(Topics in inorganic chemistry) robert b jordan reaction mechanisms of inorganic and organometallic systems oxford university press, USA (2007)

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Reaction Mechanisms of
Inorganic and Organometallic
Systems


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TOPICS IN INORGANIC CHEMISTRY

A Series of Advanced Textbooks in Inorganic Chemistry

Series Editor
Peter C. Ford, University of California, Santa Barbara

Chemical Bonding in Solids, J. Burdett
Reaction Mechanisms of Inorganic and Organometallic Systems,
3rd Edition, R. Jordan


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Reaction Mechanisms of
Inorganic and Organometallic
Systems
Third Edition

Robert B. Jordan

OXFORD
UNIVERSITY PRESS
2007


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OXPORD
UNIVERSITY PRESS

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Library of Congress Cataloging-in-Publication Data
Jordan, Robert B.
Reaction mechanisms of inorganic and organometallic systems / Robert
B. Jordan.—3rd ed
p. cm.
Includes bibliographical references and index.
ISBN 978-0-19-530100-7
1. Reaction mechanisms (Chemistry) 2. Organometallic compounds. 3. Inorganic
compounds. I. Title.
QD502J67 2006
41'.39—dc25
2006052498

9 8 7 6 5 4 3 2 1
Printed in the United States of America
on acid-free paper


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Preface
This book evolved from the lecture notes of the author for a onesemester course given to senior undergraduates and graduate students
over the past 20 years. This third edition presents an updating of the
material to cover the literature through to the end of 2005, with
occasional excursions to early 2006. As a result, the total number of
references has increased from about 660 in the second edition to over
1570 in the present one, and 140 pages of text have been added; this
seems to be a clear testament to the vitality of the subject area. A new
Chapter 9 on kinetics in heterogeneous systems has been added. This

area has long been the domain of chemical engineers, but it is of
increasing relevance to inorganic kineticists who are studying catalytic
processes, such as hydrogenation and carbonylation reactions, where
gas/liquid mass transfer is involved. This chapter also covers the kinetic
aspects of adsorption and reaction of species on solids, and the question
of whether the reaction is really homogeneous or heterogeneous.
The overall organization of the first edition has been retained. The
first two chapters cover basic kinetic and mechanistic terminology and
methodology. This material includes new sections on the analysis of
data under second-order conditions, Curtin-Hammett conditions and an
expanded discussion of pressure effects. New material has been added at
various points throughout Chapters 3 and 4. The coverage of
organometallic systems in Chapter 5 has been increased substantially,
primarily with material on metal hydrides, catalytic hydrogenation and
asymmetric hydrogenation. The inverted region and activation
parameters for electron-transfer reactions predicted by Marcus theory
have been added to Chapter 6, along with an expanded discussion of
intervalence electron transfer. The recently revised assignment of the
electronic spectra of metal carbonyls has resulted in substantial revisions
to photochemical interpretations in Chapter 7. The coverage of selected
bioinorganic systems in Chapter 8 has been extended to include
methylcobalamin as a methyl transferase and the chemistry of nitric
oxide synthase. Chapter 10 on experimental methods and their
applications is largely unchanged. Some new problems for each chapter
have been added.
There is more material than can be covered in depth in one semester,
but the organization allows the lecturer to omit or give less coverage to
certain areas without jeopardizing an understanding of other areas. It is
assumed that the students are familiar with elementary crystal field
v

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vi Preface

theory and its applications to electronic spectroscopy and energetics,
and concepts of organometallic chemistry, such as the 18-electron rule,
71 bonding and coordinative unsaturation. For the material in the first
two chapters, some background from a physical chemistry course would
be useful, and familiarity with simple differential and integral calculus is
assumed.
It is expected that students will consult the original literature to obtain
further information and to gain a feeling for the excitement in the field.
This experience also should enhance their ability to critically evaluate
such work. Many of the problems at the end of the book are taken from
the literature, and original references are given; outlines of answers to
the problems will be supplied to instructors who request them from the
author.
The issue of units continues to be a vexing one in this area. A major
goal of this course has been to provide students with sufficient
background so that they can read and analyze current research papers.
To do this and be able to compare results, the reader must be vigilant
about the units used by different authors. Energy units are a special
problem, since both joules and calories are in common usage. Both
units have been retained in the text, with the choice made on the basis of
the units in the original work as much as possible. However, within
individual sections the text uses one energy unit. Bond lengths are given
in angstroms, which are still commonly quoted for crystal structures.
The formulas for various calculations are given in the original or most
common format, and units for the various quantities are always

specified.
The author is greatly indebted to all of those whose research efforts
have provided the core of the material for this book. The author is
pleased to acknowledge those who have provided the inspiration for this
book: first, my parents, who contributed the early atmosphere and
encouragement; second, Henry Taube, whose intellectual stimulation
and experimental guidance ensured my continuing enthusiasm for
mechanistic studies. I am only sorry that I did not finish this edition
soon enough for Henry to see that I did make the changes he suggested.
Finally and foremost, Anna has been a vital force in the creation of this
book through her understanding of the time commitment, her
comments, criticisms and invaluable editorial assistance in producing the
camera-ready manuscript. However, the inevitable remaining errors and
oversights are entirely the responsibility of the author.
R.B.J.
Edmonton, Alberta
June 2006


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Contents
1

2

3

4

5

Tools of the Trade, 1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
1.10

Basic Terminology, 1
Analysis of Rate Data, 3
Concentration Variables and Rate Constants, 12
Complex Rate Laws, 15
Complex Kinetic Systems, 15
Temperature Dependence of Rate Constants, 17
Pressure Dependence of Rate Constants, 21
Ionic Strength Dependence of Rate Constants, 24
Diffusion-Controlled Rate Constants, 25
Molecular Modeling and Theory, 28

Rate Law and Mechanism, 31

2.1
2.2
2.3

2.4
2.5
2.6
2.7
2.8

Qualitative Guidelines, 31
Steady-State Approximation, 32
Rapid-Equilibrium Assumption, 34
Curtin-Hammett Conditions, 36
Rapid-Equilibrium or Steady-State?, 37
Numerical Integration Methods, 3 8
Principle of Detailed Balancing, 39
Principle of Microscopic Reversibility, 40

Ligand Substitution Reactions, 43

3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9

Operational Approach to Classification of Substitution Mechanisms, 43
Operational Tests for the Stoichiometric Mechanism, 44
Examples of Tests for a Dissociative Mechanism, 49

Operational Test for an Associative Mechanism, 54
Operational Tests for the Intimate Mechanism, 57
Some Special Effects, 73
Variation of Substitution Rates with Metal Ion, 83
Ligand Substitution on Labile Transition-Metal Ions, 94
Kinetics of Chelate Formation, 100

Stereochemical Change, 114

4.1
4.2
4.3
4.4
4.5

Types of Ligand Rearrangements, 114
Geometrical and Optical Isomerism in Octahedral Systems, 119
Stereochemical Change in Five-Coordinate Systems, 128
Isomerism in Square-Planar Systems, 130
Fluxional Organometallic Compounds, 130

Reaction Mechanisms of Organometallic Systems, 150

5.1 Ligand Substitution Reactions, 150
5.2 Insertion Reactions, 168
5.3 Oxidative Addition Reactions, 177

vii

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viii

Contents
5.4
5.5
5.6
5.7

6

7

8

9

Reductive Elimination Reactions, 188
Reactions of Alkenes, 188
Catalytic Hydrogenation of Alkenes, 195
Homogeneous Catalysis by Organometallic Compounds, 225

Oxidation-Reduction Reactions, 253

6.1
6.2
6.3
6.4
6.5

6.6

Classification of Reactions, 253
Outer-Sphere Electron-Transfer Theory, 256
Differentiation of Inner-Sphere and Outer-Sphere Mechanisms, 273
Bridging Ligand Effects in Inner-Sphere Reactions, 274
Intervalence Electron Transfer, 281
Electron Transfer in Metalloproteins, 285

Inorganic Photochemistry, 292

7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8

Basic Terminology, 292
Kinetic Factors Affecting Quantum Yields, 294
Photochemistry of Cobalt(III) Complexes, 295
Photochemistry of Rhodium(III) Complexes, 301
Photochemistry of Chromium(III) Complexes, 304
Photochemistry of Ruthenium(II) Complexes, 310
Organometallic Photochemistry, 313
Photochemical Generation of Reaction Intermediates, 327

Bioinorganic Systems, 337

8.1.
8.2
8.3
8.4
8.5
8.6

Basic Terminology, 337
Terms and Methods of Enzyme Kinetics, 338
Vitamin B12, 341
A Zinc(II) Enzyme: Carbonic Anhydrase, 356
Enzymic Reactions of Dioxygen, 361
Enzymic Reactions of Nitric Oxide, 373

Kinetics in Heterogeneous Systems, 391

9.1 Gas/Liquid Heterogeneous Systems, 391
9.2 Gas/Liquid/Solid Heterogeneous Systems, 400
9.3 Where is the Catalyst?, 409

10 Experimental Methods, 422
10.1
10.2
10.3
10.4
10.5
10.6
10.7

Flow Methods, 423
Relaxation Methods, 428
Electrochemical Methods, 431
Nuclear Magnetic Resonance Methods, 435
Electron Paramagnetic Resonance Methods, 446
Pulse Radiolysis Methods, 448
Flash Photolysis Methods, 451

Problems, 457
Chemical Abbreviations, 488
Index, 491


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Reaction Mechanisms of
Inorganic and Organometallic
Systems


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1

Tools of the Trade

This chapter covers the basic terminology and theory related to the types of
studies that are commonly used to provide information about a reaction
mechanism. The emphasis is on the practicalities of determining rate
constants and rate laws. More background material is available from
general physical chemistry texts1,2 and books devoted to kinetics.3-5 The
reader also is referred to the initial volumes of the series edited by Bamford
and Tipper.6 Experimental techniques that are commonly used in inorganic
kinetic studies are discussed in Chapter 9.

1.1 BASIC TERMINOLOGY
As with most fields, the study of reaction kinetics has some terminology
with which one must be familiar in order to understand advanced books
and research papers in the area. The following is a summary of some of
these basic terms and definitions. Many of these may be known from
previous studies in introductory and physical chemistry, and further
background can be obtained from textbooks devoted to the physical
chemistry aspects of reaction kinetics.
Rate
For the general reaction

the reaction rate and the rate of disappearance of reactants and rate of
formation of products are related by

In practice, it is not uncommon to define the rate only in terms of the
species whose concentration is being monitored. The consequences that
can result from different definitions of the rate in relation to the
stoichiometry are described below under the definition of the rate constant.
1

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2

Reaction Mechanisms of Inorganic and Organometallic Systems

Rate Law
The rate law is the experimentally determined dependence of the reaction
rate on reagent concentrations. It has the following general form:

where k is a proportionality constant called the rate constant. The
exponents m and n are determined experimentally from the kinetic study
and have no necessary relationship to the stoichiometric coefficients in the
balanced chemical reaction. The rate law may contain species that do not
appear in the balanced reaction and may be the sum of several terms for
different reaction pathways.
The rate law is an essential piece of mechanistic information because it
contains the concentrations of species necessary to get from the reactant to
the product by the lowest energy pathway. A fundamental requirement of
an acceptable mechanism is that it must predict a rate law consistent with
the experimental rate law.
Order of the Rate Law
The order of the rate law is the sum of the exponents in the rate law. For
example, if m = 1 and n = -2 in Eq. (1.3), the rate law has an overall order
of -1. However, except in the simplest cases, it is best to describe the order
with respect to individual reagents; in this example, first-order in [A] and
inverse second-order in [B].
Rate Constant
The rate constant, k, is the proportionality constant that relates the rate to
the reagent concentrations (or activities or pressures, for example), as

shown in Eq. (1.3). The units of k depend on the rate law and must give the
right-hand side of Eq. (1.3) the same units as the left-hand side.
A simple example of the need to define the rate in order to give the
meaning of the rate constant is shown for the reaction

From Eq. (1.2), and assuming the rate is second-order in [A], then

If the experiment followed the rate of disappearance of A, then the
experimental rate constant would be 2k and it must be divided by 2 to get
the numerical value of k as defined by Eq. (1.5). However, if the formation
of B was followed, then k would be determined directly from the
experiment.


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Tools of the Trade

3

Half-time
The half-time, t1/2, is the time required for a reactant concentration to
change by half of its total change. This term is used to convey a qualitative
idea of the time scale for the reaction and has a quantitative relationship to
the rate constant in simple cases. In complex systems, the half-time may be
different for different reagents and one should specify the reagent to which
the t1/2 refers.
Lifetime
The lifetime, T, for a particular species is the concentration of that species
divided by its rate of disappearance. This term is commonly used in socalled lifetime methods, such as NMR, and in relaxation methods, such as

temperature jump.

1.2 ANALYSIS OF RATE DATA
In general, a kinetic study begins with the collection of data of
concentration versus time of a reactant or product. As will be seen later,
this can also be accomplished by determining the time dependence of some
variable that is proportional to concentration, such as absorbance or NMR
peak intensity. The next step is to fit the concentration-time data to some
model that will allow one to determine the rate constant if the data fits the
model.
The following section develops some integrated rate laws for the models
most commonly encountered in inorganic kinetics. This is essentially a
mathematical problem; given a particular rate law as a differential
equation, the equation must be reduced to one concentration variable and
then integrated. The integration can be done by standard methods or by
reference to integration tables. Many more complex examples are given in
advanced textbooks on kinetics.
1.2.1 Zero-Order Reaction
A zero-order reaction is rare for inorganic reactions in solution but is
included for completeness. For the general reaction

the zero-order rate law is given by

and integration over the limits [B] = [B]0 to [B] and t = 0 to t yields


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4

Reaction Mechanisms of Inorganic and Organometallic Systems

This predicts that a plot of [B] or [B] - [B]0 versus t should be linear with a
slope of k.
1.2.2 First-Order Irreversible System
Strictly speaking, there is no such thing as an irreversible reaction. It is just
a system in which the rate constant in the forward direction is much larger
than that in the reverse direction. The kinetic analysis of the irreversible
system is just a special case of the reversible system that is described in the
next section.
For the representative irreversible reaction

the rate of disappearance of A and appearance of B are given by

The problem, in general, is to convert this differential equation to a form
with only one concentration variable, either [A] or [B], and then to
integrate the equation to obtain the integrated rate law. The choice of the
variable to retain will depend on what has been measured experimentally.
One of the concentrations can be eliminated by considering the reaction
stoichiometry and the initial conditions. The most general conditions are
that both A and B are present initially at concentrations [A]0 and [B]0,
respectively, and that the concentrations at any time are defined as [A] and
[B].
For this simple case, the rate law in terms of A can be obtained by simple
rearrangement to give

Then, integration over the limits [A] = [A]0 to [A] and t = 0 to /, gives

and predicts that a plot of In [A] versus t should be linear with a slope of
-k\. The linearity of such plots often is taken as evidence of a first-order

rate law. Since the assessment of linearity is somewhat subjective, it is
better to show that the slope of such plots is the same for different initial
concentrations of A and that the intercept corresponds to the expected
value of In [A]0.


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Tools of the Trade

5

The equivalent exponential form of Eq. (1.12) is

and it is now common to fit data to this equation by nonlinear least squares
to obtain k\.
In order to obtain the integrated form in terms of B, it is necessary to use
the mass balance conditions. For a 1:1 stoichiometry, the changes in
concentration are related by

At the end of the reaction, [A] = 0 and [B] = [B]^, and substitution of these
values into Eq. (1.14) gives

After rearrangement of Eq. (1.14) and substitution from Eq. (1.15), one
obtains

Then, substitution for [A] from Eq. (1.16) into Eq. (1.10) gives an equation
that can be integrated over the limits [B] = [B]0 to [B] and t - 0 to t, to
obtain

This equation also can be obtained by substitution for [A]0 and [A] from
Eq. (1.15) and (1.16) into Eq. (1.12) and predicts that a plot of
In ([BL - [B]) versus t should be linear with a slope of -kv
The half-time, tm, can be obtained from Eq. (1.12) for the condition
[A] = [A]case, the result is

Therefore, the half-time is independent of the initial concentrations.
An important practical advantage of the first-order system is that the
analysis can be done without any need to know the initial concentrations.
Therefore, the collection of concentration-time data can be started at any
time arbitrarily defined as t = 0. This is a significant difference from the
second-order case that is described later in this chapter.


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6

Reaction Mechanisms of Inorganic and Organometallic Systems

1.2.3 First-Order Reversible System
For a system coming to equilibrium, both the forward and reverse reactions
must be included in the kinetic analysis and one must take into account that
significant concentrations of both reactants and products will be present at
the end of the reaction. A first-order system coming to equilibrium may be
represented by

The rate of disappearance of A equals the rate of appearance of B, and
these are given by

Just as with the irreversible system, the problem is to convert this
equation to a form with only one concentration variable, either [A] or [B],
and then integrate the equation to obtain the integrated rate law. The initial
concentrations are defined as [A]0 and [B]0, those at any time as [A] and
[B], and the final concentrations at equilibrium as [A]e and [B]e. Then,
mass balance gives

To obtain the rate law in terms of B, Eq. (1.21) can be rearranged to obtain
the following expressions for [A] and [A]0:

so that

Substitution for [A] from Eq. (1.23) into Eq. (1.20) gives

Note that the initial concentrations have been eliminated.
Since Eq. (1.24) contains only one concentration variable, [B], it can be
integrated directly. However, it is convenient in the end to eliminate [A]e
by noting that, at equilibrium the rate in the forward direction must be
equal to the rate in the reverse direction:


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Tools of the Trade

1

and substitution for k,[A] into Eq. (1.24) gives

This equation can be rearranged and integrated over the limits [B] = [B]0 to
[B] and t = 0 to t to obtain

Therefore, a plot of In ([B]e - [B]) versus t should be linear with a slope of
-{fcj + fc_j). Note that the kinetic study yields the sum of the forward and
reverse rate constants. If the equilibrium constant, K, is known, then k{ and
£_! can be calculated since K = kjk^.
Just as in the irreversible first-order system, the analysis can be done
without any need to know the initial concentrations, and the collection of
concentration-time data can be started at any time defined as t - 0.
At the half-time, t = tm, [B] = l/2([B]e - [B]0) + [B]0, and substitution
intoEq. (1.27) gives

The irreversible first-order system is a special case of the reversible
system. For the irreversible system, k{ » k_{ so that (k{ + k_l) = kl.
1.2.4 Second-Order Reversible System
The second-order reversible system will be described next and the simpler
irreversible system will be developed later as a special case of the
reversible one. This reversible system can be described by

where b and c are stoichiometric coefficients defined relative to a
coefficient of 1 for the deficient reagent A. The rate of disappearance of A
is given by


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8

Reaction Mechanisms of Inorganic and Organometallic Systems

It will be assumed that there is no C present initially so that mass balance
gives the concentrations of B and C in terms of A as

and at equilibrium

It is convenient to eliminate k_2 before integrating by noting that the
forward and reverse rates are equal at equilibrium:

which, after rearrangement, gives

Substitution for [B] and [C] from Eq. (1.31) and for k_2 from Eq. (1.34)
into Eq. (1.30) gives the following equation which can be integrated
because [A] is the only concentration variable:

Integration over the limits [A] = [A]0 to [A] and t - 0 to t gives the
following solution:

A plot of the first term on the left-hand side of Eq. (1.36) versus t should be
linear with a slope related to fc2, as indicated by the right-hand side of Eq.
(1.36). It is apparent that one must know the stoichiometry coefficient, b, in
addition to the initial concentrations, [A]0 and [B]0, and the final
concentration, [A]^ in order to do the analysis and to determine the value
of k2 from the slope. In practice, all of these requirements can be difficult to
satisfy so that this an unpopular and uncommon situation for experimental
studies. The methods used to circumvent these requirements are described
in Sections 1.2.6 and 1.2.8.


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Tools of the Trade 9

1.2.5 Second-Order Irreversible System
This system can be obtained as a special case of the reversible system by
simple consideration of the stoichiometry conditions. If B is the excess
reagent and reaction (1.29) goes essentially to completion, then [A]e = 0,
and substitution of this condition into Eq. (1.36) gives

In this case, the initial concentrations of both reactants are required in order
to plot the first term on the left versus t and to determine k2 from the slope.
These conditions are not as restrictive as those for the reversible secondorder system, but they are still worse than those for the first-order system.
At the half-time for this second-order reaction, [A] = [A]g/2, and
substitution into Eq. (1.37) shows that

1.2.6 Pseudo-First-Order Reaction Conditions
The pseudo-first-order reaction condition is very widely used, but it is
seldom mentioned in textbooks. Although many reactions have secondorder or more complex rate laws, the experimental kineticist wishes to
optimize experiments by taking advantage of the first-order rate law
because it imposes the fewest restrictions on the conditions required to
determine a reliable rate constant. The trick is to use the pseudo-first-order
condition.
The pseudo-first-order condition requires that the concentration of the
reactant whose concentration is monitored is at least 10 times smaller than
that of all the other reactants, so that the concentrations of all the latter
remain essentially constant during the reaction. Under this condition, the
rate law usually simplifies to a first-order form and one gains the advantage
of not needing to know the initial concentration of the deficient reagent.
In the preceding irreversible second-order example, if it is assumed that
the conditions have been set so that [B]0 »[A]0, then [B]0 »[C] and
[C] = [A]0 - [A]. In addition, the concentration of B will remain constant at

[B]0, and the final concentration of C is [Ck = [A]0 if the reaction is
irreversible and has a 1:1 stoichiometry. Substitution of these conditions
intoEq. (1.37) gives

This equation predicts that a plot of In ([CL - [C]) versus t should be linear


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10

Reaction Mechanisms of Inorganic and Organometallic Systems

with a slope of -fc2[B]0. This is identical in form to the first-order rate law
except that k\ is replaced by fc2[B]0- The latter constant is often called the
observed, £obsd, or experimental, fcexp, rate constant. Since [B]0 is known, it
is possible to calculate k2.
In a more general case, if the rate of disappearance of reactant A is a
function of the concentrations of other species X, Y and Z, then the rate of
disappearance of A may be given by

If the conditions are such that [X]0, [Y]0, [Z]0 » [A], so that [X], [Y] and
[Z] remain essentially constant, then

and the rate law has the first-order form.
1.2.7 Comparison of First-Order and Second-Order Conditions
The differences between second-order and first-order kinetic behavior, and
the transition from second-order to pseudo-first-order conditions are
illustrated by the curves in Figure 1.1. These curves are based on the
system

and show the time dependence of the formation of C. The deficient reagent
is A, with [A]0 = 0.10 M, different initial concentrations of B of 0.50, 0.70,
0.85 and 1.0 M are used, and k2 = 3xlO"3 M~! s-1. The lower curve, with
[B]0 = 0.50 M, represents a typical time dependence for second-order
conditions. Note that this time dependence has a more gradual approach to
the final value than the others in Figure 1.1. Thus, a qualitative
examination can provide an indication of the second-order nature of the
reaction. The fact that the rate is slower under second-order conditions can
be used to advantage if the limits of the experimental method are being
strained.
As [B]0 is increased, C is formed more rapidly and the curves approach a
first-order shape. When [B]0 = 1.0 M, [B]0 > 10[A]0 and the pseudo-firstorder condition has been reached. Then, the curve calculated from Eq.
(1.37) for second-order conditions is almost indistinguishable from the line
calculated from Eq. (1.39) for pseudo-first-order conditions. The
correspondence of these curves is the rationale for the general rule that
pseudo-first-order conditions require at least a 10-fold excess of the
reagents whose concentrations are to remain constant.


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Tools of the Trade

11

Figure 1.1. The time dependence for formation of product C with the initial
deficient reagent [A]0 = 0.10 M. Results are calculated for pseudo-first-order (—)
conditions and concentrations (M) of the excess reagent B of 0.50 (•), 0.70 (n),
0.85 (•), l.O(o).

1.2.8 Initial Rate Method
This is a method for determining the concentration dependence of a rate
law that avoids the need for an integrated rate law or pseudo-first-order
conditions. It is based on the assumption that the reactant concentrations
are essentially constant during the initial -10% of reaction. The use of this
method requires that observation can begin very soon after mixing the
reactants and that the detection method is sensitive enough to provide
precise data over the small extent of reaction. The latter condition usually
means that the reaction half-time is about ten seconds or longer, so that this
method is convenient and efficient for slow reactions. Observation over a
short initial period may avoid, but also may hide, kinetic and chemical
complications that only are clearly apparent later in the reaction.
The kinetic runs simulated in Figure 1.2 are for a second-order rate law
with varying initial concentrations of the reactants A and B, with A as the
deficient reagent. The absorbance, 7A, which is proportional to the
concentration of A, has been measured at 2 s intervals. For illustrative
purposes, the concentrations have been chosen so that the initial rate,
£2[A]0[B]0, is the same for all the runs. More commonly, the reagent
concentrations are varied one at a time in a series of kinetic runs in order to
determine the rate law.


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Reaction Mechanisms of Inorganic and Organometallic Systems

Figure 1.2. The time dependence of the absorbance, /A, of reactant A during the

initial stages of its reaction with B with a rate law -d[A]/df = £2[A][B] and
k2 = 3.0xl(T2 NT1 s'1. Initial concentrations (M) are: [A]0 = 0.050, [B]0 = 0.400 (o);
[A]0 = 0.033, [B]0 = 0.600 (o); [A]0 = 0.020, [B]0 = 1.0 (o).
The simplest way of determining the initial rate is to take the slopes of
the lines through the initial points, as shown in Figure 1.2. Strictly
speaking, the slopes should be taken for lines which cover the same extent
of reaction but this is not always obvious when one just observes a small
portion of the reaction. In the Figure, the more common practice of taking
the slope over a fixed time is illustrated, even though the extent of reaction
is greatest for the fastest (lower) data set. As a result, the slopes actually
range from 0.103 to 0.112 s'1, although they should be equal. In this case,
the error probably is not too significant for the purpose of determining the
rate law. This difficulty can be minimized by fitting the data over a greater
extent of the reaction to a power series in t, such as 7, = /0 + at + P*2, in
which case a will be the initial slope. In order to determine the actual rate
constant, one must know the relationship between the property being
observed and the concentration.

1.3 CONCENTRATION VARIABLES AND RATE CONSTANTS
The rate laws have been developed in terms of concentrations, but in many
cases it is not practical or possible to determine actual molar concentrations
as a function of time. However, it is easy to measure some property that is


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13

known to be directly proportional to molar concentration, such as
absorbance in the electronic or infrared spectrum, NMR integrated peak
intensity, conductance or refractive index.
The following development shows the general relationship between the
concentration of the limiting reagent and the property being measured for
the irreversible reaction

The equation is balanced so that A has a coefficient of 1 in order to
simplify the ratios bla, yla and z/a which otherwise would appear in the
equations. The property being measured is designated as / and its
proportionality constant with concentration is e. At any time, the value of /
is given by

and the initial value is given by

If A is the limiting reagent, then the final value of / is given by

The following stoichiometric relationships can be used to express the
concentrations of B, Y and Z in terms of A:

Substitution of these values into Eq (1.44) gives

Then, taking the difference /, - /„ and solving for [A] gives

Similarly, 70 - /^ gives [A]0 as


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14

Reaction Mechanisms of Inorganic and Organometallic Systems

Combination of the last two equations gives

Substitution for [A] into the integrated rate law gives equations that can be
used to determine a k from measurements of /.
1.3.1 First-Order Irreversible System
A simple rearrangement of the previous solution for this system
(Eq. (1.12)) and substitution for [A] and [A]0 from Eq. (1.49) and (1.50)
gives

This shows that a plot of In (/, - /„) versus t should be linear with a slope of
-fc,. It is not necessary to know the concentration of A or any values of e in
order to determine the rate constant, but one does need /.„. Sometimes, it is
impossible to measure /„,, because of secondary reactions, or it is
inconvenient because the reaction is slow. Such systems can be analyzed
by nonlinear least-squares fitting of the data over as much of the reaction
as possible, or in a more classical way by the Guggenheim method,
described in more detail by Moore and Pearson3 (p 71), Mangelsdorf7 and
Espenson8 (p 25).
1.3.2 Second-Order Irreversible System
The integrated rate law for this system is given by Eq. (1.37) which may be
rewritten as

Then, substitution for [A] from Eq. (1.51) gives

Therefore, a plot of the first term on the left-hand side versus t should be
linear with a slope of ([B]0 -fe[A]0)fc2.Clearly in this case, it is necessary to
know the stoichiometry coefficient, b, as well as the initial concentrations,

[B]0 and [A]0, in order to determine k2. However, only b and the ratio of
[B]0 to [A]0 are needed in order to make the plot.