The Quantum Mechanical Atom

By the late 1800’s it was clear that classical physics was incapable of describing atoms and molecules
Experiments showed that electrons acted like tiny charged particles in some experiments and waves in others
The physics that describes objects with wave/particle duality is called quantum mechanics or quantum theory
Energy can be transferred between things as light or radiation
Radiation carries energy through space as waves or oscillations moving outward from a disturbance
Electromagnetic waves (radiation) may be characterized by their “height” or amplitude and the number that occur per second or frequency (v)
The units of frequency are the hertz (Hz)
The minimum and maximum amplitude of electromagnetic radiation are evenly spaced
The peak-to-peak distance is called the wavelength
The product of frequency and wavelength give the speed of light (c)
Electromagnetic radiation comes in a broad range of frequencies called the electromagnetic spectrum
The electromagnetic spectrum is divided into regions according to the wavelengths of radiation
What we call light is a small slice of the electromagnetic spectrum with wavelengths between about 400 and 700 nm
This is called the visible region because we can “see” these wavelengths of the electromagnetic spectrum
Gamma rays, X rays, and ultraviolet radiation have wavelengths shorter than the visible region
Microwaves, infrared radiation, and radio waves have wavelengths longer than visible light
The way a substance absorbs electromagnetic radiation can be used to characterize it
For example, each substance absorbs a uniquely different set of infrared frequencies
A plot of wavelengths absorbed versus the absorption is called an infrared absorption spectrum
It can be used to identify a substance
The oscillating magnetic and electric fields of an electromagnetic wave interact with particles that it passes
A charged particle can pick up energy at the expense of the radiation source
The energy transfer is not describe correctly by classical physics
In 1900 the German scientist Max Planck proposed that the electromagnetic radiation could be viewed as a stream of tiny energy packets or quanta we now call photons
Photons travel at the speed of light
Planck proposed, and Einstein confirmed, that the energy of a photon is proportional to its frequency
This means that both electrons and electromagnetic radiation can be represented as either waves or particles
The visible spectrum is a continuous spectrum because it contains a continuous distribution of light of all colors
Excited atoms can emit light
The atomic spectrum or emission spectrum is a series of individual lines called a line spectrum
Atomic spectra are unique for each element
In general, the line spectrum of an element is rather complicated
The line spectrum of hydrogen, with a single electron, is the simplest
The Rydberg equation can be used to calculated all the spectral lines of hydrogen
n1 and n2 are positive integers
The Rydberg constant, RH, is an empirical constant with a value of 109,678 cm-1
Atomic line spectra tells us that when an excited atom loses energy, not just any arbitrary amount can be lost
This is possible if the electron is restricted to certain energy levels
The energy of the electron is said to be quantized
The first theoretical model that successfully accounted for the Rydberg equation was proposed in 1913 by Niels Bohr
Bohr proposed that the electrons moved around the nucleus is fixed paths or orbits much like the planets move around the sun
The orbits, labeled with the integer n, have energy
This equation allows the calculation of the energy of any orbit
The lowest energy state of an atom is called the ground state (an electron with n = 1 for a hydrogen atom)
An electron that “escapes” from the nucleus has infinity for its quantum number and zero energy
Bohr’s (theoretical) equation explains the (empirical) Rydberg equation
The combination of constants, b/hc, has a value which differs from the experimentally derived value of RH by only 0.05%
Bohr’s efforts to develop a general theory of electronic structure was doomed by the wave/particle duality of electrons
De Broglie suggested that the wavelength of a particle of mass m moving at speed v is
This relation provides the link between the description as a particle and as a wave
Heavy objects have very “short” wavelengths so their matter waves and the wave properties go unnoticed
Tiny particles with small masses have “long” wavelengths so their wave properties are an important part of their behavior
Waves combine in two ways
The constructive and destructive interference is called diffraction
Electrons produce similar patterns
There are two types of waves: traveling and standing
A standing wave is produced when a guitar string is plucked: the center of the string vibrates, but the ends remain fixed
Points of zero wave amplitude are called nodes
For guitar strings the only waves are those for which a half-wavelength is repeated exactly a whole number of times
For a strength of length L with n an integer this can be written
These results can be used to show how quantum theory unites the wave and particle description of a bound electron
Consider the classical particle model of the “bead on a wire”
If the electron (particle) has mass m and speed v
The kinetic energy of the moving electron is
The De Broglie relation connects models (a) and (b)
The electron energy is quantized because it depends on the integer n
The lowest energy allowed is for n=1 or E=h2/8mL2 (the energy cannot be zero)
Electrons trapped on a wire have some residual kinetic energy, just like electrons trapped in atoms
The wave that corresponds to the electron is called a wave function
The amplitude of the wave function at a given point can be related to the probability of finding the electron there
According to quantum mechanics there are regions of the wire where the electrons will not be found
Regions of zero wave function amplitude are called nodes
It is generally true that the more nodes an electron has, the higher its energy
Erwin Schrödinger was the first to successfully apply the concept of the wave nature of matter to electronic structure
He developed an equation that can be solved to give wave functions and energy levels for electrons trapped in them
Wave functions for electrons in atoms are called orbitals
Orbitals are characterized by a set of three quantum numbers:
n = principle quantum number. All orbitals with the save principle quantum number are in the same shell. Allowed values: the set of positive integers.
l = secondary quantum number which divides the orbitals in a shell into smaller groups called subshells. Allowed values: from 0 to (n – 1).
ml = magnetic quantum number which divides the subshells into individual orbitals. Allowed values: integers from –l to +l.
The approximate energies of the subshells in an atom with more than one electron:
Electrons behave like tiny magnets
Electrons within atoms interact with a magnet field in one of two ways:
Electron spin is important in determining electronic structure
According to the Pauli exclusion principle no two electrons in the same atom can have identical values for all four quantum numbers
Thus two electron can occupy the same orbital only if they have opposite spin and are said to be paired
A substance with more spin in one direction is said to contain unpaired electrons
Substances with unpaired electrons are slightly attracted to a magnet and are called paramagnetic
Substances in which all electrons are paired are called diamagnetic
The distribution of electrons among the orbitals of an atom is called the electronic structure or electronic configuration
To indicate the ground state electron configuration we can:
List the subshells that contain electrons and indicate their electron population with a superscript.
Represent each orbital with a circle and use arrows to indicate the spin of each electron.
Electron configurations must be consistent with the Pauli principle, aufbau principle, and Hund’s rule
Example: N 1s22s22p3, Na 1s22s22p63s1
Electron configurations explain the structure of the periodic table
There are few important exceptions to the “expected” electronic figurations of commonly encountered elements
Following the rules for Cr, Cu, Ag, and Au using noble gas notation:
Apparently, half-filled and filled subshells are particularly stable
Similar irregularities occur among the lanthanide and actinide elements
The position of an electrons must be described with probabilities
Heisenberg’s uncertainty principle says that it is impossible to measure with complete precision the velocity and position of a particle simultaneously

These limitations are not important for large objects but are very important for small particles like electrons
Quantum mechanics requires that we talk about the probability of finding an electron in a particular region of space
This probability is often represented as an electron cloud about the nucleus
The probability varies with distance from the nucleus
This type of plot shows that electron density varies from place to place
Electron density variations define the shape, size, and orientation of orbitals
p orbitals are quite different from s orbitals
They posses a nodial plane which includes the nucleus and separates the “lobes” of high probability
Recall that there are three different orbitals in each p subshell
The shape and orientation of d orbitals are more complicated than for p orbitals
Shape and directional properties of the five d orbitals in a d subshell.
The f orbitals are even more complex than the d orbital
The amount of positive charge “felt” by outer electrons in atoms other than hydrogen is called the effective nuclear charge
It is lower than the atomic number because of shielding
The effective nuclear charge felt by outer electrons is determined primarily by the difference between the charge on the nucleus and the charge on the core
Effective nuclear charge controls a number of properties
Atomic size increases top to bottom in a group because of increasing n and gets smaller left to right in a groups because the effective nuclear charge increases
Variation in atomic and ionic radii. Values in picometers (10-12 m)
The size trends in ions can be summarized:
Positive ions are always smaller than the atoms they are formed
Negative ions always larger than the atoms from which they are formed
Ionization energy (IE) is the energy required to remove an electron from an isolated, gaseous atom
Successive ionizations are possible until no electrons remain
The trends in IE are the opposite of the trends in atomic size
The electron affinity (EA) is the potential energy change associated with the addition of an electron to a gaseous atom or ion in its ground state
The addition of one electron to a neutral atom is exothermic for nearly all atoms
The addition of more electrons requires energy
Consider the addition of electrons to oxygen:
The results for first electron affinities can be generalized
In general:
EA increases from left to right in a period
EA increases bottom to top in a group