The Free Electron Model of Metallic Bonding

Example 11-3. Calculate the density of silver in g/cm iftheAg atoms crystallize in a face-centered cubic and have a metallic radius of 144 pm. 

Solution. Density is mass divided by volume. There are four Ag atoms per unit cell in a face-centered cubic (cep) because each of the eight corner atoms is shared between eight unit cells (8 x /g) and each of the six face-centers is shared between two unit cells (6 X H). According to the periodic table, one mole of Ag atoms has a mass of 107.9g. The mass of one unit cell is 

In the cep lattice, the atoms touch along the diagonal of a face-center, such that a = 8 r. The volume of a cube is V=a. Hence, the volume of a unit cell is 

In the previous chapters, we discussed various models of bonding for covalent and polar covalent molecules (the VSEPR and LCP models, valence bond theory, and molecular orbital theory). We shall now turn our focus to a discussion of models describing metallic bonding. We begin with the free electron model, which assumes that the ionized electrons in a metallic solid have been completely removed from the influence of the atoms in the crystal and exist essentially as an electron gas. Freshman chemistry books typically describe this simplified version of metallic bonding as a sea of electrons that is delocalized over all the metal atoms in the crystalline solid. We shall then progress to the band theory of solids, which results from introducing the periodic potential of the crystalline lattice. 

The metallic elements lie on the left-hand side of the periodic table, where the number of valence electrons is considerably less than the number of valence orbitals available to accommodate them. As a result, the metals maximize their bonding interactions by having as many nearest-neighbor interactions as possible—often forming either a cep or hep crystalline lattice. Because the metals have a smaller effective nuclear charge than their nonmetal counterparts, the valence electrons of the metals are ionized fairly easily it is this resulting sea of delocalized electrons that serves to 

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A representation of the free-electron model of metallic bonding. This model applies to metal alloys as well as to metallic elements. 

H2, N2, or CO dissociates on a surface, we need to take two orbitals of the molecule into account, the highest occupied and the lowest unoccupied molecular orbital (the HOMO and LUMO of the so-called frontier orbital concept). Let us take a simple case to start with the molecule A2 with occupied bonding level a and unoccupied anti-bonding level a. We use jellium as the substrate metal and discuss the chemisorption of A2 in the resonant level model. What happens is that the two levels broaden because of the rather weak interaction with the free electron cloud of the metal. 

The free-electron model is a simplified representation of metallic bonding. While it is helpful for visualizing metals at the atomic level, this model cannot sufficiently explain the properties of all metals. Quantum mechanics offers a more comprehensive model for metallic bonding. Go to the web site above, and click on Web Links. This will launch you into the world of molecular orbitals and band theory. Use a graphic organizer or write a brief report that compares the free-electron and band-theory models of metallic bonding. 

Models for the electronic structure of polynuclear systems were also developed. Except for metals, where a free electron model of the valence electrons was used, all methods were based on a description of the electronic structure in terms of atomic orbitals. Direct numerical solutions of the Hartree-Fock equations were not feasible and the Thomas-Fermi density model gave ridiculous results. Instead, two different models were introduced. The valence bond formulation (5) followed closely the concepts of chemical bonds between atoms which predated quantum theory (and even the discovery of the electron). In this formulation certain reasonable "configurations" were constructed by drawing bonds between unpaired electrons on different atoms. A mathematical function formed from a sum of products of atomic orbitals was used to represent each configuration. The energy and electronic structure was then... 

This chapter begins a series of chapters devoted to electronic structure and transport properties. In the present chapter, the foundation for understanding band structures of crystalline solids is laid. The presumption is, of course, that said electronic structures are more appropriately described from the standpoint of an MO (or Bloch)-type approach, rather than the Heitler-London valence-bond approach. This chapter will start with the many-body Schrodinger equation and the independent-electron (Hartree-Fock) approximation. This is followed with Bloch s theorem for wave functions in a periodic potential and an introduction to reciprocal space. Two general approaches are then described for solving the extended electronic structure problem, the free-electron model and the LCAO method, both of which rely on the independent-electron approximation. Finally, the consequences of the independent-electron approximation are examined. Chapter 5 studies the tight-binding method in detail. Chapter 6 focuses on electron and atomic dynamics (i.e. transport properties), and the metal-nonmetal transition is discussed in Chapter 7. 

The important physical properties of simple metals and, in particular, the alkali metals can be understood in terms of a free electron model in which the most weakly bound electrons of the constituent atoms move freely throughout the volume of the metal (231). This is analogous to the free electron model for conjugated systems (365) where the electrons are assumed to be free to move along the bonds throughout the system under a potential field which is, in a first approximation, constant (the particle-in-a-box model). The free electron approach can be improved by replaeing the constant potential with a periodic potential to represent discrete atoms in the chain (365). This corresponds to the nearly free electron model (231) for treating electrons in a metal. 

The transition metals are not describable by the conventional near-free electron model of the metalhc bond since the valence d electrons remain relatively tightly bound to their parent atoms forming unsaturated bondings with their neighbors. These d bondings are responsible for the structural and cohesive properties of transition metals. 

This molecular-orbital model of metallic bonding (or band theory, as it is also called) is not so different in some respects from the electron-sea model. In both models Ihe electrons are free to move about in the solid. The molecular-orbital model is more quantitative than the simple electron-sea model, however, so many properties of metals can be accounted for by quantum mechanical calculations using molecular-orbital theory. 

A theory for the metallic state proposed by Drude at the turn of this century explained many characteristic features of metals. In this model, called the free-electron theory, all the atoms in a metallic crystal are assumed to take part collectively in bonding, each atom providing a certain number of (valence) electrons to the bond. These free electrons belong to the crystal as a whole. The crystal is considered to be... 

The above model of an sp-valent metal as a gas of free electrons would exhibit no bonding because the only contribution to the energy is the repulsive kinetic energy. It takes an average value per electron, which is given by... 

The free-electron gas model is a good starting point for the sp-valent metals where the loosely bound valence electrons are stripped off from their ion cores as the atoms are brought together to form the solid. However, bonding in the majority of elements and compounds takes place through saturated... 

Fig. 17. Pictorial representation of intercalated superconducting compound of 2-dimensional graphite (carbon atoms interconnected with solid lines each line represents a pair of covalent bond) interleaved with potassium (circles) which ionizes easily to K. and provide free electrons . According to the model, COVALON conduction takes place within the graphite plane and affects the COVALON on the adjacent graphite plane through plasmon waves provided by the free electrons from the potassium metal.

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