Fundamentals in Chemical Physics

319 Pages·1998·16.543 MB·English

by  Franco Battaglia, Thomas F. George (auth.)

FUNDAMENTALS IN CHEMICAL PHYSICS
Fundamentals in
Chemical Physics
by
Franco Battaglia
Department ofP hysics 'Edouardo Amaldi ',
Universita degli Studi 'Roma Tre ',
Rome, Italy
and
Thomas F. George
Department of Chemistry,
University of Wisconsin at Stevens Point,
Stevens Point, Wisconsin, U.S.A.
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-5082-3 ISBN 978-94-017-1636-9 (eBook)
DOI 10.1007/978-94-017-1636-9
Printed on acid-free paper
This is a completely revised and updated
translation of the original Italian work
Lezoine de Chimica Fisica by
F. Battaglia, CEDAM, Padova, 1997.
Translated by the authors.
AII Rights Reserved
© 1998 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1998
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
Table of Contents
Introduction ix
1 Classical Physics 1
1.1 Newtonian mechanics 1
1.2 Lagrangian mechanics 2
1.3 Hamiltonian mechanics . 4
1.4 Constants of the motion 5
1.5 Applications . . . . . . . 6
1.5.1 Harmonic oscillator . 6
1.5.2 Central potential . 7
1.5.3 Rigid rotator ... 8
1.5.4 Two-body problem 9
1.6 Electrodynamics ..... 10
2 Quantum Physics 17
2.1 Mathematical formalism 18
2.1.1 State space ... 18
2.1.2 Linear operators 21
2.1.3 Eigenvalue equation 24
2.1.4 Observables . . . . . 26
2.2 Physical interpretation . . . 27
2.2.1 Probabilistic interpretation 28
2.2.2 Coordinate representation 31
2.2.3 Schrodinger equations 33
2.2.4 Angular momentum 35
2.2.5 Many-body systems 41
2.3 Applications . . . . . . . . . . 43
2.3.1 One-dimensional system 43
2.3.2 Free particle .. 44
2.3.3 Particle in a box 45
v
vi TABLE OF CONTENTS
2.3.4 Harmonic oscillator . 47
2.3.5 Rigid rotator .... 49
2.3.6 Central potential . . 49
2.3.7 Two-body problem . 51
2.4 Approximation methods . . 51
2.4.1 Variational method . 51
2.4.2 Perturbation theory 53
2.5 Electrodynamics 57
• 0 ••••
3 Thermodynamics 63
3.1 Internal energy and entropy 63
3.2 Intensive variables ..... 67
3.3 Partition functions ..... 71
3.3.1 Canonical ensemble 71
3.3.2 Isothermal-isobaric ensemble 73
3.3.3 Grand canonical ensemble . 74
3.4 Thermodynamic potentials 75
3.4.1 Helmholtz free energy 76
3.4.2 Gibbs free energy . 77
3.4.3 Grand potential 78
3.4.4 Enthalpy 79
3.4.5 Jacobians 80
3.5 Processes 82
0 ••••
3.5.1 Equilibrium 82
3.5.2 Stability .. 86
3.5.3 Work and heat 89
3.5.4 Principles of thermodynamics . 93
3.6 Applications . . . . . . . . . . 96
3.6.1 Equipartition theorem 96
3.6.2 Ideal crystal ..... 98
3.6.3 Ideal quantum gases 103
3.6.4 Ideal classical gas . . . 118
3.6.5 Examples of processes 125
3.6.6 Mixture of ideal gases 133
4 Quantum Chemistry 141
4.1 Hydrogen-like atom. 142
•••• 0 0 •
4.2 Many-electron atoms . . . . . . . 148
4.3 Electronic structure of molecules 152
4.3.1 Diatomic molecules . . 163
4.3.2 Polyatomic molecules 166
4.4 Nuclear motion in molecules . 169
TABLE OF CONTENTS vii
4.4.1 Diatomic molecules . . 169
4.4.2 Polyatomic molecules 174
5 Molecular Spectroscopy 183
5.1 Microwave spectroscopy 189
5.2 IR spectroscopy . . . 192
5.3 UV spectroscopy .. 195
5.4 Raman spectroscopy 200
5.5 NMR spectroscopy . 203
5.5.1 Chemical shift 206
5.5.2 Spin-spin coupling 208
5.5.3 First-order spectra 216
5.5.4 Conclusions . 218
6 States of Aggregation 221
6.1 Real gases . . . . . . . . . . . . . . 221
6.1.1 Intramolecular structure . . 221
6.1.2 Intermolecular interactions 232
6.2 Liquids ........ . 244
6.3 Crystals .......... . 247
6.3.1 Lattice structure .. 247
6.3.2 Electronic structure 252
6.4 Phase equilibria . 257
6.5 Solutions . . . . . . . . . . 265
1 Chemical Reactions 275
7.1 Equilibrium ........ . 275
7.2 Kinetics .......... . 283
7.2.1 Descriptive kinetics . 283
7.2.2 Reaction mechanisms 290
7.2.3 Conclusions ..... . 302
Index 307
Introduction
Each science has its own fundamental program to pursue, although the
fields pertaining to a given science overlap those in another in a wide va
riety of interdisciplinary ways. In this regard, the fundamental program of
chemical physics consists in understanding chemical phenomena in terms
of the most fundamental laws of physics. The purpose just stated can be
pursued by adopting two major methodologies - an experimental and a
theoretical one. The place of action for experimentalists is a laboratory,
where they try to keep under controlled and repeatable conditions the
chemical phenomenon to be studied. The place of action for theoreticians
is a desk (possibly equipped with computing facilities), where they try
either to fit a chemical phenomenon within the known physical theories, or
to tailor a specific theory for it from which emerges the essential features
that make the phenomenon intelligible.
Besides their own technical skills, both experimentalists and theoreti
cians need a clear understanding, at a quantitative level, of the fundamental
ideas of the subject. Chemical physics is a science on its own. Its main
concern is chemistry, whose phenomena it wishes to describe using the lan
guage of physics and mathematics. In this book we shall not consider any
of the tools of a chemical physicist, i.e., laboratory apparata or theoretical
techniques. Rather, we shall focus on concepts, which are presented at the
quantitative level, i.e., carefully discussing the equations (their origin and
strength of predictability) governing the phenomena of chemical-physics
interest.
We believe that an approach which maintains itself at this quantitative
level is the best suited for removing, as much as possible, the conceptual
ambiguities that inevitably arise from such a wide and complex subject.
To pursue our goal, we present the subject starting from the main ideas
of physics (classical, quantum and statistical physics) relevant to the sub
sequent description of chemically-interesting phenomena. These ideas are
analyzed systematically, trying to avoid any confusion between assumptions
and logical conclusions. Our task has been to write a book which is not
a mere explanation of apparently unrelated results, but an individual
presentation of the essence of a connected theory, with no claim of com
pleteness. In fact, our hope is that the book will not be used or judged for
completeness, but for organization, clarity and economy.
The question arises whether the book is suited to students. A potential
concern about our approach might be that it could encourage students to
doubt their intuition. However, chemical-physics concepts often require
X INTRODUCTION
very simple mathematics and logic, as opposed to intuition. Of course,
intuition is important in the subject, but, in our opinion, it is unwise to
avoid mathematics and logic, since they strengthen rather than compete
with intuition. Moreover, the very essence of chemical physics is the task
of eventually fitting any result or concept relying on intuition within the
more satisfactory rigor of a formal conceptual structure. Therefore, the an
swer to the above question is that the book is indeed intended to be useful
to students, not only in chemistry, but also in those fields (physics, mate
rials science, engineering, biology) where there is a need for a knowledge
of chemistry which includes, besides the presentations of the facts, their
explanation in terms of general principles.
We have found some major weaknesses in the available literature on
the subject. First of all, there appears to be a lack of continuity between
introductory books and the sophisticated concepts needed for the more
quantitative approach we have been looking for. The applied books give
at most a brief discussion of background material, since their objective
is not to use the applications to strenghten the reader's understanding
of basic concepts, but instead to provide a detailed discussion of special
tepics. As for the most advanced books, they are often lengthy one
thousand-page treatises, more suitable as informative reference books than
formative tools.
The book has been written to be read in consecutive order. However, two
alternatives may be followed: (i) sections 3.4-3.6 may be postponed after
chapter 5, and (ii) sections 6.4-7.2.1 may be read right after chapter 3.
Consistent with the spirit of this book (which focuses on fundamentals
rather than applications), we suggest that the reader, as a useful exercise,
fills in the missing steps as we go from a given equation to the next one (and
continues to do this for all the equations or, at least, for most ofthem). The notation
used is very common, and should create no difficulty to the reader. For
instance, vectors are written in boldface and their components are labeled
by Latin letters (j, k, ... = 1, 2, 3).
Most of the assertions in chapters 3-7 have been proved, whereas many
in chapters 1-2 have been only quoted. The reason is that all assertions
of the first two chapters have been proved in our book Lecture Notes in
Classical and Quantum Physics (Blackwell Scientific Publications, Oxford,
1990), to which the interested reader might refer.
One of us (FB) aknowledges Professors Arieh Ben-Naim, Augusto Rastel
li and Luisa Schenetti for valuable discussions, and the Department of
Chemistry at the Universita di Modena, Italy, for their kind hospitality while
this work was completed.
Chapter 1
Classical Physics
Classical physics distinguishes in the Universe two components: matter and
radiation. Matter consists of localized particles whose state, specified by
their position and velocity coordinates, evolves in time according to the
laws of Newtonian mechanics. The state ofradiation is given, at each time,
by the components, at each point of space, of the electric and magnetic
fields satisfying the Maxwell equations. The interaction between particles
and radiation is described by the Lorentz force.
1.1 Newtonian mechanics
The fundamental equation of classical dynamics is the Newton equation,
= =
F ma mr(t), (1.1)
where F, the force acting on the particle, is, in general, a known function
of position, r, velocity, v :::: r, of the particle, and, possibly, of time, t. The
mass, m, and the acceleration, a, may be defined independently of eq. 1.1,
which, therefore, can be regarded as the equation defining the force, i.e.,
the system itself.
For instance, the one-dimensional harmonic oscillator (HO) is a particle,
with mass m, moving along a segment of straight line (along the x-axis,
say), subjected to a force proportional to and in the opposite direction of
the position coordinate: F = -kx. Therefore, eq. 1.1 for the HO becomes
x(t) + w2x(t) = 0, (1.2)
where
(1.3)
1
F. Battaglia et al., Fundamentals in Chemical Physics
© Springer Science+Business Media Dordrecht 1998
2 CHAPTER 1. CLASSICAL PHYSICS
is the angular frequency.
Equation 1.1 is a second-order differential equation and has a unique
solution if the initial conditions are given (i.e., the position, x(O), and the
velocity, v(O), of the particle at a given time, t0, that can be chosen as the
origin of time: t0 = 0). In the case of the HO there is an analytic solution:
x(t) =A sin(wt + 4>), (1.4)
where the amplitude A and the phase 4> are two constants, which are deter
mined by the initial conditions.
It is important to stress that the dynamical equation has a unique solu
tion if, at a given time, t (that from now on we shall set equal to zero), the
0
coordinates of position, x(O), and velocity, v(O), of all the particles in the
system are known. We say that the collection of all coordinates of position
and velocity at a given time defines the state of the system at that time.
The fundamental problem of classical dynamics is, therefore, the following:
from the state of the system at time t = 0, find the state at any other time.
Such a problem is solved once eq. 1.1 is solved.
1.2 Lagrangian mechanics
A limitation of the Newton equation is that it preserves its form 1.1 only in
Cartesian coordinates. Let us consider, for instance, a particle, with mass
m, subjected to a force which depends only on the distance between the
point where the particle is located and a given point in space (called the
center of the force and that we shall choose to be the origin of the reference
system):
F = J(r)r, (1.5)
where
(1.6)
The symmetry of the system allows us to simplify the solution of the prob
lem if one chooses spherical polar coordinates, rather than Cartesian ones:
X r sin (} cos cp (1. 7a)
y r sin (}sin cp (1.7b)
z rcosO. (1.7c)
It can easily be shown that in spherical polar coordinates, the equations
of motion do not preserve the form 1.1. For instance, the equation for the
radial coordinate is
mr = J(r) + (23 , (1.8)
mr