100 Years of Physical Chemistry: A Collection of Landmark Papers
384
100 Years of Physical Chemistry: A Collection of Landmark Papers
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| ISBN-13: | 9780854049875 |
|---|---|
| Publisher: | RSC |
| Publication date: | 03/26/2003 |
| Pages: | 384 |
| Product dimensions: | 6.14(w) x 9.21(h) x (d) |
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100 Years of Physical Chemistry
By The Royal Society of Chemistry
The Royal Society of Chemistry
Copyright © 2003 The Royal Society of ChemistryAll rights reserved.
ISBN: 978-0-85404-987-5
CHAPTER 1
Intermolecular Forces
A. D. Buckingham
Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge, UK CB2 1EW
Commentary on: The general theory of molecular forces, F. London, Trans. Faraday Soc., 1937, 33, 8-26.
The origin of the substantial attractive forces between nonpolar molecules was a serious problem in the early 20th century. While much was known of the strength of these forces from the Van der Waals equation of state for imperfect gases and from thermodynamic properties of liquids and solids, there was little understanding. The difficulty can be illustrated by the fact that the binding energy of solid argon is of the same order of magnitude as that of the highly polar isoelectronic species HCl. Debye suggested in 1921 that argon atoms, while known to be non-dipolar, may be quadrupolar; however, after the advent of quantum mechanics in 1926, it was clear that the charge distribution of an inert-gas atom is spherically symmetric. In 1928 Wang showed that there is a long-range attractive energy between two hydrogen atoms that varies as R-6 where R is their separation. Soon afterwards, London presented his 'general theory of molecular forces' and gave us approximate formulae relating the interaction energy to the polarizability of the free molecules and their 'internal zero-point energy'. London showed that these forces arise from the quantum-mechanical fluctuations in the coordinates of the electrons and called them the dispersion effect. He demonstrated their additivity and estimated their magnitude for many simple molecules. The paper points out the important role of the Pauli principle in determining the overlap-repulsion force (on p. 21 it associates the Coulomb interaction of overlapping spherical atomic charge clouds with an incomplete screening of the nuclei, causing a repulsion; actually the enhanced electronic charge density in the overlap region between the nuclei would lead to an attraction, so the strong repulsion at short range is due to the Pauli principle).
A feature of London's paper is its emphasis on the zero-point motion of electrons: it is the intermolecular correlation of this zero-point motion that is responsible for dispersion forces. London's Section 9 extends the idea of zero-point fluctuations to the interaction of dipolar molecules. If their moment of inertia is small, as it is for hydrogen halide molecules, then even near the absolute zero of temperature when the molecules are in their non-rotating ground states, there are large fluctuations in the orientation of the molecules and these become correlated in the interacting pair.
London's eqn. (15) for the dipole-dipole dispersion energy is not a simple product of properties of the separate atoms. A partial separation was achieved in 1948 by Casimir and Polder who expressed the R-6 dispersion energy as the product of the polarizability of each molecule at the imaginary frequency iu integrated over u from zero to infinity. The polarizability at imaginary frequencies may be a bizarre property but it is a mathematically well behaved function that decreases monotonically from the static polarizability at u = 0 to zero as u]IT [right arrow] ∞.
Casimir and Polder also showed that retardation effects weaken the dispersion force at separations of the order of the wavelength of the electronic absorption bands of the interacting molecules, which is typically 10-7 m. The retarded dispersion energy varies as R-7 at large R and is determined by the static polarizabilities of the interacting molecules. At very large separations the forces between molecules are weak but for colloidal particles and macroscopic objects they may add and their effects are measurable. Fluctuations in particle position occur more slowly for nuclei than for electrons, so the intermolecular forces that are due to nuclear motion are effectively unretarded. A general theory of the interaction of macroscopic bodies in terms of the bulk static and dynamic dielectric properties has been presented by Lifshitz. Proton movements in hydrogen-bonded solids and liquids may contribute to the binding energy as well as to the dielectric constant, electrical conductivity and intense continuous infrared absorption.
If one or both of the molecules in an interacting pair lacks a centre of symmetry, e.g. CH4 ... CH4, Ar ... CH4, or Ar ... cyclopropane, there is, in addition to the dispersion energy terms in R-6, R-8, R-10, ..., an orientation-dependent contribution that varies as R-7. It could be significant for coupling the translation and rotation in gases and liquids and for the lattice energy of solids.
London illuminated the origin of dispersion forces by considering the dipolar coupling of two three-dimensional isotropic harmonic oscillators. He obtained the exact energy and showed that it varies as R-6 for large R. Longuet-Higgins discussed the range of validity of London's theory and used a similar harmonic-oscillator model to show that at equilibrium at temperature T there is a lowering of the free energy A(R), though not of the internal energy E(R), through the coupling of two classical harmonic oscillators, so their attraction is entropic in nature and vanishes at T = 0. It is the quantization of the energy of the oscillators that leads to a lowering of E(R) through the dispersion force. If one of a pair of identical oscillators is in its first excited state the interaction lifts the degeneracy and leads to a first-order dipolar interaction energy proportional to R-3 this is an example of a resonance energy which may be considered to arise from the exchange of a photon between identical oscillators.
Since the dispersion energy arises from intermolecular correlation of charge fluctuations, it is not accounted for by the usual computational techniques of density functional theory (DFT) which employ the local density and its spatial derivatives. Special techniques are needed if DFT is to be used for investigating problems where intermolecular forces play a significant role.
THE GENERAL THEORY OF MOLECULAR FORCES.
By F. London (Paris).
Received 31st July, 1936.
Following Van der Waals, we have learnt to think of the molecules as centres of forces and to consider these so-called Molecular Forces as the common cause for various phenomena: The deviations of the gas equation from that of an ideal gas, which, as one knows, indicate the identity of the molecular forces in the liquid with those in the gaseous state; the phenomena of capillarity and of adsorption; the sublimation heat of molecular lattices; certain effects of broadening of spectral lines, etc. It has already been possible roughly to determine these forces in a fairly consistent quantitative way, using their measurable effects as basis.
In these semi-empirical calculations, for reasons of simplicity, one imagined the molecular forces simply as rigid, additive central forces, in general cohesion, like gravitation; this presumption actually implied a very suggestive and simple explanation of the parallelism observed in the different effects of these forces. When, however, one began to try to explain the molecular forces by the general conceptions of the electric structure of the molecules it seemed hopeless to obtain such a simple result.
[section] 1. Orientation Effect.
Since molecules as a whole are usually uncharged the dipole moment μ was regarded as the most important constant for the forces between molecules. The interaction between two such dipoles μI and μII depends upon their relative orientation. The interaction energy is well known to be given to a first approximation by
U = - μIμII/R3 (2 cos θI cosθII - sin θI sin θII cos(φI - φII (I)
where θI, πI; φI; θII, πII are polar co-ordinates giving the orientation of the dipoles, the polar axis being represented by the line joining the two centres, R = their distance. We obtain attraction as well as repulsion, corresponding to the different orientations. If all orientations were equally often realised the average of ITLμITL would be zero.
But according to Boltzmann statistics the orientations of lower energy are statistically preferred, the more preferred the lower the temperature. Keesom, averaging over all positions, found as a result of this preference:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
For low temperatures or small distances ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) this expression does not hold. It is obvious that the molecules cannot have a more favourable orientation than parallel to each other along the line joining the two molecules, in which case one would obtain as interaction energy (see (1)):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)
which gives in any case a lower limit for this energy. (2) and (3) represent an attractive force, the so-called orientation effect, by which Keesom tried to interpret the Van der Waals attraction.
[secdtion] 2. Induction Effect.
Debye remarked that these forces cannot be the only ones. According to (2) they give an attraction which vanishes with increasing temperature. But experience shows that the empirical Van der Waals corrections do not vanish equally rapidly with high temperatures, and Debye therefore concluded that there must be, in addition, an interaction energy independent of temperature. In this respect it would not help to consider the actual charge distribution of the molecules more in detail, e.g. by introducing the quadrupole and higher moments. The average of these interactions also would vanish for high temperatures.
But by its charge distribution alone a molecule is, of course, still very roughly characterised. Actually, the charge distribution will be changed under the influence of another molecule. This property of a molecule can very simply be described by introducing a further constant, the polarisability [varies]. In an external electric field of the strength F a molecule of polarisability [varies] shows an induced moment
M = α x F.... (4)
(in addition to a possible permanent dipole moment) and its energy in the field F is given by
U = 1/2 [varies] x F2.... (5)
Now the molecule I may produce near the molecule II an electric field of the strength
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)
This field polarises the molecule II and gives rise to an additional interaction energy according to (5)
U = -1/2 [varies]IIF2 = [varies]II/2 μI2/R6 (I + 3 cos2 θI) .. (7)
which is always negative (attraction) and therefore its average, even for infinitely high temperatures, is also negative. Since cos2 θ = 1/3 we obtain:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
A corresponding amount would result for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], i.e. for the action of μII upon [varies]I. As total interaction of the two molecules we obtain:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ... (8)
If the two molecules are of the same kind (μI = II = μ and [varies]i = [varies]I = [varies]II = [varies]) we have
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].... (8')
This is the so-called induction effect.
In such a way Debye and Falckenhagen believed it possible to explain the Van der Waals equation. But many molecules have certainly no permanent dipole moment (rare gases, H2, N2, CH4, etc.). There they assumed the existence of quadrupole moments τ, which would of course also give rise to a similar interaction by inducing dipoles in each other. Instead of (8) this would give:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].... (9)
Since no other method of measuring these quadrupoles was known, the Van der Waals corrections (second Virial coefficient) were used in order to determine backwards ITLτITL which, after μ and [varies], has been regarded as the most fundamental molecular constant.
[section] 3. Criticism of the Static Models for Molecular Forces.
The most obvious objection to all these conceptions is that they do not explain the above mentioned parallelism in the different manifestations of the molecular forces. One cannot understand why, for example, in the liquid and in the solid state between all neighbours simultaneously practically the same forces should act as between the occasional pairs of molecules in the gaseous state. All these models are very far from simply representing a general additive cohesion:
Suppose that two molecules I and II have such orientations of their permanent dipoles that they are attracted by a third one ; then between the two former molecules very different forces are usually operative, mostly repulsive forces. Or, if the forces are due to polarisation, the acting field will usually be greatly lowered, when many molecules from different sides superimpose their polarising fields. One should expect, therefore, that in the liquid and in the solid state the forces caused by induced or permanent dipoles or multipoles should at least be greatly diminished, if not by reasons of symmetry completely cancelled.
The situation seemed to be still worse when wave mechanics showed that the rare gases are exactly spherically symmetrical, that they have neither a permanent dipole nor quadrupole nor any other multipole. They showed none of the mentioned interactions. It is true, that for H2, N2, etc., wave mechanics, too, gives at least quadrupoles. But for H2 we are now able to calculate the value of the quadrupole moment numerically by wave mechanics. One gets only about 1/100 of the Van der Waals forces that were attributed hitherto to suitably chosen quadrupoles.
On the other hand, wave mechanics has provided us with a completely new aspect of the interaction between neutral atomic systems.
[section] 4. Dispersion Effect; a Simplified Model.
Let us take two spherically symmetrical systems, each with a polarisability [varies], say two three-dimensional isotropic harmonic oscillators with no permanent moment in their rest position. If the charges e of these oscillators are artificially displaced from their rest positions by the displacements
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
respectively, we obtain for the potential energy:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].... (10)
Classically the two systems in their equilibrium position
(xI = XII = … = zII = 0)
would not act upon each other and, when brought into finite distance (R >3 [square root of [2 [varies], remain in their rest position. They could not influence a momentum in each other.
However, in quantum mechanics, as is well known, a particle cannot lie absolutely at rest on a certain point. That would contradict the uncertainty relation. According to quantum mechanics our isotropic oscillators, even in their lowest states, make a so-called zero-point motion which one can only describe statistically, for example, by a probability function which defines the probability with which any configuration occurs; whilst one cannot describe the way in which the different configurations follow each other. For the isotropic oscillators these probability functions give a spherically symmetric distribution of configurations round the rest position. (The rare gases, too, have such a spherically symmetrical distribution for the electrons around the nucleus.) We need not know much quantum mechanics in order to discuss our simple model. We only need to know that in quantum mechanics the lowest state of a harmonic oscillator of the proper frequency v has the energy
E0 = 1/2 hv..... (11)
the so-called zero-point energy. If we introduce the following co-ordinates ("normal"-co-ordinates):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
the potential energy (10) can be written as a sum of squares like the potential energy of six independent oscillators (while the kinetic energy would not change its form):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10')
The frequencies of these six oscillators are given by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)
Here v0 = e/[square root of] [m[varies]] is the proper frequency of the two elastic systems, if isolated from each other (R [right arrow] ∞), and m is their reduced mass. Assuming [varies] [much less than] R3, we have developed the square roots in into powers of ([varies]/R3.
The lowest state of this system of six oscillators will therefore be given, according to (II), by:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
The first term 3hv0] is, of course, simply the internal zero-point energy of the two isolated elastic systems. The second term, however,
U = 3/4 hv0[varies]2/R6.... (13)
depends upon the distance R and is to be considered as an interaction energy which, being negative, characterises an attractive force. We shall presume that this type of force, which is not conditioned by the existence of a permanent dipole or any higher multipole, will be responsible for the Van der Waals attraction of the rare gases and also of the simple molecules H2, N2, etc. For reasons which will be explained presently these forces are called the dispersion effect.
[section] 5. Dispersion Effect; General Formula.
Though it is of course not possible to describe this interaction mechanism in terms of our customary classical mechanics, we may still illustrate it in a kind of semi-classical language.
If one were to take an instantaneous photograph of a molecule at any time, one would find various configurations of nuclei and electrons, showing in general dipole moments. In a spherically symmetrical rare gas molecule, as well as in our isotropic oscillators, the average over very many of such snapshots would of course give no preference for any direction. These very quickly varying dipoles, represented by the zero-point motion of a molecule, produce an electric field and act upon the polarisability of the other molecule and produce there induced dipoles, which are in phase and in interaction with the instantaneous dipoles producing them. The zero-point motion is, so to speak, accompanied by synchronised electric alternating field, but not by a radiation field: The energy of the zero-point motion cannot be dissipated by radiation.
(Continues...)
Excerpted from 100 Years of Physical Chemistry by The Royal Society of Chemistry. Copyright © 2003 The Royal Society of Chemistry. Excerpted by permission of The Royal Society of Chemistry.
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Table of Contents
Intermolecular Forces; Clusters; Molecular Spectroscopy; Magnetic Resonance; Quantum Chemistry; Photochemical Dynamics; Gas-Phase Kinetics; Ultrafast Processes; Molecular Reaction Dynamics; Atmospheric Chemistry; Astrophysical Chemistry; Theoretical Dynamics; Statistical Thermodynamics; Polymers; Colloids; Liquid Crystals; Liquid-Solid Interfaces: Liquid-Liquid Interfaces; Electrochemistry; Gas-Solid Surface Science; Biophysical Chemistry; Solid State Chemistry; Catalysis.What People are Saying About This
This volume illustrates magnificently how important these Faraday journals have been to the development of physical chemistry. May the next 100 years prove equally fruitful.