Atomic Struct.
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Readings for this section

Petrucci: Chapter 8

Quantum Mechanics

• Mathematical Treatment necessary for the understanding of the structure of atoms.
• Necessary because Newtonian Physics, a.k.a., classical mechanics, is insufficient for this purpose.

Newtonian physics deals with macroscopic (bigger than microscopic) objects and their interactions. It can deal with such things as inertia, gravity, collisions between large objects, etc.

Atoms are made of small (sub atomic) particles that do not obey Newtonian physics. Over the first half of this century, the theory now referred to as Quantum Mechanics, a.k.a., quantum chemistry, was developed. There are now several kinds of quantum mechanics theories. The earliest ones are now referred to as Classical Quantum Mechanics as opposed to more recent theories like Relativistic Quantum Mechanics.  The most fundamental of these sub atomic particle is the electron.  It's thought to be indivisible.  All of Quantum mechanics is based on this premise.  In last month's New Scientist, a report was published of an experiment which successfully split an electron into two smaller particles.  If this report holds up under scrutiny, then Quantum Mechanics as we know it will need to be replaced with a new theory which allows for this to happen.

Duality of nature

There are two basic types of physical properties that can be used to categorize nature.  These are outlined here.

• Particles:
  Objects with mass and possibly charge.
  Can always measure position and momentum but not necessarily at the same time.
• Waves:
  Periodic function (oscillatory motion)
  Cannot exist at a point, displaced over time and space.

Both matter and light have properties that can be categorized as "wave-like" and others of which can be called "particle like". 

• Electrons can be diffracted like light waves (electron diffraction patterns through metal foils) yet we think of them as particles.
• Light waves can behave like particles (photo-electron effect) yet we think of them commonly as waves.

Nature of light

• Wave Picture
  
  Wavelength l l ·n = c
  reassess Frequency n (NOTE) m ·s^-1 = ms^-1
  Speed c

Electromagnetic spectrum

We can see that the electromagnetic spectrum consists of a wide range of frequencies from radio-waves with wavelengths on the scale of mountain ranges and continents to gamma rays with wavelengths smaller than an atom. The visible spectrum makes up a very small part of this continuous range of wavelengths.

Wave-like behaviour:

• Shows up in:
  • Interference patterns (addition/subtraction of waves)
  • diffraction (objects ~ same size as l)
  • refraction (change in speed of light in different medium)

Particle-like behaviour

• Photons: E = hn

  h = Planck’s constant = 6.626176(36) x 10^-34 Js

  Frequency can also be expressed in terms of the numbers of radians per second w.

  h = h/2p and w = 2p x n.

  thus, E = h w is also true.

Photo-electron effect

• Light of sufficient energy strikes metal.

  

  Electrons are ejected.  The energy from the photon is used to extract the electron from the metal (this is called the work function (something akin to the ionization energy) and is given the symbol f.  Any excess energy goes into kinetic energy in the electron.  Thus, if we measure the kinetic energy of the electrons that are ejected as a function of frequency, we find a relationship as follows.

  
  Kinetic energy of electrons depends on frequency of light and on the type of metal.  At a certain minimum photon energy (E=hn) the electrons are being supplied with just enough energy to be ejected from the metal.  At this point, the photon energy is exactly equal to f.  As the energy of the photon increases (higher frequency) the ejected electrons have increasing kinetic energy.  Plotting KE versus frequency gives a line with slope = h that crosses the KE=0 axis at what is called the threshold (minimum) frequency.  We can illustrate this with a simple energy-level diagram:

Energy levels in metal

• The zero in energy is defined to be when the electron is very far from atom (no interaction with the nucleus).
  • f = e^- binding energy, a.k.a. metal's work function, a.k.a. ionization energy.
  • hn = photon energy
  • hn = KE + f.
  • KE=0 u f = hn₀.

Quantization effects in nature

We normally experience the macroscopic world (larger than a microscope) where our observations are of measurables that are merely averages of the microscopic state of the matter.  For example, we can easily measure the average kinetic energy of a gas (temperature), the momentum of a moving object, etc.  Using light, we can probe the quantization inherent in atoms and molecules that we cannot otherwise detect with our own senses.

Hydrogen atom

We begin with the hydrogen atom, the simplest of all atoms and the only one for which we have exact (analytical) equations that describe the energy states.

• Emission spectrum revealed information regarding the structure of atom.
• Discrete frequencies of light are emitted from excited atoms.
• Explained as a consequence of the existence of discrete ‘quantized’ energy levels.

Emission spectrum H-atom

	<--- Hydrogen Spectrum
	<--- Energy levels with transitions shown lined up with the spectral lines.

• Spectrum (top)
  series of ‘lines’, discrete frequencies
• attributed to energy-level changes of electron as shown.

Rydberg equation

This equation is useful to determine the energy difference between two different energy levels in a hydrogen atom.  When an electron jumps from one level (n_i) to a different one (n_f), there is an energy transfer involved.  This energy could be thermal or it could be photon energy.  It's easier to measure photon energy on an individual atomic basis since DE = hn.  Where n is the frequency (s^-1) of the photon emitted or absorbed by the hydrogen atom and h is Planck's constant.  Since n = c/l, we can use the reciprocal of wavelength (called wave number) as a measure of energy directly.  Thus, 1/l is directly proportional to energy.

R_H is the Rydberg constant and has a value of 1.09678×10⁷ m^-1.  The absolute value symbols are because, depending on which way we set up the n₁ and n₂ values, we could get a negative value for reciprocal wavelength.  This is not a physical possibility.  We should realize, however, that if the electron dropped from a high energy level to a low one that the wavelength is for a photon that is emitted by the atom while if the electron jumped from a low level to a high one then the wavelength is for a photon that the atom absorbed.

The Rydberg equation is a phenomenological equation.  That means that it models the discrete energy pattern observed in the H-atom spectrum (and other one-electron ions); but that there is no theory behind it. It simply works! We need to try to develop models that explain the observed behaviour as described in the Rydberg equation.  One such model is the Bohr model.

Bohr Model

Bohr proposed a model, which explained the Rydberg equation. (See Reference Here)

• It consisted of circular discrete orbits labelled n=1,2,... infinity
• n is called the principal quantum number.
• Each orbit represents one energy level.

Bohr was able to use classical mechanics to describe the energy levels.  Using Columbic interactions between the electron and the nucleus and using the concept that the electron's angular momentum is quantized and must be an multiple of h/2p, he was able to calculate the Rydberg constant using fundamental constants.

What Bohr's model did not explain was why the electrons did not simply spiral down to the centre (nucleus) and continuously loose energy.  It also fails to accurately calculate energies for anything other than a one-electron atom or ion.  We'll also see that it fails on a few more experiments as well.

Example

What is Energy and wavelength of a photon emitted when e^- drops from the n=4 to n=2 energy level in a H atom?

now we calculate the energy in one photon of this wavelength:

E = hn = hc/l = (6.62618×10⁻³⁴Js × 2.9979×10⁸ ms⁻¹) / (486₂₇₁₉ nm)

(the subscript 2719 are extra digits I carried
in the calculation but are not significant
digits.  I wanted to avoid round off errors)

E = 4.09×10⁻¹⁹ J (for one atom, assuming one atom absorbs one photon)

E =  4.09×10⁻¹⁹ J/atom * 6.626×10²³ atoms/mol
    = 246 kJ/mole of atoms

Other elements

Rydberg and Bohr did a good job explaining the hydrogen atom.  Unfortunately, their model falls apart for anything else.

While other atoms also show ‘line’-type emission spectra, they are much more complex spectra that are not easily explained as was done for hydrogen.  This is because there are more electrons and hence the calculations are more complex. To this day, there is no simple equation to describe the pattern of lines for any atom other than hydrogen.

Each atom displays a unique pattern of frequencies.  This allows allows for a technique called Atomic Absorption (AA) to detect specific atoms.  AA can detect atoms in stars and interstellar space. 

Quantum numbers

A more complete model needs more quantum numbers to fully define all the electrons in an atom:

• n: principal quantum number (q.n.)
• l: orbital q.n.
  l = 0,1,2,…,n-1
• m_l: orbital angular momentum (or magnetic) q.n.
  -l £ m_l £ l
• m_s: electron spin q.n.
  m_s = ± ½

Spherical Harmonics and Quantum numbers

To better understand the concept involved here, and to try to develop a picture of the concept discussed herein, let's look at a series of macroscopic harmonic oscillators and see if we can find any trends.

1-dimensional harmonics 

(The following is also called a "particle in a 1-d box")

nodes:

If the string vibrates with one continuous motion over it's whole length, then it is vibrating in the lowest energy mode.  We have a standing wave with length of 2l.  If the string is prevented from vibrating in certain positions (nodes) the string will be forced to vibrate in a different mode. Each of these modes differs by the number of nodes in the wave function and in the frequency that results. (Listen).  Note that to excite a particular mode of vibration, it is best to pluck the strung near the maximum in the vibrational wave function.

2-dimensional Harmonics (drum skin)

Did you ever listen to a steel drum player?  By tapping on different parts of the drum surface, (s)he can make different notes sound out of the instrument.  That's because there are carefully placed dents in the surface of the drum that tends to damp out certain vibrations and which set up vibrational modes in the surface with maxima located in different places around the surface. By tapping in the location of a maximum of a particular vibrational mode the player can excite just that one mode and hence sound one note.  By tapping a different place, a different vibrational mode is excited and hence a different note.  Again the basic principal that the higher the number of nodes, the higher the vibration frequency is found to be true here in 2-d just like in 1-d sound.

3-D HARMONICS

Harmonics in 3-d are not quite as easy to relate to musical instruments since most musical instruments are designed to vibrate in 1 or 2 dimensions, not three. Imagine a sound wave in a spherical container.  Certain 3-d standing waves would tend to get established with high and low sound intensities at different regions and for different modes of vibration. 

The basic types of nodes in the 3-d case are:

• spherical nodes, resulting in a spherically symmetrical sound wave whose intensity is always the same at any given distance from the centre (radial function only, no angular dependence) and
• planar nodes (3 of them, actually) which would have a radial  dependence and an angular one too. Hence, if we're listening on the node, no sound is heard;  but if we're listening at some position perpendicular to the node then the sound would be maximized for a given distance from the centre.  Note that there would be three linearly independent planar nodes, one perpendicular to each of the three Cartesian coordinate axes.  Any other planar node in any random orientation can always be constructed as a linear combination of these three. 
• higher-order shaped nodes also exist but much more difficult to picture or describe.  In general, the more complex the node, the higher the energy of vibration.

It turns out that the properties of standing waves in a spherical system are very well described by a system of mathematics called spherical harmonics.  This same mathematics works well to describe the electronic wave functions of hydrogen atoms.

Summary:

Notice that in all the previous cases, 1-d, 2-d and 3-d  the following was true.

The lowest energy vibrational mode had no nodes (mode  level 1)

The second level of vibration had one node (level 2) and in higher than 1-d, there were the possibility of more than one type of node.

The third level of vibration had 2 nodes, etc.  Using the 3-d example, we can relate the nodes and their types to the quantum numbers discussed earlier.  From now on, I'll be discussing spherical harmonics as they relate to electron wave functions and I'll be using the word 'orbital' to mean wave function.   In this case, the type of orbital, obviously is a function of the type (or lack) of nodes.

We have now seen that spherical harmonics describe the number of orbitals and also their shape. Let’s recap this information and relate it to the symbols commonly used to indicate the type of orbital.

The orbital quantum number l describes the type of orbital. While the number m_l details its orientation. The shape of the orbital nodes can be determined from l as follows:

• l =0 => spherical nodes (and spherical orbitals)

• l =1 => planar nodes (and dumb-bell-shaped orbitals)
• l ³2 => More complex nodes (and more complex-shaped orbitals)

We can also use these numbers to count the total number of orbitals for any given value of the n.

Electron spin

Once we try to establish the actual electron count, we need yet one more quantum number. The electron spin is designated by the quantum number m_s which can have a value of +½ or –½.

According to the Pauli Exclusion Principle, no two electrons can exist in an atom with the same set of these four quantum numbers.

Another way of saying it… every orbital can have at most two electrons in it and these two must have opposite spin.

Explains the electron count in the various ‘shells’ of the atom

Combining the functions which describe the electron with other energy-calculating functions (hamiltonian) we can calculate the energy expected for any given function. The energy levels for an electron in a hydrogen atom can be calculated. These different calculated energy levels can be enumerated using quantum numbers n and l. If we add q.n. m_l, we now have a complete enumeration of the orbitals and q.n. m_s finally enumerates the electrons.

Periodic table

The periodic table shows the elements grouped according to the quantum numbers of their outermost valence electron.

• The rows correspond to the principal quantum number n.

  the columns correspond
  (in groups) to l.

The groups (blocks) of elements in the periodic table are indicated as follows.

Electronic Configurations

Aufbau principle

• Build up electronic configuration for atoms
  • Lowest energy levels first

The energy of the various quantum states of the hydrogen atom are shown graphically on the left. We would fill up electrons by placing them (two per orbital) starting with the lowest energy levels and building up the electron configuration. A way to remember the sequence of orbitals without the need to draw the energy levels out in diagram form is to write the orbital identifiers (1s, 2s, etc.) as shown on the right and then draw arrows as shown. Starting at the top, the levels are filled in sequence as directed by the arrows. The number of electrons in any given set of orbitals is given as a superscript (1s¹ means one electron in the 1s orbital)

Thus, a 15 electron atom (P) would be filled to give the following electron configuration.

1s² 2s² 2p⁶ 3s² 3p³.

Alternatively, we can follow the rows of the periodic table to determine the order of filling the orbitals. Unfortunately, we don’t always have such a reference with us and hence the memory technique is good to know too.

Note that levels 4s and 3d are very close in energy. This also holds for 5s,4d and 6s,5d combinations. This leads to some exceptions to the rules normally used to assign the electronic configuration as we shall see later.

• Using aufbau process and Pauli exclusion principle, determine the electron configuration for the following:

H (1e–) 1s¹

He (2e–) 1s²

O (8e–) 1s² 2s² 2p⁴

S (16e–) 1s² 2s² 2p⁶ 3s² 3p⁴

Ti (22e–) 1s² 2s² 2p⁶ 3s² 3p⁶ 4s² 3d²

Cr (24e–) 1s² 2s² 2p⁶ 3s² 3p⁶ 4s¹ 3d⁵
or               1s² 2s² 2p⁶ 3s² 3p⁶ 3d⁵ 4s¹

Cr is one of the exceptions. The 3d shell prefers to be half filled (5 electrons) since this is a perfectly spherical configuration. There is significant relaxing of the energy levels in this configuration, such that the energy for the 3d electrons actually drops below the 4s energy. This type of situation arises whenever one can exactly half-fill or completely fill the d-shell.

Exactly filled or half-filled shells are spherically symmetrical and are more stable. In the case of the d-shell, since the s electrons of the next period are so close in energy, it is possible to promote an electron from the s shell to the d shell of the previous period, thus (half) filling the d shell. This filled (or half-filled) state is more quite stable and hence, the d shell becomes lower in energy than the s shell. Hence, In these cases, the normal sequence of filling the orbitals is changed.

Use box diagrams

The Chromium atom has its 3d shell exactly half-filled in its most stable configuration. We would expect similar exceptions from the sequencing given above in other atoms in the Cr family. Also, exceptions occur in Cu and it's family where the d10 configuration with one lone s electron is favoured over the alternative where one d orbital is half filled but the s has two electrons. Cu = [Ar]3d104s1.

Electronic configurations for the metals in the fourth row of the periodic table (Ca-Zn). * exceptions to the normal sequencing are indicated.

Trends in the Periodic Table

Now that we've seen the way that we can build up the electronic configuration of the atoms and how to organize the atoms into a periodic Table, according to their electronic configurations (quantum numbers) we can now look at some trends

Atomic Radius

The first trend of import is in the atomic radius. The force of attraction between the outermost electron(s) and the nucleus affects the atomic radius. The force of attraction of two charges is given classically by Coulombs Law.

.

In the case of quantum mechanics, we cannot calculate a value for r exactly. And must rely on some mean value <r> of the distance between the nucleus and the electron. Charges q1 and q2 represent the charge of the electron and the nucleus, respectively. For the purpose of discussing trends it is convenient to use the concept of core charge. The core charge represents the charge experienced by the outermost electrons of an atom, assuming the inner electrons shield the nucleus from their 'view'. For example, in the atom Li there are two electrons in the 'core' shell (1s) and one in the 'valence' shell. The outer electron (2s1) sees the nucleus (Z=3) only through the screen created by the two core electrons. Thus, it sees a core charge of Z – 2 = 1. More generally, the Core charge can be defined as

Core Charge = Z – # inner electrons.

It turns out that, If we ignore the transition metals, we can simply take the column number to be the core charge, for example, Oxygen is in column # 6 and has a core charge of +6.  Thus, along a given row, the orbitals are all approximately the same size but the the attraction of the electron for the nucleus (core) in creases from left to right.  Since the core charge increases from left to right along any one row, we expect the atoms on the right side to be smaller than those on the left. Similarly, as one descends a given column the core charge remain the same but the size of the orbitals increase.

Core Charge assumes that inner electrons shield the nuclear charge completely from the outer electrons.  In reality, the orbitals in which these outer electrons exist actually penetrate the inner orbitals somewhat such that the effective shielding is not 100%.  In addition, electrons in the same shell can partially shield each other somewhat.  The amount of shielding and the final effective nuclear charge depends on differences in orbitals, and how they interact with each other.  For example, s orbitals have more electron density near (or even on) the nucleus than a p orbital with the same principal quantum number and so shields the nucleus better (and is less effected by shielding from other orbitals) than the p.

Ionization energy

The First ionization energy of an atom is the energy required to remove one electron from the neutral atom as described by the equation below.

M -----> M⁺ + e^-                            DH = ionization energy

Arguments similar to the ones given for atomic size play a roll here. This time, the larger the force of attraction, the larger will be the amount of energy required to pull the electron away. If we plot actual energies as a function of atomic number (or column in the periodic table, we can see this trend more explicitly.

Note how the ionization energy increases as we go from left to right along the different rows of the table. This follows the concept of Coulombic attraction. However, there are a few dips in the curve which bare considering. Be is higher than B (and Mg > Al, Ca > Ga as well). This makes sense since the former of each of these pairs represents the atom with a filled s orbital. The next orbital in the list is a p that is higher in energy (closer to the zero mark called completely removed). Thus the drop in ionization energies even though the force of attraction would be higher according to our first hand-waving discussion.

There is a similar drop in ionization energies as we move from column 5 to column 6. Here, we are transitioning from an exactly half-filled p-shell to one in which one extra electron exists. Since the half-filled shell is particularly stable, we expect a high ionization energy (for N, P, As…) whereas that for O, S, Se will be slightly lower (trend wise).

For second, third, etc, ionizations, the general trend within an atom is that each subsequent ionization is harder than the previous.  This makes sense since each subsequent ionization represents the removal of a negatively charged electron from an increasingly positive ion (more attraction for the electron) and each subsequent electron is in a lower orbital on the given atom. For example,  the second ionization energy is the energy required to remove the second electron from the atom, i.e., the energy to remove one electron from a +1 cation.  Arguments similar to those for the first ionization energy hold when discussing the relative ionization energies of the various atoms.

Lets compare the relative ionizations of Na with Mg.  Their first ionizations as plotted in the graph above show that Mg holds it's outermost electron more tightly than Na since f₁(Na) is lower than f₁(Mg).   However, when we look at the +1 ions resulting from these ionizations, we see that the Mg⁺¹ still has one electron in the 3^rd shell (It's isoelectronic with Na) whereas the Na⁺¹ has its outer electron in shell 2 (It's isoelectronic with Ne).  It's much harder to remove an electron from the second shell than from the third since the second is closer and more strongly held by the nucleus.  So while there has been a large jump in difficulty in removing the electron from the Na⁺¹, compared to the neutral, there's only been a small jump in energy required to remove the second third-shell electron from the Mg⁺¹.  Hence, the second ionization energy of Na is higher than the second ionization energy of Mg. 

It's dangerous to try to carry this type of argument too far into the levels of ionization of all atoms.  There are too many factors involved to be able to perfectly explain all relative energy comparisons using any simple theory.  More complete atomic orbital calculations would be required to precisely determine these energies.

Electron Affinity

We can see a similar trend in the electron affinity comparisons. Electron affinity is the energy of attraction of the neutral atom for one extra electron. (How much energy does it take to add one electron to an already neutral atom as given by the formula below.

M + e^-  -----> M^-                           DH  =  electron affinity

(Note that DH can be thought of as negative of the the ionization energy of the anion)

   Again, There is interplay between the relative size of the atom and the relative charge of the nucleus. The electron affinities listed here are for the first three rows in the periodic table (excepting the transition metals). We see that the highest affinities are displayed by the elements F and Cl (from the 7^th column). These belong to the Halogens. The Halogens as a family have the highest electron affinity of the elements. That also makes sense using Coulomb's law wherein the Halogens experience the highest 'core charge'.

One might ask: 'Why do the Nobel gases not have a higher electron affinity than the Halogens?'. The answer lies in the fact that the Nobel gases have a completely filled outer shell and any extra electron added thereon would have to go onto the next higher shell which is of course higher in energy (not as attractive). These are not even included in the diagram above since they have virtually no tendency to form negative ions.

Reactivity

The reactivity of the elements in any one column (family) will tend to have similarities. 

1. The elements in the first column are called the Alkali metals.  The members of this family can all easily loose one electron to form a positive ion.  These elements can often be found in salts such as LiCl, NaCl, KCl, ... Their reactivity, although similar, will also differ since as they get lower in the periodic table, their outer electron is further and further from the nucleus and also, they are getting larger and larger.  For example, Li(s) will react with water vigorously, Na(s) more so, K(s) will burst into flames on dropping into water (even explode if too large a chunk is added).
   
2. The elements of the second column belong to the Alkali-Earth family.  These tend to form +2 ions and salts of these metals combine with a different ratio than that of the alkali metals.  Such compounds as BeF₂, MgCl₂, ... belie the +2 charge on the metal.
   
   We'll skip the transition metals for now. Follow the columns designated by A (see the diagram below)
   
3. The elements of the third column tend to form +3 ions except for the B case where a +3 ion of B would be very unstable.  AlCl₃ is an example of a +3 ion combining with three -1 chloride ions.  Here, we see a significant break in the reactivity of the elements as we go from the first to the second row of the table.  This tends to demark the boundary between metals (which form cations easily) and non-metals, which do not form cations easily.
   
4. In the fourth column, C is quite different from Si just as we saw in the third column. (See the article in the CHEMBOOK website).  Yet there are still similarities.  Both elements can form tetrahedral structures, as we'll see later.  Si tends to stand alone in that it does not act quite like C but it is also different from the lower elements in Column 4.  The lower elements can all easily form cations (and are therefore metals).  Si has semblance to the metals lower in the column and is there for classified as a semi-metal or metalloid.
   
5. In column 5, N and P both are non metals and can form -3 anions  while As and Sb are classified as metalloids.
   
6. The 6th column (headed by O) is the Chalcogens family.  Here, the first three elements are non-metals and can all form -2 anions.  Te is the metalloid and Po is a metal.  This is the only one of the elements in column 6 or greater to be classified as a metal.
   
7. In the 7th column (the Halogens), the elements tend to form -1 anions easily.  They need only one electron to fill their outer shell and hence the strong tendency to -1 charge.
   
8. Finally, in the last column (the noble gases) the elements tend to be quite unreactive.  Their outer shells are filled.

Michael J. Mombourquette.
Copyright © 1997
Revised: September 02, 2006.

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