Atomic and Molecular Structure

Because so much of the chemistry of atoms and molecules is related to their structures, the study of descriptive chemistry begins with a consideration of these topics. The reasons for this are simple and straightforward. For example, many of the chemical characteristics of nitrogen are attributable to the structure of the N2 molecule, :N ≡ N:. The triple bond in the N2 molecule is very strong, and that bond strength is responsible for many chemical properties of nitrogen (such as it being a relatively unreactive gas). Likewise, to understand the basis for the enormous difference in the chemical behavior of SF4 and SF6 it is necessary to understand the difference between the structures of these molecules, which can be shown as

Moreover, to understand why SF6 exists as a stable compound whereas SCl6 does not, we need to know something about the properties of the S, F, and Cl atoms. As another illustration, it may be asked why the PO43− ion is quite stable but NO43− is not. Throughout this descriptive chemistry book, reference will be made in many instances to differences in chemical behavior that are based on atomic and molecular properties. Certainly not all chemical characteristics are predictable from an understanding of atomic and molecular structure. However, structural principles are useful in so many cases (for both comprehension of facts and prediction of properties) that a study of atomic and molecular structure is essential.

What follows is a nonmathematical treatment of the aspects of atomic and molecular structure that provides an adequate basis for understanding much of the chemistry presented later in this book. Much of this chapter should be a review of principles learned in earlier chemistry courses, which is intentional. More theoretical treatments of these topics can be found in the suggested readings at the end of this chapter.

2.1 Atomic Structure

A knowledge of the structure of atoms provides the basis for understanding how they combine and the types of bonds that are formed. In this section, we review early work in this area, and variations in atomic properties will be related to the periodic table.

2.1.1 Quantum Numbers

It was the analysis of the line spectrum of hydrogen observed by J. J. Balmer and others that led Neils Bohr to a treatment of the hydrogen atom that is now referred to as the Bohr model. In that model, there are supposedly “allowed” orbits in which the electron can move around the nucleus without radiating electromagnetic energy. The orbits are those for which the angular momentum, mvr, can have only certain values (they are referred to as quantized). This condition can be represented by the relationship

vr=nh2π (2.1)

where n is an integer (1, 2, 3, …) corresponding to the orbit, h is Planck’s constant, m is the mass of the electron, v is its velocity, and r is the radius of the orbit. Although the Bohr model gave a successful interpretation of the line spectrum of hydrogen, it did not explain the spectral properties of species other than hydrogen and ions containing a single electron (He+, Li2+, etc.).

In 1924, Louis de Broglie, as a young doctoral student, investigated some of the consequences of relativity theory. It was known that for electromagnetic radiation, the energy, E, is expressed by the Planck relationship,

=hυ=hcλ (2.2)

where c, ν, and λ are the velocity, frequency, and wavelength of the radiation, respectively. The photon also has an energy given by a relationship obtained from relativity theory,

=mc2 (2.3)

A specific photon can have only one energy, so the right-hand sides of Eqs. (2.2) and (2.3) must be equal. Therefore,

cλ=mc2 (2.4)

and solving for the wavelength gives

=hmc (2.5)

The product of mass and velocity equals momentum, so the wavelength of a photon, represented by h/mc, is Planck’s constant divided by its momentum. Because particles have many of the characteristics of photons, de Broglie reasoned that for a particle moving at a velocity, ν, there should be an associated wavelength that is expressed as

=hmv (2.6)

This predicted wave character was verified in 1927 by C. J. Davisson and L. H. Germer who studied the diffraction of an electron beam that was directed at a nickel crystal. Diffraction is a characteristic of waves, so it was demonstrated that moving electrons have a wave character.

If an electron behaves as a wave as it moves in a hydrogen atom, a stable orbit can result only when the circumference of a circular orbit contains a whole number of waves. In that way, the waves can join smoothly to produce a standing wave with the circumference being equal to an integral number of wavelengths. This equality can be represented as

πr=nλ (2.7)

where n is an integer. Because λ is equal to h/mv, substitution of this value in Eq. (2.7) gives

πr=nhmv (2.8)

which can be rearranged to give

vr=nh2π (2.9)

It should be noted that this relationship is identical to Bohr’s assumption about stable orbits (shown in Eq. 2.1)!

In 1926, Erwin Schrödinger made use of the wave character of the electron and adapted a previously known equation for three-dimensional waves to the hydrogen atom problem. The result is known as the Schrödinger wave equation for the hydrogen atom, which can be written as

2Ψ+2mℏ2(E−V)Ψ=0 (2.10)

where Ψ is the wave function, ħ is h/2π, m is the mass of the electron, E is the total energy, V is the potential energy (in this case the electrostatic energy) of the system, and ∇2 is the Laplacian operator:

2=∂2∂x2+∂2∂y2+∂2∂z2 (2.11)

The wave function is, therefore, a function of the coordinates of the parts of the system that completely describes the system. A useful characteristic of the quantum mechanical way of treating problems is that once the wave function is known, it provides a way for calculating some properties of the system.

The Schrödinger equation for the hydrogen atom is a second-order partial differential equation in three variables. A customary technique for solving this type of differential equation is by a procedure known as the separation of variables. In that way, a complicated equation that contains multiple variables is reduced to multiple equations, each of which contains a smaller number of variables. The potential energy, V, is a function of the distance of the electron from the nucleus, and this distance is represented in Cartesian coordinates as r = (x2 + y2 + z2)1/2. Because of this relationship, it is impossible to use the separation of variables technique. Schrödinger solved the wave equation by first transforming the Laplacian operator into polar coordinates. The resulting equation can be written as

r2∂∂rr2∂Ψ∂r+1r2sinθ∂∂θ(sinθ∂Ψ∂θ)+1r2sin2θ∂2Ψ∂φ2+2mℏ2(E+e2r)Ψ=0 (2.12)

Although no attempt will be made to solve this very complicated equation, it should be pointed out that in this form the separation of the variables is possible, and equations that are functions of r, θ, and ϕ result. Each of the simpler equations that are obtained can be solved to give solutions that are functions of only one variable. These partial solutions are described by the functions R(r), Θ(θ), and Φ(ϕ), respectively, and the overall solution is the product of these partial solutions.

It is important to note at this point that the mathematical restrictions imposed by solving the differential equations naturally lead to some restraints on the nature of the solutions. For example, solution of the equation containing r requires the introduction of an integer, n, which can have the values n = 1, 2, 3, … and an integer l, which has values that are related to the value of n such that l = 0, 1, 2, … (n − 1). For a given value of n, the values for l can be all integers from 0 up to (n − 1). The quantum number n is called the principal quantum number and l is called the angular momentum quantum number. The principal quantum number determines the energy of the state for the hydrogen atom, but for complex atoms the energy also depends on l.

The partial solution of the equation that contains the angular dependence results in the introduction of another quantum number, ml. This number is called the magnetic quantum...