by excess O
2
there is a second-order loss of Fe(II). The
second-order rate constant k is linearly dependent on [O
2
] with a zero intercept for the
k/
[O
2
] plot. An overall third-order rate law therefore holds.
1 4
= Ar,[Fe(II)P
4 +
]
2
[O
2
]
(1.43)
Wh e n a = b = 1 in (1.42) the overall reactio n is secon d-ord er. E ven a qui te small excess
of one reagent (here B) can be used and pseudo first-order conditions will still pertain.
3 3
As
the reaction proceeds, the ratio of concentration of the excess to that of the deficient reagent
progressively increases so that towards the end of the reaction, pseudo first-order conditions
certainly hold. Even if [B] is maintained in only a two-fold excess over [A], the error in the
comp uted second-order rate constant is ^ 2% for 60% co nversion.
34
(c) The value of
k
may be linear with respect to [B] *bu t have a resid ual valu e at zero [B]
(Fig. 1.4(c)). This behavior is compatible with concurrent reactions of A. The oxidation of
Fe(phen)3
+
by excess Tl(III) conforms to this behavior leading to the rate law
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7.5Monophasic Reversible Reactions 13
= k[Fe(phen)
2+
] = k
x
[Fe(phen)
2+
] + A:
2
[Fe(phen)f
+
] [Tl(III)]
(1.44)
The second-order redox reaction, giving rise to the rate constant k
2
, is accompanied also
by loss of the iron(II) complex by hydrolysis, which leads to the
k
l
term. The latter can be
more accurately measured in the absence of Tl(III).
35
The kinetics of substitution of many
square-planar complexes conform to behavior (c), see Sec. 4.6. It is important to note that an
intercept might be accurately defined and conclusive only if low concentrations of
B
are used.
In the base catalyzed conversion
Co(NH
3
)
5
ONO
2+
-> Co(NH
3
)
5
NO
2+
(1.45)
-rf[Co(NH
3
)
5
ON0
2+
]A# = (/t, + Ar
2
[OH-])[Co(NH
3
)
5
ONO
2+
] (1.46)
the k
x
term can be established only if low OH~ concentrations of 0.01-0.1 M are used.
36
1.5
Monophasic Reversible Reactions
The only reactions considered so far have been those that proceed to all intents and purposes
(>95%) to completion. The treatment ofreversiblereactions is analogous to that given above,
although now it is even more important to establish the stoichiometry and the thermodynamic
characteristics of the reaction. A number of reversible reactions are reduced to pseudo first-
order opposing reactions when reactants or products or both are used in excess
A
^ X k
x
,k_
x
,K (1.47)
of A and X. The order with respect to these can then be separately determined. The approach
to equilibrium for (1.47) is still first-order, but the derived first-order rate constant
k
is the
sum of k
1
(the forward rate constant) and k_
x
(the reverse rate constant):
a 4 8 )
This equation resembles (1.26) but includes [A]
e
, the concentration of A at equilibrium,
which is not now equal to zero. The ratio of rate constants, &,/_, = K, the so-called
equilibrium constant, can be determined independently from equilibrium constant
measurements. The value ofk,or the relaxation time or half-life for (1.47), will all be indepen-
dent of the direction from which the equilibrium is approached, that is, of whether one starts
with pure A or X or even a nonequilibrium mixture of the two. A first-order reaction that
hides concurrent first-order reactions (Sec.1.4.2) can apply to reversible reactions also.
The scheme
A + B ^ X k
l
,k_
i
(1.49)
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14
1 The Determination of the Rate Law
can be redu ced to (1.47) by using B in excess, and creating thereby a p seud o first-order reversi-
ble reaction. The rate law that arises is
37
V=k[A] =
(1.50)
where k
x
[B] and k_
x
are the forward an d reverse first-order rate consta nts. Such a situation
arises in the interaction of V(H
2
O )|
+
with SCN~ ions,
N C S "
^ V(H
2
O )
5
N C S
2 +
+ H
2
O k
x
,k_
x
,K
(1.51)
which is studied using a large excess of V
3 +
ions, although an equilibrium position is still at-
tained. A plot of the pseudo first-order rate constant k for the approach to equilibrium vs
[V
3 +
] is linear (Fig.1.5).
38
The slope is k
x
and the intercept is k_
x
:
k = k
x
[V
3+
] + k_
x
(1.52)
10 20
|V
3
*](mM)
Fig. 1.5 P lot of fr(s-') vs [K
3 +
] for reac-
tion (1.51) at 25C. [H
+
]
= 1.0 M
(circles); 0.50 M (triangles); 0.25 M
(diamonds) and 0.15 M (squares).
38
It is not always easy to obtain an accurate value for k_
x
from such a plot. However, com -
bination of k
x
with K(= k
x
/k ^
x
) obtain ed from spectral me asurem ents, yields a mean ingful
value for
k_
x
.
Th e plots in Fig. 1.5 show the indepe nden ce of the values of
k
x
and _, on the
acid concentrations in the range 0.15 to 1.0 M. There are slight variations to this approach,
which have been delineated in a number of papers.
3 9
Since the k vs [B] plot illustrated in Fig. 1.5 is identical to that obtained with unidirectional
concurrent first- and second-order reactions of A (Fig. 1.4 (c)) confusion might result if the
equilibria characteristics are not carefully assessed. The pseudo first-order rate constant
k
for
the reaction
P t (d ie n )H
2
O
2 +
+ C\' ^ Pt(dien)Cl
+
+ H
2
O k
x
,k_
x
(1.53)
has been determine d using excess [Cl ]. The variatio n of k with [Cl ] resembles tha t shown
in Fig.1.4(c), i.e.
k = a + b[Cl~]
(1.54)
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1.5
Monophasic
Reversible Reactions
15
The results were interpreted
as a
two-term rate law
for an
irreversible r eaction ,
b
represent-
in g
the
second-order rate constant
for
attack
by Cl~ ion andaan
unusual dissociative path
(Sec.
4.6). More recent w ork indicates th at
at the low [Cl~]
concentrations used, reaction
(1.53)
is
reversible
and Eqn.
(1.54)
is
better interpreted
in
terms
of a
reversible reaction
as
depicted
in
(1.53)
in
which
in
(1.54),
b= k
{
anda= k_
x
.
4 0
As a
check
the
value
ofb/a = K
is close to that estimated for reaction (1.53).
The conversionofsecond-order reversible reactionstoreversible first-order kinetics by using
allbut one of thereactants and all but one of theproducts in excess is a valuable ploy.The
reversible reaction
(1.55)
is studied
by
using excess Fe
3 +
and
F e
2 +
.
The
plot
of
In (Z?
e
D
t
) vs
t ime
is
linear, where
D
e
andD
t
represent
the
absorbance
of the
highly colored radical
at
equilibrium
and
t ime
t,
respectively.
The
slope
of
this plot
isk
ohs
,
which from (1.48)
is
given
by
k
obs
=
Ar,[Fe
3+
]
+
Ar_,[Fe
2+
]
(1.56)
Thus aplot of
o b s
/ [ F e
2 +
] vs [Fe
3 +
] is linear with slopek
l
andintercept k_
1
.
41
Forother
examples of this approach seeRefs.42 and43.The full treatm ent for
A
+ B ^ X + Y
is quite tedious
41
'
44
'
45
as it is for
(1.58) Prob.
10
46
'
47
A + B ^ X
and (1.59)
48
'
49
A
= X + Y
(1.57)
(1.58)
(1.59)
but simplified approaches,
45
'
49
-
5 0
relaxation treatment
and
compute rs
5 1
have largely removed
the pain.
1.5.1 Conversion
of
Reversible
to
Unidirectional Reactions
An often useful approachis toeliminate the elementsofreversibility from areactionandforce
it
to
completion, either
by the use of a
large excess
of
reactant
or by
rapid removal
of one
of theproducts . A good illustration is afforded by thestudyof
P tC l (P P h
3
)
2
C O
+
+ ROH P tCl(PPh
3
)
2
C O O R + H
4
(1.60)
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16 / The Determination of the Rate Law
Since the equilibrium quotient K
l
is small, a nonnucleophilic base is added to the reaction
mixture to react with liberated protons and drive the reaction to completion (left to right).
Using an excess of ROH then ensures simple unidirectional pseudo first-order (rate con-
stant kf ) kinetics:
-din [PtCl(PPh
3
)
2
CO
+
]/ X process
by scavenging X rapidly and preventing its return to A. Thus the intramolecular reversible elec-
tron transfer in modified myoglobin (Sec. 5.9)
i P R O T E I N -
R u
2 +
i
F e
3 +
;=
i P R O T E I N -
^ R u
3 +
i
F e
2 +
-
iPROTEIN-
^ > R u
3 +
i
F e
2
'
h
C O
(1.63)
can be converted into a unidirectional process controlled by k
x
, by carrying ou t the ex-
periments in the presence of CO which binds strongly to the ferrous heme.
54
Care should be exercised that excess of one reactant does in fact promote irreversible reac-
tion if this is the desired object, otherwise invalid kinetics and mechanistic conclusions will
result. Consideration of the reduction potentials for cytochrome-c Fe(III) and Fe(CN)g"
(0.273 V and 0.420V respectively) ind icates th at even by using a 10
2
-1 0
3
fold excess of
Fe(CN)|j~,
reduction of cytochrome-c Fe(III) will still not be complete. An equilibrium
kinetic treatment is therefore necessary.
45
1.6 Multiphasic Unidirectional Reactions
Atten tion is now directed to reactions that show a nonlinear plot of the approp riate function
or that have rate laws that are altered with changes in the concentration of the species involved
in the reaction. Such deviations are usually associated with concurrent and consecutive reac-
t ions.
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1.6 Multiphasic Unidirectional Reactions
17
1.6.1 Concurrent Reactions
A single species A (produced for example by radiolytic or photolytie means) may often disap-
pear by concurrent first- (or pseudo first-) order (,) as well as by a second order process
(2 k
2
) already alluded to (Sec.
1.4.3).
Thus
-d[A]/dt = 2k
2
[A ]
2
(1.64)
At higher concentrations of A, the second-order process is more important and loss of A
is second-order. A s the conc entration of A decreases, so the first term in (1.64) becom es d om i-
nant and decay of A is a purely first-order process (it may, for example, represent decomposi-
tion by solvent
24
) (Fig.
1.6).
55
The rate curve may be analyzed by a relatively straightforward
linearizing method.
2 3
0.4
a
^ b
Time (ms)
Fig. 1.6 Decay of (NH
3
)
5
Co ( m b p y ' )
3 +
at pH 7.2 and
25C. The transient was generated by using equivalent
amounts (10 uM) of (NH
3
)
5
Co ( m b p y )
4 +
and COf.
The decay of the transient was nicely second-order up
to 85% reaction (a). After this, when the Co(III)
radical concentration is small (1-2 uM), there is a
slight
deviation for the expected second-order plot
(not shown) and a first-order reaction (b) remains
(k =
5.4 x 1 0
2
s - ' ) -
5 5
The difference between the
(steep) second-order decay and the (extrapolated) first-
order loss is apparent.
If a mixture of A and B undergoes parallel first-order or pse udo first-order reactions to give
a common product C, and A and B do not interconvert readily compared with the reaction
under study,
[C ]
e
- [C], = [A]
o
e xp ( -
[B ]
0
exp(-A:
2
Z )
(1.65)
where k
x
and k
2
are the first-order rate constants for conversion of A and B, respectively.
12
The resultant semilog plot of [C] vs time will in general be curved, and can be disected
algebraically or by a computer program.
5 6
The hydrolysis of the anions in the complexes
Na[Co(medta)Br] and K[Co(medta)Br] has been examined (medta = N-methyl-ethylene-
diaminetriacetate). For the semilog plots the Na salt gives marked curvature, whereas the plot
of the K salt is linear over four half-lives (Fig. 1.7). It is considered that the sodium salt is a
mixture of isomers, in which the bromine is either in an equatorial or an axial position of the
cobalt(III) octahedron. The biphasic plot can be separated into a fast component and a slow
one. Significantly, the fast portion matches exactly the semilog plot for the K salt, which is
considered isomerically pure.
5 7
A similar concurrent hydrolysis pattern has been observed
with other complex ions.
5 8
It is not always easy to distinguish concurrent from consecutive
reactions, as we shall see in the next section.
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18
1 The Determination of the Rate Law
0.02 -
50
Time (min)
Fig. 1.7 Semilog plot of optica l a bsor banc e
at 540 nm vs time for hydrolysis of
Co (m edt a)B r at 78.8 C and pH 2.6. Mix-
ture of isomers (+); fast component derived
from mixture by subtraction of slow compo-
nent (o); experimental points for fast isomer
C).
5 7
1.6.2 Consecutive Reaction s with no Elem ents of Reversibility
Consecutive reactions figure prominently in Part II. Since complex ions have a number of
reactive centers, the product of one reaction may very well take part in a subsequent one. The
simplest and very common, but still surprisingly involved, sequence is that of two irreversible
first-order (1.66) or pseudo first-order (with X and Y in large excess) (1.67) reactions,
B
=*,'[X];
C (1.66)
(1.67)
(1.68)
If k
{
> k
2
then both steps in (1.66) can be analyzed separately as described previously. If
k
2
> k
]
, then only the first step is observed a nd
-d[A]/dt = d[C]/dt =
(1.69)
The only way for determining k
2
will be through isolation and separate examination of B
(see later). If B is not isolable however and its properties are unkno wn , real difficulties might
arise.-
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1.6 Multiphasic Unidirectional Reactions 19
The rate equations for (1.66) are
-d[A]/dt = A:, [A] (1.70)
d[B]/dt = A-JA] - k
2
[B] (1.71)
d[C]/dt = k
2
[B]
(1.72)
Integrating these equations, and assuming [A] = [A]
o
, and [B] = [C] = 0 at
t =
0, we
obtain the concentrations of A, B and C at any time / in terms of the concentration, [A]
o
:
[A] = [A]
o
exp
(-*,*)
(1.73)
[ A ] i-
[B] =
Jo
' [exp(-A-
1
f)-exp(-A:
2
?) (1.74)
k
2
k
t
[C]
=
[A]
o
|
1
-
k
\
exp(-A-,0 + *' exp(- *
2
f) (1.75)
Often
k
x
will be comparable in value to A:
2
and in this event the distribution of A, B and
C with time is illustrated for example by the stepwise hydrolysis of a cobalt(III) complex 6,
(X = -OC
6
H
3
(NO
2
)
2
)
O
0 H
2
+ /y
//
N >
NP.r-0
(H
3
N)
5
CoOP X 2 t i _(H
3
N)
4
Co^ ^ P ( + X ' ) - ${ &
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20
/ TheDetermination of the RateLaw
30
Time imin)
Fig.
1.8
D istribut ion
of
6,
7
and
8
vs time
from integrated
31
P nmr spectrain0.35M
O H - , n = 1.0 M at 5C. Signals (ppm)
are
at
6.5 (6), 22-31 (7, dependent
on
[OH-[ ) and7.1 ppm (8) .
60
(Reprinted with
permission from P. Hendry and A.
M.
Sargeson, Inorg. Chem. 25, 865 (1986).
1986, Am erican C hemical Society)
Such systems
as
(1.66)
or
(1.67) app ear
to
have been examined mainly
by
spectral m ethod s,
and sodiscussion will center around this monitoring me thod. Theoccurrence of steps with
similar ratesis indicated by the lackof isosbestic points (Sec. 3.9.1) over some portion of th e
reaction.
In
addition the appro priate kinetic plots are biphasic althou gh perhap s only slightly
so ,or
only
at
certain wavelengths. D ata o btained
at a
wavelength that monitors only the con-
centrationofA will givea perfect first-order plot (or pseudo first-order inthe presenceofex-
cess X) rate constantk
t
(1.73). Considering the optical absorbanceD,of the reacting solution
at time
t, in a
cell
of
path length 1
cm,
D, =E
A
[A]+
s
B
[B]
+E
C
[ C]
(1.80)
where
e
A
is the
mo lar absorptivity
of A
and
so on
(Sec. 3.9.1). By sub stituting (1.73) thr ou gh
(1.75) into (1.80)
and
rearranging terms,
it is not
difficult
to
derive
the
expression
65
'
66
D, - D
e
=
c?,exp
(-,? ) + a
2
exp{-k
2
t)
(1.81)
whereD
t
= s
c
[ A ]
0
anda
l
anda
2
are com posed ofrate constan ts and molar absorp tivities:
6 7
k
2
- h
]
0
(E
B
- Ec)
(1.82)
(1.83)
It
is not
always easy
by
inspection
to be
certain tha t two reactions
are
involved.
The use
of
a
semi-log plot helps
68
" since
it
shows better the deviation from linearity that
a
biphasic
reaction demandsbut a computer t reatment of the data isn ow thedefinitive appro ach.
71
~
77
Less apparent is thefact that having resolved the curve into two rate cons tants /t
fast
and A:
slow
we cannot simply assign these
tofc,andk
2
respectively
(in
1.66) without ad ditional inform al
tion since (1.81)
is
derived without specifying
ak^Jk
x
ork
bu
/k
2
condit ion.
7 1
'
78
The
most
popular method for resolving this ambiguity is byresorting to spectral considerations.The
spectrum of theintermediate B(which is a collection of s
b
at various wavelengths) can be
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1.6
Multiphasic
Unidirectional Reactions 21
calculated from (1.81), (1.82)and(1.83)on the twopremises that (a)Ar
fast
=k
{
and
stow
=k
2
or (b)Ar
fast
= k
2
andA:
slow
= k
x
. One of thetwo spectra estimated by this means will usually
be much more plausible than
the
other (which may even have negative mola r absorp tivities )
and
the
sequence which leads
to the
unreasonable spectrum
for B can be
discarded.
An ap-
proach
of
this type
was
first m ade
7 9
in a
study
of
Cr(en)
3 +
-^->
cis
-Cr (e n )
2
( e n H)(H
2
O )
4 +
-^->
cis
-Cr (e n )
2
(H
2
O )l
+
+ enH|+
(1.84)
and has since been adopted onnumerous occasions. '
73
'
76
'
80
In themajority ofcasesthefast
and slow rate constants areassigned to k
x
andk
2
indicating fast formation of a relatively
weakly absorbing intermediate.
70
'
73
'
7 6
'
8 0
A few
instances
are
known however
61
'
81
where
the
reverse pertains
i.e. a
highly absorbing intermediate arises which decays rapidly compared
with itsformation. NowAr
fast
= k
2
.
An intense purple-blue species forms within
a few
seconds when Cr
2 +
,
in
excess,
is
added
to 4,4 '-bipyridinium ion, bpyH|
+
.
The
color fades slowly over man y m inutes.
The
formation
and disappearance of theintermediate, k
tast
andk
i]ow
respectively, are assigned tok
2
andk
x
respectively from considerations of the assessed spectrum of the intermediate.
81
Thereac-
tions involvedare
(1.85)
highly colored
HN /
= =
\
N H
0-
8 6
)
colorless
and
the
rate equations
are
V= a [ C r
2 +
] [ b p y H l
+
] (1.87)
V
= \b +
c[Cr
2+
]
[H+ ]][b p y H
2
t] (1.88)
The radical intermediate
is
always less than
1% of
b p y H |
+
but
shows
up
because
of its
high absorbance.
If the spectrumof anintermediateisknown,achoiceof observation wavelength may allow
isolation
of
each stage.
In the
reaction
(A) -> (C)
using concentrated H C1:
Re(en)
2
(OH)
2
3+
~^> Re(en)Cl
2
(OH)J + en H
2 +
(1.89)
(A ) (B)
Re(en)Cl
2
(OH)
2
+
- ^ > ReCl
4
(OH)2 + e n H
2 +
(1.90)
(C )
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22 / TheDetermination of the Rate Law
examination at 395 or 535 nm (where
B
= s
c
) gives -d[A]/dt and hence a value for , .
Observation at 465 nm (where e
A
= s
B
) allows a value for k
2
to be obtained after a short in-
duction period. Absorbance changes at 685 nm (s
A
= e
c
) reflect d[B]/dt, and specifically
allow an estimate of t
max
, the time for the concentration of B to reach a maximum. A confir-
mation of k
t
and k
2
is then possible by using (1.79).
82
Finally we return to the situation mentioned at the beginning of the section of a system of
two consecutive reactions still yielding a single linear plot of In (D, D
t
) vs time. A condi-
tion which is not obvious is (1.91) which emerges from (1.81)
65
'
78
'
83
~
85
(1.91)
ER
A
This may appear to be an unlikely situation to encounter until one recalls that there are a
number of reactions involving two isolated and independently reacting centers. One might
then anticipate that statistically &, =
2k
2
and that e
B
= l /2 ( s
A
+ s
c
). These are precisely
the conditions demanded by (1.91). It is worth noting that in this case the observed first-order
rate constant is
k
2
.
Eigh t-iron ferredoxin cont ains two 4-Fe-clusters separated by about 12 A. Each cluster can
undergo a one-electron redox reaction (r and o represent the reduced and oxidized forms)
8-Fe(rr) - ^ * 8-Fe(or) (1.92)
8-Fe(or) - ^ > 8-Fe(oo) 1.93)
Oxidation of 8-Fe(rr) to 8-Fe(oo) by a number of one-electron oxidants gives a single first-
order process. After ruling out the more obvious reasons for this observation, it is concluded
that condition (1.91) holds,
8 6
one often referred to as "statistical kinetics". For other ex-
amples, see Refs. 87 and 88. Even if (1.91) is not strictly satisfied, linear plots may still be ob-
ta ined.
8 4
'
8 8
Further complexities may be anticipated. For example in the consecutive reaction (1.66),
direct conversion of A to C may occur as well as conversion that proceeds via B
A
Jl+
B
- ^ C (1.94)
I i
The SCN anation of Co (N H
3
)
5
H
2
O
3 +
has only recently been recognized to form
(N H
3
)
5
C o S C N
2 +
in parallel with the stable N-bonded Co(NH
3
)
5
N C S
2 +
. The system has
been fully analyzed and illustrates well the difficulties in detecting biphasic behavior.
89
Co(NH
3
)
5
SCN
2+
C o ( N H
3
)
5
H
2
O
3 +
\ 1 (1.95)
^ Co(NH
3
)
5
N CS
2 +
For another example, see Ref. 90.
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1.6 MultiphasicUnidirectional Reactions 23
As the number of reaction steps increases so, of course, does the complexity. A novel exam-
ple is the six consecutive steps for hydrolysis of a cyanobridged polynuclear complex 9
91
[ R u ( b p z ) |
+
( F e ( C N ) ^ )
6
]
1 6
- + 6H
2
O -> Ru(bpz)2+ + 6Fe(CN)
5
H
2
O
3
- (1.96)
Fe{CN)
5
1 6
"
The successive rate constant s vary only from 1.3 x 10 "
3
s "
1
to 4.6 x 10~
4
s"
1
at n =
0.1 M and th e mu ltistep kinetics are analyzed w ith a com puter. A large excess of dm so is used
to aid the dissociation of the polynuclear complex by reacting irreversibly with the product,
F e ( C N )
5
H
2
O
3
-
F e ( C N )
5
H
2
O
3
- + dmso - Fe (CN )
5
(d ms o )
3
- + H
2
O (1.97)
The ambiguities which have been alluded to in this section may sometimes be circumvented
by changing the cond itions, the concentrations of excess reactants, temperature, pH and so on.
1.6.3 Two-Step Reactions with an Elem ent of Reversibility
Suppose that an irreversible reaction between A and B leading to >95% product or products,
designated D, is examined in the usual way. One of the reactants, B, is held in excess and the
loss of A monitored. It is likely that the loss will be a first-order process (rate constant
k) .
At low concentrations of B (but still >[A]), the value of k may be proportional to the concen-
tration of B. At higher conc entrations of B however this direct proportiona lity may disappear
and eventually k will become independent of [B]. Obviously, a second-order reaction at low
reactant concentrations has lost its simplicity at higher reactant concentrations and eventually
turned over to first-order in A alone. Such a situation is accommodated by a rate law of the
form
This behavior is sometimes referred to as saturation kinetics. When
b
[B] < 1, the obse rved
second-order is easily understood (rate constant =
a).
When
b
[B] ~ 1 there is a mix ed-order
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24 ; TheDetermination of the Rate Law
and eventually, when b[B] > 1, the reaction is first-order in A, rate constant = a/b. The
kvs [B]plotisdescribedby ahyperbolaand(1.98)can be treated directlyby acomputerpro-
gram.A favorite approachin thepast, and still useful, is to convert (1.98) into a linear form
(1.99).
A plot of Ar
1
vs [B]"
1
k a[B] a
yields I/a (slope)andb/a (intercept).An exampleof theratelaw(1.98)isshownin theredox
reaction (1.100).
92
Co(edta)
2
~ (= B) isused inexcessanda = 4 .5 M~
1
s~
1
andb = 8 3 1 M "
1
at 25C.
C o " ( e d t a )
2
- + F e (CN) | " -> C o " ' ( e d t a ) - + F e( C N )^ (1.100)
1.6.4
Reaction Schemes Associated with (1.98)
Thereare anumber of possible schemes whichmayexplain the rate behavior associated with
(1.98).A single stepcan be ruledout. At least two consecutive or competitive reactions in-
cluding one reversible step must be invoked,
(a) Consider the scheme
(1.101)
which
is a
very important
one in
chemistry.
93
If we do not see
deviations from
a
single first-
order processit islikely that the first reversible stepis much more rapid than the second.At
higher concentrationsof B the rapid formation of substantial amounts of C may be discern-
able.If the first stepis themore rapidone, Cwillbe in equilibrium withA and B throughout
the reaction and
will be continually maintained. We shall be monitoring the loss of bo th A and C or the
equivalent gain in one of theproducts(D).
d[D]/dt = k([A] + [C]) = k
2
[C] = k
2
K
x
[A][B] (1.103)
From (1.102) and (1.103),
=
M T . I B ]
( U 0 4 )
1
+ K
x
[B]
whichis of the form (1.98) witha = k
2
K
x
andb = K
x
. In reaction (1.100),C would represent
some adduct,nowbelievedto be a cyano bridged Co(III)Fe(II) species, withaformationcon-
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1.6
Mu ltiphasic
Unidirectional Reactions 25
stant K
x
= b = 8 3 1 M "
1
. This adduct would break down to products with a rate constant
k
2
= a/b
= 5.4 x l O ^ s "
1
Co (e d ta )
2
- + Fe(CN)^- ^ ( e d t a ) C o
n l
N C F e
n
(CN)i- (1.105)
(e d ta )Co
n l
N C F e " ( C N ) | ~ ^ - Co(ed ta )" + Fe(C N) i~ (1.106)
The rate constant k_
2
can be ignored at present.
(b)A related scheme to tha t of (a) is one in whichA and B react directly to form D, but
are alsoin a rapid "dead-end" or nonproductive equilibriumtogiveC (competitive reactions)
A + B ^ C
K
x
(1.107)
A + B ^ D k
3
,k_
3
(neg ligible) (1.108)
d[D]/dt
= A:([A] + [C]) =
k
3
[A ][B] (1.109)
whence
=
*