TOPICS
1.5 An introduction to wave mechanics
52.93 pm. This value is called the Bohr radius of the H atom and is given the symbol a0.
An increase in the principal quantum number from n¼ 1 to n¼ 1 has a special significance; it corresponds to the ionization of the atom (equation 1.9) and the ionization energy, IE, can be determined by combining equations 1.5 and 1.7, as shown in equation 1.10. Values of IEs are quoted per mole of atoms:
One mole of a substance contains the Avogadro number, L, of particles:
L¼ 6:022 1023mol1
HðgÞ "HþðgÞ þ e ð1:9Þ
IE¼ E1 E1¼hc
¼ hcR
1 12 1
12
ð1:10Þ
¼ 2:179 1018J
¼ 2:179 1018 6:022 1023J mol1
¼ 1:312 106J mol1
¼ 1312 kJ mol1
Although the SI unit of energy is the joule, ionization energies are often expressed in electron volts (eV) (1 eV ¼ 96:4853 96:5 kJ mol1).
Impressive as the success of the Bohr model was when applied to the H atom, extensive modifications were required to cope with species containing more than one electron; we shall not pursue this further here.
CHEMICAL AND THEORETICAL BACKGROUND
Box 1.2 Determination of structure: electron diffraction The diffraction of electrons by molecules illustrates the
fact that the electrons behave as both particles and waves.
Electrons that have been accelerated through a potential difference of 50 kV possess a wavelength of 5.5 pm and a monochromated (i.e. a single wavelength) electron beam is suitable for diffraction by molecules in the gas phase. The electron diffraction apparatus (maintained under high vacuum) is arranged so that the electron beam interacts with a gas stream emerging from a nozzle. The electric fields of the atomic nuclei in the sample are responsible for most of the electron scattering that is observed.
Electron diffraction studies of gas phase samples are concerned with molecules that are continually in motion, which are, therefore, in random orientations and well separated from one another. The diffraction data therefore mainly provide information about intramolecular bond parameters (contrast with the results of X-ray diffraction, see Box 5.5). The initial data relate the scattering angle of the electron beam to intensity. After corrections have been made for atomic scattering, molecular scattering data are obtained, and from these data it is possible (via Fourier transformation) to obtain interatomic distances between all possible (bonded and non-bonded) pairs of atoms in the gaseous molecule. Converting these distances into a three-dimensional molecular structure is not trivial, particularly for large molecules. As a simple example, consider electron diffraction data for BCl3 in the gas phase. The results give bonded distances B–Cl¼ 174 pm (all bonds of equal length) and non-bonded distances Cl—Cl¼ 301 pm (three equal distances):
By trigonometry, it is possible to show that each Cl–B–Cl bond angle, , is equal to 1208 and that BCl3is therefore a planar molecule.
Electron diffraction is not confined to the study of gases.
Low energy electrons (10–200 eV) are diffracted from the surface of a solid and the diffraction pattern so obtained provides information about the arrangement of atoms on the surface of the solid sample. This technique is called low energy electron diffraction(LEED).
Further reading
E.A.V. Ebsworth, D.W.H. Rankin and S. Cradock (1991) Structural Methods in Inorganic Chemistry, 2nd edn, CRC Press, Boca Raton, FL – A chapter on diffraction methods includes electron diffraction by gases and liquids.
C. Hammond (2001) The Basics of Crystallography and Diffraction, 2nd edn, Oxford University Press, Oxford – Chapter 11 covers electron diffraction and its applications.
Fig. 1.4 Definition of the polar coordinates (r, , ) for a point shown here in pink; r is the radial coordinate and and are angular coordinates. and are measured in radians (rad). Cartesian axes (x, y and z) are also shown.
Chapter 1 . An introduction to wave mechanics 7
CHEMICAL AND THEORETICAL BACKGROUND Box 1.3 Particle in a box
The following discussion illustrates the so-called particle in a one-dimensional boxand illustrates quantization arising from the Schro¨dinger wave equation.
The Schro¨dinger wave equation for the motion of a particle in one dimension is given by:
d2
dx2þ82m
h2 ðE VÞ ¼ 0
where m is the mass, E is the total energy and V is the potential energy of the particle. The derivation of this equation is considered in the set of exercises at the end of Box 1.3. For a given system for which V and m are known, we can use the Schro¨dinger equation to obtain values of E (the allowed energies of the particle) and (the wavefunction).
The wavefunction itself has no physical meaning, but 2is a probability (see main text) and for this to be the case, must have certain properties:
. must be finite for all values of x;
. can only have one value for any value of x;
. andd
dxmust vary continuously as x varies.
Now, consider a particle that is undergoing simple-harmonic wave-like motion in one dimension, i.e. we can fix the direction of wave propagation to be along the x axis (the choice of x is arbitrary). Let the motion be further constrained such that the particle cannot go outside the fixed, vertical walls of a box of width a. There is no force acting on the particle within the box and so the potential energy, V, is zero; if we take V¼ 0, we are placing limits on x such that 0 x a, i.e. the particle cannot move outside the box. The only restriction that we place on the total energy E is that it must be positive and cannot be infinite.
There is one further restriction that we shall simply state: the boundary conditionfor the particle in the box is that must be zero when x¼ 0 and x ¼ a.
Now let us rewrite the Schro¨dinger equation for the specific case of the particle in the one-dimensional box where V¼ 0:
d2
dx2¼ 82mE h2
which may be written in the simpler form:
d2
dx2¼ k2 where k2¼82mE h2
The solution to this (a well-known general equation) is:
¼ A sin kx þ B cos kx
where A and B are integration constants. When x¼ 0, sin kx¼ 0 and cos kx ¼ 1; hence, ¼ B when x ¼ 0. How-ever, the boundary condition above stated that ¼ 0 when x¼ 0, and this is only true if B ¼ 0. Also from the boundary condition, we see that ¼ 0 when x ¼ a, and hence we can rewrite the above equation in the form:
¼ A sin ka ¼ 0
Since the probability, 2, that the particle will be at points between x¼ 0 and x ¼ a cannot be zero (i.e. the particle must be somewhere inside the box), A cannot be zero and the last equation is only valid if:
ka¼ n
where n¼ 1, 2, 3 . . .; n cannot be zero as this would make the probability, 2, zero meaning that the particle would no longer be in the box.
Combining the last two equations gives:
¼ A sinnx a and, from earlier:
E¼ k2h2 82m¼ n2h2
8ma2
where n¼ 1, 2, 3 . . .; n is the quantum number determining the energy of a particle of mass m confined within a one-dimensional box of width a. So, the limitations placed on the value of have led to quantized energy levels, the spacing of which is determined by m and a.
The resultant motion of the particle is described by a series of standing sine waves, three of which are illustrated below. The wavefunction 2 has a wavelength of a, while wavefunctions 1 and 3 possess wavelengths ofa
2and2a 3 respectively. Each of the waves in the diagram has an amplitude of zero at the origin (i.e. at the point a¼ 0);
points at which ¼ 0 are called nodes. For a given particle of mass m, the separations of the energy levels vary according to n2, i.e. the spacings are not equal.
and this is represented in equation 1.14 where RðrÞ and Að; Þ are radial and angular wavefunctions respectively.†
Cartesianðx; y; zÞ radialðrÞ angularð; Þ ¼ RðrÞAð; Þ ð1:14Þ Having solved the wave equation, what are the results?
. The wavefunction is a solution of the Schro¨dinger equation and describes the behaviour of an electron in a region of space called the atomic orbital.
. We can find energy values that are associated with parti-cular wavefunctions.
. The quantization of energy levels arises naturally from the Schro¨dinger equation (seeBox 1.3).
A wavefunction is a mathematical function that contains detailed information about the behaviour of an electron. An atomic wavefunction consists of a radial component, RðrÞ, and an angular component, Að; Þ. The region of space defined by a wavefunction is called an atomic orbital.