The amazing world of the Atom and the Ultimate Periodic ...€¦ · The Amazing World of the Atom and the Ultimate Periodic Table of the Elements To our Sons, Life is filled with

The Amazing World of the Atom and the Ultimate

Periodic Table of the Elements

To our Sons, Life is filled with hard work and good times.

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The Amazing World of the Atom and the Ultimate Periodic Table of the Elements Authors: Mihaela Rizescu, Costel Rizescu ISBN: 978-1-947641-16-7 FIRST EDITION Copyright © 2018 by Shutter Waves Text copyright © Mihaela Rizescu, Costel Rizescu, 1996 - 2018 Illustrations copyright © Mihaela Rizescu, Costel Rizescu, 1996 - 2018 Photos copyright © Mihaela Rizescu, Costel Rizescu, 1996 - 2018 All rights reserved. No part of this book may be used or reproduced or transmitted in any form or by any means whatsoever, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system and Internet usage, without permission in writing from the authors and publisher. All rights reserved. Printed in the United States of America

Preface Written in an introductory style, this book is addressed to undergraduate and graduate students taking physics and chemistry courses at the introductory level, to STEM high-school students, and to more experienced professors and researchers who need to stay up-to-date with the most recent advances on data, as well. Also beneficial to those involved in materials science, medical physics, biotechnology, astronomy, quantum chemistry, Earth and space science, and other related fields.

The book consists of two parts. The first part is devoted to the quantum mechanics theory of atomic structures as discovered and verified by physics whereas the second part presents and analyzes a large amount of the latest critically evaluated data for the properties of all elements on the periodic table of the elements. The first part of this book begins with the general properties of atoms. In chapter 2, the authors present the Bohr’s model of the atom – orbits and energy levels, electron transitions, emission spectrum of hydrogen along with fundamentals of spectrophotometry – and, at the end, it succinctly summaries its shortcomings. Chapters 3 and 4 discuss the wave particle duality and the uncertainty principle of quantum events, phenomena that serve as a foundation in building an understanding of quantum mechanics. In chapter 5, the authors discuss the wave functions that are derived from Schrödinger equation and present the quantum mechanical model of the atom. Chapter 6 deals with the electronic configuration of atoms. Terms and concepts associated to Pauli Exclusion Principle, Aufbau principle, as well as Hund and Madelung rules of occupation of the atomic orbitals by electrons are presented. Chapter 7 gives a brief discussion of the shielding and penetration effects incorporated in Slater’s rules for atoms, which is largely used to explain the periodic trends in the properties of elements.

The second part of the book starts with introduction to electron configuration, block classification, groups and periods on the periodic table. In second chapter it discusses families of elements and their general properties. In chapter 3 the authors present the data on their unique layout of the periodic table of the elements along with associated explanatory notes. Chapter 4 deals with periodic trends for ionization energy, electronegativity, atomic radius, density and melting points. Additional data of ionic radius, covalent radius, van der Waals radius, and 12-coordination atomic radius for almost all elements on the periodic table are also presented. The authors discussed over 30 review questions, problems and their solutions to help deepen the insight in this subject area to make this book a real study text. Plenty of problems are given to elucidate the material.

This book represents the culmination of authors’ many years of research and teaching medical physics, instrumentation, and materials science. It is also a crystallization of their intense passion and strong interest in the history of atomic physics and the philosophy of science. The authors worked with certain information and collected data for atomic physics and atomic processes, material sciences and biophysics in the course of professional activity. The data presented in this book has been checked in the course of the authors’ professional activity and sparingly published in various publications. Recently the authors have decided to publish this data set by its entirety, which is mainly comprised of atomic data, useful information about the periodic table of the elements and atomic systems, including molecules, metals and condensed systems of elements in crystalline state form.

From the beginning, it was authors’ intention to include information in this book that is not easily located elsewhere or is not found completely in a single book or publication. Thus, while this book can be used as a text, the authors hope that it will be used as a useful reference too. To the end, the authors merged data that is difficult to find in other books and, indeed, in the specialized literature. This book was meant from the beginning to be a constantly evolving work on progress. Each chapter and sub-chapter of this book capitalizes on the strengths, comments, feedback and criticism that the authors expect to have from students, faculty and working professionals. As such, all readers sending any comments, suggestions, or notification of errors to the authors at [email protected] will be greatly appreciated.

The authors

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Table of Contents Part 1: The Atom 1. INTRODUCTION to ATOMIC STRUCTURE ................................................................................................................... 1 2. BOHR MODEL of the ATOM ....................................................................................................................................... 4

2.1 Orbits and Energy Levels in the Bohr Model of the Atom ................................................................................... 5 2.2 Electron Transitions ........................................................................................................................................... 10 2.3 Emission Spectrum of Hydrogen ....................................................................................................................... 11 2.4 Introduction to Spectroscopy and Spectrophotometry .................................................................................... 13 2.5 Shortcomings of Bohr Atomic Model ................................................................................................................ 19

3. WAVE-PARTICLE DUALITY ........................................................................................................................................ 19 4. THE UNCERTAINTY PRINCIPLE ................................................................................................................................. 23 5. SCHRÖDINGER EQUATION and the QUANTUM MODEL of the ATOM .................................................................... 27

5.1 The Wave Function ............................................................................................................................................ 28 5.2 The Complete Set of the Atomic Quantum Numbers ....................................................................................... 30

5.2.1 Principal Quantum Number, 𝑛 ................................................................................................................... 30 5.2.2 Orbital Angular Momentum Quantum Number, 𝑙 ..................................................................................... 31 5.2.3 Orbital Magnetic Quantum Number, 𝑚 ................................................................................................... 32 5.2.4 Shape, Orientation and Size of 𝑠, 𝑝, 𝑑, 𝑓 Orbitals ....................................................................................... 36 5.2.5 Spin Magnetic Quantum Number, 𝑚 ....................................................................................................... 37

6. ELECTRONIC CONFIGURATION OF ATOMS .............................................................................................................. 40 6.1 The Pauli Exclusion Principle ............................................................................................................................. 40 6.2 The Aufbau Principle ......................................................................................................................................... 42 6.3 The Hund Rule ................................................................................................................................................... 43 6.4 The Madelung Rule ........................................................................................................................................... 45

7. SCHIELDING and PENETRATION EFFECTS, SLATER’s RULES ..................................................................................... 49 SUMMARY ................................................................................................................................................................... 51 Review Questions and Problems ................................................................................................................................. 54

Part 2: The Periodic Table of the Elements 1. INTRODUCTION to the PERIODIC TABLE of the ELEMENTS ..................................................................................... 57

1.1. Electron Configuration and Block Classification on the Periodic Table ............................................................ 57 1.2 Groups and Periods ........................................................................................................................................... 59

2. FAMILIES of ELEMENTS and their GENERAL PROPERTIES ........................................................................................ 60 2.1 Metals and Nonmetals ...................................................................................................................................... 60 2.2 Families of Elements ......................................................................................................................................... 61 2.3 About the Definition of Transition Metals ........................................................................................................ 63

3. ELEMENTAL DATA on the PERIODIC TABLE ............................................................................................................. 64 4. PERIODIC TRENDS .................................................................................................................................................... 73

4.1 Ionization Energy ............................................................................................................................................... 73 4.2 Electronegativity ............................................................................................................................................... 77 4.3 Sizes of Atoms ................................................................................................................................................... 80

4.3.1 Atomic Radius in Crystalline Solid – Metallic Radius in Crystals ................................................................ 81 4.3.2 Covalent Radius ......................................................................................................................................... 84 4.3.3 Van der Waals Radius ................................................................................................................................ 88 4.3.4 Ionic Radius ................................................................................................................................................ 90

4.4 Periodic Trends for Atomic Radius in Crystalline Solid State Form ................................................................... 94 4.5 Periodic Trends for Density in Crystalline Solid State Form ............................................................................ 100 4.6 Periodic Trends for Melting Points .................................................................................................................. 102

SUMMARY ................................................................................................................................................................. 105 Review Questions ...................................................................................................................................................... 107 Selected References .................................................................................................................................................. 109

Part 1: The ATOM

1. INTRODUCTION to ATOMIC STRUCTURE The atom is the smallest unit of matter that still maintains the identity and retains all chemical and physical properties of the element. An atom consists of a central, very small and dense nucleus that is surrounded by one or more negatively charged subatomic particles called electrons. The core of the atom – the nucleus - consists of protons and neutrons, both called also nucleons (see Figure 1-1). Protons and neutrons are held tightly together by short-distance strong attractive nuclear forces. A proton has a positive charge equal to 1.6 × 10 𝐶, representing the elementary charge for electric charges, while a neutron has no charge. Table 1-1 presents a few physical properties of some subatomic particles.

Figure 1-1. Schematic representation of helium atom (left) and silicon nucleus (right).

The mass of a neutron is slightly heavier than of proton, as shown in Table 1-1. All atoms are made of three types of subatomic particles: protons, electrons, and neutrons.

TABLE 1-1. Physical properties of subatomic particles that make up an atom.

Mass Charge Decay Lifetime

Proton

1.67262 × 10 𝐾𝑔 𝑚 = 1838.15 · 𝑚

1.6 × 10 𝐶

Proton decays into a positron and a neutral pion that itself immediately decays into 2 gamma ray photons: 𝑝 → 𝑒 + 𝜋 𝜋 → 2𝛾

Quantum mechanics theories predict that a free proton should

decay with a half-life on the order of no less than 10 years. Experimentally not proved yet.

Neutron

1.67493 × 10 𝐾𝑔 𝑚 = 1838.68 · 𝑚

No charge

A free neutron decays to a proton, an electron and an electron antineutrino,

a process known as beta decay: 𝑛 → 𝑝 + 𝑒 +

881.5 ± 0.2 𝑠

Electron 9.1 × 10 𝐾𝑔

−1.6 × 10 𝐶

There is no evidence of its decay because it would violate the law of

the charge conservation 𝑛/𝑎

According to today’s quantum physics theories, protons and neutrons – the building blocks of nuclei – are believed to be made up of smaller elementary particles called quarks and gluons. Quarks are considered the

1. INTRODUCTION to ATOMIC STRUCTURE

M. & C. Rizescu, Copyright © 2018 Shutter Waves, All rights reserved 2

fundamental constituents of matter. Quarks combine to form particles called hadrons, the most stable of which are protons and neutrons, the constituents of atomic nuclei. Quarks were never directly observed or found in isolation and they are the only elementary particles to experience all four fundamental interactions of the electromagnetic, gravitational, strong and weak forces, as well as the only known particles whose electric charges are not integer multiples of the elementary charge. Gluons are considered elementary particles that act as the exchange particles for the strong force between quarks, analogous to the exchange of photons in the electromagnetic force between two charged particles. Figure 1-2 shows simplified spherical configurations of hydrogen, neon and aluminum atoms, illustrating “circular orbits” for electrons “orbiting” the nucleus. Electrons occupy most of the volume of the atom.

Figure 1-2. Simplified representation of atomic configurations for hydrogen, neon and aluminum atoms

Electrons are subatomic particles, extremely lightweight, that “orbit” the nucleus in a cloud having a radius that is about four orders of magnitude greater than the diameter of atom’s nucleus, which has a radius of about 10 𝑚 (see the hydrogen atom illustrated in Figure 1-3). The charge of an electron is taken by convention as negative and has a magnitude of 1.6 × 10 𝐶 (Coulombs). Free atoms have the same number of electrons as protons to be electrically neutral. The opposite charge of the electrons and protons binds the atom together through electrostatic forces of attraction (this theoretical aspect will be discussed in a larger detail in next chapter). This force is relatively strong to hold the electrons and nucleus together but not so strong that an atom cannot lose or gain electrons.

Figure 1-3. A few typical characteristics of a hydrogen atom – the most abundant element, making up for almost 75% of the mass of the universe. A hydrogen atom is around one hundred thousand (10 ) times larger than the proton that forms its nucleus. The diameter of hydrogen atom is roughly equal to 1.1 Angstroms (1 𝐴𝑛𝑔𝑠𝑡𝑟𝑜𝑚 = 10 𝑚), which is about twice the Bohr radius of hydrogen. The electron, which radius was only calculated but still not experimentally measured yet, moves around the nucleus in definite domains called orbitals in an enormous volume compared to the proton. We will see later that this large volume is the result of balancing the electrostatic attraction occurring between electron and proton with the highly kinetic energy of the electron that results from confining the electron in a small region around nucleus, according to the Heisenberg uncertainty principle.

In chemistry, the structure of an atom is usually specified by indicating how many electrons, protons and neutrons it contains. For a neutral atom the number of protons - specified by the atomic number 𝑍 - must equal the number of electrons. The number of nucleons - protons and neutrons - in an atom is specified by the mass number 𝐴. If 𝑁 is the number of neutrons in an atom then the mass number is indicated as follows: 𝐴 = 𝑍 + 𝑁

The symbol of a particular element in the Periodic Table of the Elements contains the atomic number and mass number written as a superscript and subscript preceding the chemical symbol of the atom, as follows:

1. INTRODUCTION to ATOMIC STRUCTURE


𝑆𝑦𝑚𝑏𝑜𝑙 For example, a neutral silicon atom, whose structure is represented in Figure 1-4, has 14 protons, 14 neutrons and 14 electrons. Hence the mass number 𝐴 = 𝑍 + 𝑁 = 14 + 14 = 28 and silicon atom is represented by 𝑆𝑖. An ion is an atom (or molecule) that has a different total number of electrons than the total number of protons. An ion with fewer electrons than protons, hence having a net positive charge, is called cation and an ion with more electrons than protons, thus having a net negative charge, is named anion. When writing the chemical formula for an ion, its net charge is written in superscript to the right of the chemical symbol by placing the magnitude before the sign. For example, a double positive iron ion, an iron atom which lost 2 electrons, will be indicated as follows: 𝐹𝑒

Isotopes are atoms with the same atomic number but different mass numbers. Isotopes may be either stable or unstable. Unstable isotopes, known as radioisotopes, are radioactive and decay by spontaneous emission of energy and particles. For example iron has four stable isotopes; 𝐹𝑒 (5.85% natural abundance), 𝐹𝑒 (91.75% natural abundance), 𝐹𝑒 (2.12% natural abundance), and 𝐹𝑒 (0.28% natural abundance). Uranium has three major isotopes; 𝑈 (99.28% natural abundance), 𝑈 (0.71% natural abundance) and 𝑈 (0.01% natural abundance). All three isotopes are radioactive, emitting alpha particles ( 𝛼). Uranium-238 isotope, 𝑈, is the most stable one for uranium family, with a half-time of about 4.468 billion years (about the same as Earth’s life).

Radioactive decay occurs when an unstable isotope transforms to a more stable isotope, generally by emitting a subatomic particle such as an alpha or beta particle (electron) and sometimes gamma radiation as an excess energy of that decay reaction. Gamma radiation is a highly energetic electromagnetic radiation emitted from the nucleus of the atom during its decay. For example, 𝐶𝑜, which is an isotope of cobalt that is produced artificially in a nuclear reactor by neutron activation of the isotope 𝐶𝑜, decays by beta decay to a stable isotope nickel, 𝑁𝑖. The life-time of cobalt-60 radionuclide is 5.27 years and its decay reaction is as follows: 𝐶𝑜 → 𝑁𝑖 + 𝑒 + 𝛾

Example Problem #1 Determination of atomic and mass numbers How many protons, neutrons, and electrons does a neutral chromium atom 𝐶𝑟 contain? Solution: Let us identify the atomic and mass numbers of chromium as follows: 𝐴 = 𝑍 + 𝑁 52 = 24 + 𝑁 𝑁 = 28

Therefore, a neutral chromium atom has 24 protons, 28 neutrons and 24 electrons.

Example Problem #2 Determination of the number of protons, neutrons and electrons in a molecule How many protons, electrons and neutrons does a molecule of calcium nitrate 𝐶𝑎(𝑁𝑂 ) contain? Solution: Let us look up in the periodic table of the elements for the atomic and mass number of each element included in the calcium nitrate molecule and then re-write the formula of calcium nitrate with atomic and mass number notations. Therefore, we have 𝐶𝑎( 𝑁 𝑂 )

As seen above, the calcium nitrate molecule contains one calcium atom, 2 nitrogen atoms and 6 oxygen atoms. We also see that each atom has the same number of neutrons and electrons as protons. Therefore the calcium citrate molecule contains 20 + 2 × 7 + 3 × 2 × 8 = 82 protons, 82 neutrons and 82 electrons.

2. BOHR MODEL of the ATOM


2. BOHR MODEL of the ATOM The electron configuration of an atom is defined as the spatial arrangement (distribution) of electrons in that atom. The Danish physicist Niels Bohr (1885-1962) firstly formulated the electron configuration of an atom in 1913. Bohr described the atom as a small, positively charged nucleus surrounded by electrons that travel in stationary circular orbits at discrete set of distances from the nucleus due to electrostatic forces of attraction - similar in structure to the solar system (see Figure 1-4). Bohr postulated that electrons were confined in clearly defined and quantized orbits associated with definite energies, called allowed energy levels or shells, where electrons do not loose energy as they travel a particular orbit. Electrons occupy different orbits (marked as 𝐾-shell for 𝑛 = 1, 𝐿-shell for 𝑛 = 2, 𝑀-shell for 𝑛 = 3 in Figure 1-4) based on the energy level of the orbits with the lower energy level orbits being filled with electrons first.

Figure 1-4. The simplified three-dimensional (3-D) and two dimensional (2-D) representations of electron configuration

and nucleus composition of a silicon atom, 𝑆𝑖, as we are so familiar with from science logos.

Bohr imagined an atom as a very small nucleus surrounded by electrons orbiting the nucleus in a much larger volume. The space around the nucleus is divided into regions (like layers or shells), where electrons have a well-defined energy (kinetic and potential energy). These regions were called energy levels, numbered 1, 2, 3, 4, 5, ... outward from the nucleus. The energy levels are called shells by chemists as follows: 𝐾-shell for 𝑛 =1, 𝐿-shell for 𝑛 = 2, 𝑀-shell for 𝑛 = 3, 𝑁-shell for 𝑛 = 4, 𝑂-shell for 𝑛 = 5, and so on (as pictured in Figure 1-5). The energy associated with each energy level in an atom increases as the distance from its nucleus increases, similar to planets orbiting the Sun; the farther the planet is orbiting the Sun the higher the energy of that planet bound to that orbit becomes.

The binding energy of the electron in the atom is defined as the minimum amount of energy required to release that electron from the atom, and thus the energy with which it is bound in the atom. To release an electron from the 𝐿-shell (second innermost shell, 𝑛 = 2) of the atom is lower than that needed to release an electron from 𝐾 -shell ( 𝑛 = 1 ). For example, later in this chapter we will see that the energy level of the first “orbit” in hydrogen is equal to −13.606 𝑒𝑉, where the reference for energy (0 𝑒𝑉 ) is taken at infinite (). In literature the terms binding energy and energy level are often used as interchangeable.

Figure 1-5. Energy levels in atom



2.1 Orbits and Energy Levels in the Bohr Model of the Atom Niels Bohr proposed his atomic model based on the following postulates:

An electron executes circular motion around the nucleus under the influence of the electrostatic force of attraction between the positive nucleus and the negative electron in accordance with the laws of classical physics.

An electron can occupy only certain allowed orbits or stationary states having defined energies for which its orbital angular momentum, 𝐿, is quantized as an integral multiple of reduced Planck constant (ℏ =ℎ/2𝜋) according to the following relationship: 𝐿 = 𝑛 ℎ2𝜋 = 𝑛ℏ, 𝑤ℎ𝑒𝑟𝑒 𝑛 = 1, 2, 3, … 𝑎𝑛𝑑 ℎ 𝑖𝑠 𝑡ℎ𝑒 𝑃𝑙𝑎𝑛𝑐𝑘′𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

Any electron in a stationary state does not radiate electromagnetic energy.

An atom absorbs or emits energy when an electron transitions (moves) from one stationary state to another stationary state. The difference in energy between the initial and final stationary states is equal to the energy of the emitted or absorbed photon and is quantized according to the Planck relationship: ∆𝐸 = 𝐸 − 𝐸 = ℎ𝜈, 𝑤ℎ𝑒𝑟𝑒 𝜈 𝑖𝑠 𝑡ℎ𝑒 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑜𝑓 𝑡ℎ𝑒 𝑒𝑚𝑖𝑡𝑡𝑒𝑑 𝑝ℎ𝑜𝑡𝑜𝑛

Using Bohr postulates let us determine the radii of allowed orbits and energies of an electron in the Bohr model of the atom by analyzing the forces acting on and energies of an electron that surrounds a nucleus with 𝑍 protons on a circular path of radius 𝑟 . Illustrated in Figure 1-6, for an electron orbiting the nucleus the centripetal force that causes the uniform circular motion is the electrostatic force of attraction between the negatively charged electron and positively charged nucleus. The direction of the centripetal force is toward the center of the curvature, the same as the direction of the centripetal (radial) acceleration and it is perpendicular to velocity. According to Newton’s third law of motion, for every action there is an equal and opposite reaction. A reaction force called the centrifugal force balances the centripetal force - the action. The centripetal and centrifugal forces are equal in magnitude and opposite in direction. The centrifugal force does not act on the body in motion; it is just the inertial force acting on the source of the centripetal force to displace the body in a radial way from the center of the path. Thus, in orbiting a nucleus, the centripetal force (i.e. the electrostatic force) pulls in on the electron to keep it in its circular path, while the centrifugal force pulls outward. The centripetal force is equal in magnitude to the electrostatic force, having the same direction, as pictured in Figure 1-6, 𝐹 = 𝐹 where 𝐹 is pictured as 𝐹 and 𝐹 is pictured as 𝐹( ) , representing the electrostatic force experienced by the negatively charged electron (𝑒) from the action of the positively charged nucleus (𝑍𝑒). As depicted in Figure 1-6, 𝐹 ( ) represents the electrostatic force experienced by the positively charged nucleus (𝑍𝑒) from the action of the negatively charged electron (𝑒) and it is equal in magnitude and opposite direction to 𝐹( ) . Therefore, for the 𝑛 allowed orbit we have the following relation:

𝐹 = 𝑚 𝑣𝑟 = 14𝜋 ∙ (−𝑒)𝑍𝑒𝑟 = 14𝜋 ∙ 𝑍𝑒𝑟 = 𝐹

where 𝑚 is the mass of electron at rest, 𝑟 is the radius of 𝑛 orbit, 𝑣 is the constant speed of electron orbiting the 𝑛 orbit and is the permittivity of the material ( = , where is the relative permittivity of the material, and = 8.85 ∙ 10 𝐹/𝑚 is the permittivity of vacuum). Here 𝑛 is an integer and 𝑛 is the allowed orbit. The allowed orbits are numbered 1, 2, 3, 4, …, according to the value of 𝑛, which is called the quantum number of the orbit.



Figure 1-6. The Bohr model of the atom: illustration of an electron orbiting the nucleus on a circular path under the influence of the electric force of attraction between the negatively charged electron and positively charged nucleus. The electric force is the centripetal force causing a uniform circular motion for the negatively charged electron spinning around a positively charged nucleus (𝑍𝑒) on an allowed orbit of radius 𝑟 . The vector 𝑅 ⃗ is used in the computation of the orbital angular momentum 𝐿 , and is defined as 𝑅 ⃗ = 𝑟 ∙ 𝑢 , where 𝑢 represents the unit vector and 𝑟 represents the distance between the center of nucleus, O, and electron’s position, P.

Figure 1-7. The allowed energy levels and electron orbits in hydrogen atom in Bohr’s model of the atom. The radius of every orbit increases by the square of the quantum number associated with the energy level. The radius of the first orbit – called Bohr radius - has been experimentally confirmed for the hydrogen atom. It has a value of 0.529 Angstroms (0.0529 nanometers or 52.9 picometers).

The angular momentum of the electron, according to the Bohr quantum condition, is as follows: 𝐿 = 𝑚 𝑣 𝑟 = 𝑛 ℎ2𝜋 , 𝑤ℎ𝑒𝑟𝑒 𝑛 = 1, 2, 3, …

Solving the angular momentum equation for 𝑣 and substituting it into the centripetal force equation yields the following expression for 𝑟 :

𝑣 = 𝑛ℎ2𝜋𝑚 𝑟

𝑚𝑟 ( 𝑛ℎ2𝜋𝑚 𝑟 ) = 14𝜋 ∙ 𝑍𝑒𝑟 ⇒ 𝑚𝑟 ∙ 𝑛 ℎ4𝜋 𝑚 𝑟 = 14𝜋 ∙ 𝑍𝑒𝑟

𝑟 = 𝑛 ∙ ℎ𝜋𝑚 𝑍𝑒

The above highlighted formula provides the radii of the allowed orbits for an electron orbiting a nucleus with 𝑍 protons. As noticed the radii of allowed orbits increase by the square of 𝑛. According to the Bohr postulates an electron can exist only in the allowed orbits where the electron has a definite energy.

Therefore, the electron orbit with the smallest radius is for 𝑛 = 1 and for hydrogen atom (𝑍 = 1). Substituting the values in the above formula then we have the following value for the radius of the first orbit in hydrogen:

r1 = 0.529 Å The radius of the smallest orbit for the electron in hydrogen atom, 𝑟 , is called Bohr radius; it is equal

to 0.529 Angstroms (an Angstrom is equal to 10 meters and its symbol is Å). The radii of larger orbits



increase by the square of the quantum number, so for the second, third and fourth orbit in hydrogen atom we have the following radii: 𝑟 = 2 · 𝑟 = 4𝑟 = 4 ∙ 0.529 × 10 𝑚 = 2.116 × 10 𝑚 = 2.116 Å 𝑟 = 3 · 𝑟 = 9𝑟 = 9 ∙ 0.529 × 10 𝑚 = 4.761 × 10 𝑚 = 4.761 Å 𝑟 = 4 · 𝑟 = 16𝑟 = 16 ∙ 0.529 × 10 𝑚 = 8.464 × 10 𝑚 = 8.464 Å

Figure 1-7 depicts the first four allowed energy levels as well as the scaled size of the orbits in hydrogen atom. Bohr’s atomic model specifies that when an electron moves around a nucleus it moves in a certain orbit having a defined energy. There are many orbits surrounding the nucleus; the nearest orbit has the lowest energy and the farthest one has the highest energy.

Let us now determine the energy of an electron orbiting the nucleus in the 𝑛 orbit. The total energy of the electron is equal to the sum of its kinetic energy and electric potential energy as follows: 𝐸 (𝑡𝑜𝑡𝑎𝑙 𝑒𝑛𝑒𝑟𝑔𝑦) = 𝐾𝐸 (𝑘𝑖𝑛𝑒𝑡𝑖𝑐 𝑒𝑛𝑒𝑟𝑔𝑦) + 𝑃𝐸 (𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐 𝑝𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙 𝑒𝑛𝑒𝑟𝑔𝑦)

The electric potential energy is the energy of an electrically charged particle (at rest) in an external electric field produced by a punctual distribution of electric charges and is defined as the work that must be done to move that charged particle from an infinite distance to its current location. Any charged particle (at rest) placed in an external electric field poses electric potential energy. If a point positive charge 𝑄 is fixed at some point in space, any other positive charge 𝑞 (test charge) being brought close to it will experience a repulsive force and will therefore have electric potential energy (see Figure 1-8). For example, if an alpha-particle (a nucleus of the element Helium, which has 2 protons and 2 neutrons), is fixed at some point in space, any proton brought close to it will experience a repulsive force and will therefore have electric potential energy (see also Figure 1-8). The electric potential energy of a test charge 𝑞 in the vicinity of the source point-charge 𝑄 will be: 𝑃𝐸 = 𝑈 = 14𝜋𝜀 ∙ 𝑄𝑞𝑟

where 𝑄 is the point-charge creating the external electric field, 𝑞 is the test charge, 𝑟 is the distance between point-charge 𝑄 and test charge 𝑞 and and is the permittivity of the medium.

Figure 1-8. Potential energy of a test charge 𝑞 in close vicinity of a pointcharge 𝑄, both at rest. The point charge 𝑄 = 2𝑒 is represented by an alphaparticle (helium nucleus having two protons and two neutrons) and testcharge 𝑞 = 𝑒 is represented by a proton. The test charge 𝑞 will experience arepulsive force, 𝐹 ⃗, from the point charge 𝑄, with a magnitude which isequal to the repulsive force, 𝐹 ⃗, that point charge 𝑄 is experiencing fromtest charge 𝑞. Therefore the test charge 𝑞 will posses potential energy 𝑈 in the electric field produced by the point charge 𝑄.

In electricity, the electric potential is defined as the electric potential energy per unit charge, representing the work that must be done to move a unit of charge from infinite to its current position in the external electric field produced by a point-charge 𝑄. Thus the electric potential has the following expression: 𝑉 = 𝑊𝑞 = 14𝜋𝜀 ∙ 𝑄𝑟



Figure 1-10. Energy-level diagram for hydrogen showing the Lyman, Balmer, and Paschen series of possible electron transitions. In an energy-level diagram, each horizontal line represents one of the allowed energies.

Atomic spectra are discrete (quantized) because only certain orbits are allowed; the energy carried away from an atom by a photon comes from the electron transitioning from one allowed orbit to another and is thus quantized.

Figure 1-11. Illustration of theplanetary model of the hydrogenatom with quantized orbits and

electron transitions that formLyman series (wavelengths inultraviolet spectrum), Balmer

series (wavelengths in visible andultraviolet spectrum) and

Paschen series (wavelengths ininfrared spectrum). The four

visible hydrogen emission lines inthe Balmer series are sketched on

the right side the way theyappear on a spectrograph

recording (film).



There are other three series associated to the transition energies in hydrogen atom; Brakett series with transitions from higher energy levels to the third excited state (𝑛 → 4, 𝑖 = 5, 6, … ) , Pfund series with transitions from higher energy levels to the fourth excited state (𝑛 → 5, 𝑖 = 6, 7, … ), Humphreys series with transitions from higher energy levels to the fifth excited state (𝑛 → 6, 𝑖 = 7, 8, … ).

2.4 Introduction to Spectroscopy and Spectrophotometry Spectroscopy is defined as the study of the interaction between matter and radiation. Historically, spectroscopy originated through the use of emission and absorption spectra to provide information about the structure and the composition of a substance or an object based on the spectrum of color it emits or absorbs. A spectrograph (i.e. a spectrometer, spectroscope or spectrophotometer) is an instrument used to measure properties of incident light over a specific portion of the electromagnetic radiation spectrum in order to identify materials that emitted or absorbed specific spectral lines by measuring their wavelengths and intensities through what is called spectroscopic analysis (see the schematic diagram of an imaginary spectroscope presented in Figure 1-13). When white light (which contains all visible colors) emitted by an incandescent bulb is passed through a narrow slit and split up by an optical prism, the spectrum of light will show the familiar rainbow of colors from red to violet. All component wavelengths are spread out into a continuous rainbow spectrum, as illustrated in Figure 1-12.

Figure 1-12. Schematic diagram of a spectroscope (spectrograph). White light, emitted by an incandescent bulb, passes through a narrow slit of an opaque barrier to form a thin parallel beam of light. When the beam of light strikes a dispersive prism, its component colors (wavelengths) are bent at different angles, allowing them to reach the screen or photodetector at different locations. Above, the spectrum of visible light emitted by an incandescent bulb is illustrated.

When a cool gas sample of an element (for example hydrogen) is placed in the path of white light produced by a lamp, the gas absorbs the light of certain characteristic wavelengths while light of the rest of the wavelengths get transmitted. Analyzing the white light with a spectrograph or spectroscope a series of dark lines on a continuous colored (rainbow like) background are seen (see Figure 1-13, upper left). This spectrum is called the absorption spectrum of the gas sample. Conversely, when the same gas sample is heated to a high temperature, the atoms of the gas sample get energized, thus the electrons populate excited states in their respective atoms. All energized atoms will eventually return to the ground state by emitting photons of certain definite wavelength. A series of bright lines separated from each other by dark spaces is obtained with a spectrograph (see Figure 1-13, upper right). This spectrum is called the atomic emission spectrum of that gas.

3. WAVE-PARTICLE DUALITY


French physicist Louis de Broglie (1892-1987) and German physicist Albert Einstein (1879-1955) were the first ones that proved the electromagnetic waves exhibit both dual wave-particle properties, which can be described independently in terms of photons (“particles”) and electromagnetic waves (“waves”). Louis de Broglie went further proposing in 1923 that all small subatomic particles like electrons or protons also exhibit a wave-particle dual nature. The wave particle duality is one of those major principles that paved the way to the development of what we call today Quantum Physics. The formation of an interference pattern when a beam of particles passes through a double slit represents the signature of the wave-particle duality in quantum physics (see Figure 1-16). Wave-particle duality has been observed with subatomic particles (electrons, protons, and neutrons), atoms, ions, and small molecules. Most recently, in 1988 during the so-called Vienna experiment, two Austrian scientists, Markus Arndt and Anton Zeilinger, observed the wave-particle duality in a beam of carbon-60 molecules, which represented an order of magnitude larger than any other particles for which quantum interference effects have been observed.

Nowadays researchers have successfully performed the quantum double slit experiment with much larger and more massive molecules than ever before. In 2011 Thomas Juffmann et al. (at Arndt Group, University of Vienna, Austria) have successfully demonstrated wave-particle duality in relatively large phthalocyanine (𝐶 𝐻 𝑁 ) and its derivative molecule (𝐶 𝐻 𝐹𝑒 𝑁 𝑂 ) that have more mass than anything in which quantum interference has previously been observed. Molecular masses of 𝐶 𝐻 𝑁 and 𝐶 𝐻 𝐹𝑒 𝑁 𝑂 are approximately equal to 514 and 1,298 atomic mass units, respectively. Juffmann’s experiment approached the range where the macroscopic and quantum physics overlap. To fire such “massive particles” and have wavelengths that are relatively large compared to their sizes these molecules need to move very slowly in order to build-up an interference pattern.

Figure 1-16. Schematic illustration of the wave-particle duality of matter in a double slit experiment.

When a source of electrons (see the particle cannon with the firing point centered in O pictured in Figure 1-16,) with a definite kinetic energy (speed) shoots electrons toward two extremely narrow slits, then the electrons that passed through both slits exhibit diffraction; therefore, interference fringes with maximums and minimums will appear on the fluorescent screen. If we were to consider the electrons as just classical particles then we were expecting them to hit the fluorescent screen in a two slit pattern, i.e. two vertical bright bands centered on points A and B behind each slit. When scientists sent electrons through both slits to the fluorescent screen one at the time, the fringes of interference still appeared. So they concluded the electrons

3. WAVE-PARTICLE DUALITY


were acting as waves, interfering with themselves only. It looked like each electron must be passing through both slits at the same time! The scientists also confirmed another mystery in physics when they went further by individually observing one by one all electrons passing through both slits; the electrons started acting like ordinary particles, providing a very predictable two slit pattern (two vertical bright bands) on the fluorescent screen centered on A and B behind each slit, with no interference pattern. The scientists had collapsed the wave function of the electrons, thus wiping out the interference pattern, just by simply observing them when passing through both slits! From such experimental evidence, the scientists concluded that a particle in motion with a constant momentum is associated with a monochromatic wave with a definite wavelength.

Louis de Broglie postulated that the wavelength, , associated to a particle in motion is inversely proportionally to its momentum:

= ℎ𝑝 = ℎ𝑚𝑣

where ℎ is Planck constant, 𝑝, 𝑚 and 𝑣 are particle’s momentum, mass and speed. This wavelength is very small for electrons of low energies as well as subatomic particles, but it is almost insignificant for larger particles.

Example Problem #8: Computation of de Broglie wavelength What is the de Broglie wavelength associated to an electron having a kinetic energy of 25 eV? Solution: The relation between kinetic energy and momentum is as follows: 𝐸 = 𝑚 𝑣2 = (𝑚 𝑣)2𝑚 = 𝑝2𝑚 ⇒ 𝑝 = 2𝑚 𝐸

Knowing that 1 𝑒𝑉 = 1.602 ∙ 10 𝐽 , then 𝑝 = √2 ∙ 9.1 × 10 ∙ 1.602 ∙ 25 × 10 = 2.7 × 10 𝑘𝑔 ∙ 𝑚/𝑠 . Therefore, the wavelength associated with the electron is then

= ℎ𝑝 = 6.626 × 10 𝐽 ∙ 𝑠2.7 × 10 𝑘𝑔 ∙ 𝑚/𝑠 = 2.45 × 10 𝑚 = 245 𝑝𝑚

Notice that electrons with an energy of 25 𝑒𝑉 have a wavelength comparable to the size of atoms of natural substances, like Cobalt atoms ( 𝐶𝑜 has an atomic radius of about 125 𝑝𝑚). This theoretical result shows that electrons can be used in diffraction type of experimental setups to determine inter-atomic distances in crystal structures.

Example Problem #9 Calculate the de Broglie wavelength associated to a bullet having a mass of 10 𝑔 and a speed of 350 𝑚/𝑠. Solution: The linear momentum of the bullet is as follows 𝑝 = 𝑚𝑣 = 10 × 10 𝑘𝑔 ∙ 350 𝑚/𝑠 = 3.5 𝑘𝑔 ∙ 𝑚/𝑠 Therefore, the wavelength of the bullet is

= ℎ𝑝 = 6.626 × 10 𝐽 ∙ 𝑠3.5 𝑘𝑔 ∙ 𝑚/𝑠 = 1.89 × 10 𝑚

This example shows us that such wavelength associated with the bullet is insignificant and it cannot be measured accurately by any current experimental means.

Example Problem #10 Calculate the de Broglie wavelength associated to a carbon-60 molecule (𝐶 ) having a speed of 210 𝑚/𝑠. Solution: A carbon-60 molecule contains 60 atoms of 𝐶. So the total mass of a C molecule would be equal to 720 atomic mass units. Thus 𝑚 = 60 ∙ 12 ∙ 1.661 × 10 𝑘𝑔 = 11.96 × 10 𝑘𝑔 The linear momentum of the 𝐶 molecule is as follows 𝑝 = 𝑚𝑣 = 11.96 × 10 𝑘𝑔 × 210 𝑚/𝑠 = 25.12 × 10 𝑘𝑔 ∙ 𝑚/𝑠

4. THE UNCERTAINTY PRINCIPLE


microscope’s objective) perceives the reflected light. As a result of the photon-electron collision, the position as well as the velocity (momentum) of the electron will be disturbed. According to the laws of classical optics, the particle position can be resolved by a microscope up to an uncertainty ∆𝑥 that depends upon the wavelength of the incoming light and the angle of the cone of light that reaches the objective, 𝜃, as follows ∆𝑥 = 𝑠𝑖𝑛𝜃

the incoming photon has a long wavelength (such as visible light with wavelengths between 400 nm and 800 nm) and low momentum, the collision with the electron will not result in a significant change of the momentum of the electron, but the scattering of the photon will be dramatically significant, thus the position of the electron will be very poorly resolved. Then Heisenberg imagined an experiment trying to measure the position and momentum of an electron with a microscope which used gamma rays rather than visible light, as depicted in Figure 1-19.

Figure 1-19. Experimental illustration of the uncertainty principle: Heisenberg’s famous microscope “thought” experiment for locating an electron with gamma rays. The incoming gamma ray (shown in blue) is scattered by the electron up into the microscope’s aperture angle 𝜃. The scattered gamma-ray is shown in red. According to the laws of classical optics, the particle position can be resolved by a microscope up to an uncertainty ∆𝑥 that depends upon the wavelength of the incoming light and the angle of the cone of light that reaches the objective, 𝜃. As illustrated, there is a change of momentum and position of an electron on impact with an incident photon. Heisenberg demonstrated that a photon that collides with a moving electron lets the measurement be taken but also changes the measured system.

Gamma rays have a much smaller wavelength than the size of an electron, but they are much more energetic than visible light. A photon of such short wavelength with large momentum will provide a very good accuracy in measuring the position of the electron (see the above formula), but the photon will transfer a large and uncertain amount of momentum to the electron, thus the velocity – hence the momentum of the electron – will be changed in unpredictable and uncontrollable way. As a conclusion, increasing or decreasing the momentum of incoming photons (of shorter or longer wavelengths) will decrease or increase the uncertainty in position, but will also increase or decrease the uncertainty in the velocity of the electron. So any combination of these trade-offs will provide the same result for the product of the measured two pair of conjugate variables (uncertainties), which is greater or equal to a certain quantity, known as the Plank constant ℎ.

5. SCHRÖDINGER EQUATION and the QUANTUM MODEL of the ATOM


the electron in it is to the nucleus and the more difficult it is to remove the electron from the atom; hence the ionization energy increases as the electron belongs to a lower energy level (closer to the nucleus). As a rule, each energy level (shell) in an atom contains no more than 2𝑛 electrons, where 𝑛 denotes the principal quantum number of a particular energy level. Therefore, the first energy level in an atom can contain no more than 2(1 ) = 2 electrons, the second energy level no more than 2(2 ) = 8 electrons, the third energy level up to 2(3 ) = 18 electrons, and the fourth energy level no more than 2(2 ) = 32 electrons (see Figure 1-21).

This rule works for the first four shells (𝐾, 𝐿, 𝑀, 𝑁), but no known elements have 50 electrons in the 𝑂-shell or 72 electrons in the 𝑃-shell if we were to apply the aforementioned formula for the maximum number of electrons in all shells of large atoms. Later on in this chapter we will see that those numbers of electrons in the 𝑂 shell (50 electrons) or 𝑃 shell (72 electrons) will never be achieved because of the Aufbau principle. Looking at the Periodic Table of the Elements (see Figure 2-4, presented later in next section) we see that up to eight energy levels are filled with electrons in atoms of all natural elements as well as the artificial ones (synthetically produced in research laboratories) we actually know.

Figure 1-21. A schematic representation of the principal energy levels, numbered 1, 2 … outward from the nucleus along with 𝐾, 𝐿, 𝑀 … shell notation. The maximum number of electrons populating each energy level is also illustrated.

5.2.2 Orbital Angular Momentum Quantum Number, 𝒍 The orbital angular momentum quantum number (also known as the azimuthal quantum number, or angular quantum number or orbital quantum number), 𝑙, describes the energy sublevels (sub-shells) in atom and is related to the angular momentum of the electron. Each value of the principal quantum number 𝑛 has multiple values of 𝑙 ranging from 0 to 𝑛 − 1. The magnitude of the orbital angular momentum 𝐿 of an electron in atom is related to the quantum number 𝑙 as follows 𝐿 = 𝑙(𝑙 + 1)ℏ = 𝑙(𝑙 + 1) ∙ ℎ2𝜋 𝑙 = 0, 1, 2, … , 𝑛 − 1 Earlier we mentioned that the orbital angular momentum of an electron is zero, 𝐿 = 0, when 𝑙 = 0, which is completely different than what Bohr model postulated for the orbital angular momentum of the electron in the ground state (𝑛 = 1) to be one reduced Planck constant, ℏ. In the quantum-mechanics interpretation, it means that the electron cloud (or orbital) for the 𝐿 = 0 state is spherically symmetric and has no fundamental axis of revolution (see Figure 1-22). The orbital associated to the electron cloud for 𝑙 = 0 is called “𝑠 orbital”. So the orbital angular momentum quantum number 𝑙 essentially determines the “shape” of the electron cloud (orbital). In chemistry and spectroscopy, there are names for each value of 𝑙 as follows:

𝑙 = 0 corresponds to an 𝑠 orbital. The 𝑠 orbitals are spherical and centered on the nucleus. 𝑙 = 1 corresponds to a 𝑝 orbital. The 𝑝 orbitals are usually polar and form a teardrop petal shape with

the point toward the nucleus (see Figure 1-22). Other authors specify 𝑝 orbitals as having a dumbbell shape.

𝑙 = 2 corresponds to a 𝑑 orbital. The 𝑑 orbitals are similar to the 𝑝 orbital shape, but with more “petals” like a clover leaf. They also have ring shapes around the base of the petals (see Figure 1-29, presented later in this section).

𝑙 = 3 corresponds to an 𝑓 orbital. The 𝑓 orbitals are similar to the 𝑑 orbitals, but they have even more ”petals” and look rather complex.

higher values of 𝑙 have names that follow in alphabetical order.

5. SCHRÖDINGER EQUATION and the QUANTUM MODEL of the ATOM


5.2.4 Shape, Orientation and Size of 𝒔, 𝒑, 𝒅, 𝒇 Orbitals The 1𝑠 orbital (𝑛 = 1, 𝑙 = 0, 𝑚 = 0) is spherically symmetrical about the nucleus and has only one

unidirectional orientation, meaning that the probability of finding the electron(s) is the same in all directions (see Figure 1-29). Successive 𝑠 orbitals (𝑛 = 2, 3, …, 𝑙 = 0, 𝑚 = 0) are also spherical but increase in size as distance from nucleus increases; the 2𝑠 orbital is a larger sphere than the 1𝑠 orbital, the 3𝑠 orbital is a larger sphere than the 2𝑠 orbital and so on.

Figure 1-29. The shapes and orientations of 1𝑠, 2𝑝 and 3𝑑 orbitals in an atom

The 2𝑝 orbitals (𝑛 = 2, 𝑙 = 1, 𝑚 = −1, 0, +1) are dumbbell-shaped (or teardrop petal shape) with the nucleus at the center of the dumbbell in which the two lobes on opposite side are separated by the nodal plane. There are three different orientations for 𝑝 orbitals, since the orbital magnetic quantum number 𝑚 has three possible values (𝑚 = −1, 0, +1), as seen in Figure 1-29. The 2𝑝 orbitals are oriented at right angles to one another along the 𝑥, 𝑦, and 𝑧 axes. Each of the three 2𝑝 orbitals is identical in energy, size and shape, but it is labeled differently and is named due to which plane the polar orbital is resting on. The orbital that rests along the 𝑥 axis is labeled as 2𝑝 , the orbital resting along the 𝑦 axis is labeled as 2𝑝 , and the orbital resting along the 𝑧 axis is labeled as 2𝑝 . Like the 𝑠 orbitals, the 𝑝 orbitals increase in size as the quantum number of the energy level increases; thus, a 3𝑝 orbital is larger than a 2𝑝 orbital. As pictured in Figure 1-29 the 2𝑝 orbitals have directional properties and an electron belonging to a particular 2𝑝 orbital will be found where the lobes are found. The effect of the angular momentum possessed by the electron is to concentrate probability density along one axis.

The 3𝑑 orbitals (𝑛 = 3, 𝑙 = 2, 𝑚 = −2, −1, 0, +1, +2) are more complex in shape (double-dumbbell shape with or without rings). There are five different orientations for 𝑑 orbitals, since the orbital magnetic

6. ELECTRONIC CONFIGURATION OF ATOMS


TABLE 1-6. Sublevels and orbitals of the first four energy levels.

6.2 The Aufbau Principle The Aufbau Principle describes a simple process of hypothetically building up one atom with 𝑍 protons from the atom of the preceding atomic number (𝑍 − 1). As a conventional notation for electronic structure chemists use the sub-shell notation (called also the spectroscopic notation or sometimes called the 𝑠𝑝𝑑𝑓 notation), which is a combination of a shell and sub-shell identification followed by a superscript indicating the number of electrons in that particular sub-shell, as follows: 𝑥 𝑡𝑦𝑝𝑒 where 𝑥 stands for the energy level (shell) corresponding to the principal quantum number 𝑛, 𝑡𝑦𝑝𝑒 denotes the subshell type ( 𝑠, 𝑝, 𝑑, or 𝑓 ) corresponding to the orbital angular momentum quantum number 𝑙 , and 𝑦 represents the number of electrons in that subshell. For example, we would write the electron configurations for the first two elements in the Periodic Table of the Elements, Hydrogen (𝑍 = 1) and Helium (𝑍 = 2) in their ground states as follows:

for 𝐻 is written as 1𝑠 (pronounced “one ess one”) because hydrogen has one electron and is in its lowest energy level (𝑛 = 1) with an orbital angular momentum number of 𝑙 = 0 (an 𝑠 orbital).

for 𝐻𝑒 is written as 1𝑠 because helium has two electrons in its lowest energy level (𝑛 = 1) and has an orbital angular momentum number of 𝑙 = 0 (an 𝑠 orbital with maximum two electrons).

There are other two types of notations used for electron configurations besides the conventional one: the noble-gas-core abbreviated notation and the orbital diagram notation (or orbital box notation). A noble-gas-core notation is an abbreviation in an atom’s electronic configuration where the previous noble gas’s electron configuration is replaced with the noble gas’s element symbol in brackets. For a silicon atom (𝑍 = 14), the electron configuration written in conventional notation is 1𝑠 2𝑠 2𝑝 3𝑠 3𝑝 (see Figure 1-34), while written in a noble-gas-core abbreviated notation is [𝑁𝑒]3𝑠 3𝑝 because the previous noble gas on the periodic table of the elements is neon with an electron configuration of 1𝑠 2𝑠 2𝑝 . If this configuration is replaced by neon’s symbol in brackets, [𝑁𝑒], in silicon’s electron configuration then we have silicon’s noble-gas-core

Ener

gy 𝒏

Shel

ls 𝒍

Sub-

shel

ls 𝒎𝒍 = −𝟑 𝒎𝒍 = −𝟐 𝒎𝒍 = −𝟏 𝒎𝒍 = 𝟎 𝒎𝒍 = 𝟏 𝒎𝒍 = 𝟐 𝒎𝒍 = 𝟑

Num

ber

of

elec

tron

s To

tal

num

ber

of

elec

tron

s

𝑛 = 4 𝑁 𝑙 = 3 4𝑓 14

32 𝑙 = 2 3𝑑 10 𝑙 = 1 2𝑑 6 𝑙 = 0 1𝑠 2 𝑛 = 3 𝑀 𝑙 = 2 3𝑑 10 18 𝑙 = 1 2𝑝 6 𝑙 = 0 1𝑠 2 𝑛 = 2 𝐿 𝑙 = 1 2𝑝 6 8 𝑙 = 0 1𝑠 2 𝑛 = 1 𝐾 𝑙 = 0 1𝑠 2 2



notation as [𝑁𝑒]3𝑠 3𝑝 . An orbital diagram consists of a box representing each orbital and an arrow for each electron belonging to that orbital (see Figure 1-33).

Figure 1-34. Orbital diagram notation for electron configuration of a silicon atom, 𝑆𝑖, in its ground state.

The arrow pointing up, , represents an electron with the spin magnetic quantum number 𝑚 = (spin-up),

while an arrow pointing down, , represents an electron with 𝑚 = − (spin-down).

Figure 1-34. A simplified two-dimensional (2-D) schematic representation of electron configuration in a silicon atom, 𝑆𝑖 . The conventional electronic structure is written as: 1𝑠 2𝑠 2𝑝 3𝑠 3𝑝 . As a noble-gas-core abbreviated notation, the silicon’s electronic structure is written as: [𝑁𝑒]3𝑠 3𝑝 . Sub-shells 1𝑠, 2𝑠, 2𝑝, and 3𝑠 are completely occupied (filled) with electrons while the 3𝑝 sub-shell contains only two electrons out of a maximum possible six.

6.3 The Hund Rule According to Pauli’s Exclusion Principle if two electrons occupy the same orbital, they must have opposite spins. Two such electrons are said to be spin paired and are often represented by arrows pointing in different directions, i.e. by the symbol . An unpaired electron in an orbital is one not accompanied by a partner of opposite spin. As pictured in Figure 1-34, there are two unpaired 3𝑝 electrons that occupy two different orbitals singly. Electrons arranged in this way are said to have parallel spins. By occupying different orbitals in a singly way, the electrons remain as far as possible from one another, thus minimizing electron-electron repulsion. This is the basis of the Hund’s rule stating that in order to attain the lowest available energy in an atom degenerate orbitals are each occupied singly with electrons of parallel spin before double occupation occurs.

TABLE 1-7. Electronic Configuration of the first eighteen elements in their ground state.

𝑍 Symbol Conventional notation Noble-gas-core notation Orbital diagram 1 H 1𝑠

2 He 1𝑠



the electrons in singly occupied orbitals. The state is lower in energy if the total spin is maximum, which coincides with the maximum number of unpaired electrons. Armed with Pauli’s exclusion Principle, Hund’s rule and Aufbau Principle, it is possible to predict probable electron configurations for many of the elements.

Figure 1-35. Illustration of cross over between some of energy levels of sub-shells within different shells in a complex atom (left) and the energy diagram of the atomic orbitals (right).

Table 1-7 contains the electron configuration of the first eighteen elements in the Periodic Table of Elements written in all three notations (conventional, noble-gas-core and orbital diagram) based on the aforementioned rules. Clearly illustrated in Table 1-7 we notice that the sub-shells of each shell are filled firstly with spin-up electrons and then with pairing electrons. For example neon (𝑍 = 10) is a stable element (noble gas) because its ten electrons complete the first two shells, with two and eight electrons, respectively (1𝑠 2𝑠 2𝑝 ). The next element is sodium (𝑍 = 11) with eleven electrons; thus, we must add one electron to the third shell (𝑀), which goes in the 3𝑠 sub-shell and so on. The next noble gas, argon (𝑍 = 18), fills the 3𝑝 sub-shell in the 𝑀-shell with a configuration of 1𝑠 2𝑠 2𝑝 3𝑠 3𝑝 . After argon, the order of sub-shells in electronic configuration changes because a cross over between some of energy levels of sub-shells within different shells occurs due to the electron-electron repulsion, which causes the different sub-shells to be at different energies (see Figure 1-35 at left). For example, the 4𝑠 energy level is a little less energetic that the 3𝑑 energy level, so it fills first with electrons. Looking at the energy diagram illustrated in Figure 1-35 (at right), the energy of atomic orbitals follows the pattern of increasing energy in the order of: 1𝑠 < 2𝑠 < 2𝑝 < 3𝑠 < 3𝑝 < 4𝑠 < 3𝑑 < 4𝑝 < 5𝑠 < 4𝑑 < 5𝑝 < 6𝑠 < 4𝑓 < 5𝑑 < 6𝑝 < 7𝑠 < 5𝑓 < 6𝑑 < 7𝑝 < 8𝑠 … 𝑎𝑛𝑑 𝑠𝑜 𝑜𝑛 6.4 The Madelung Rule According to the order of the energy of atomic orbitals in an atom, the Aufbau principle – which states that the electrons in an atom are so arranged that they occupy orbitals in the order of their increasing energy - should include a simple rule that helps remember the exact order in which the atomic orbitals are filled first with electrons. Since the energy of an orbital in a particular shell depends on both 𝑛 and 𝑙 quantum numbers then

7. SCHIELDING and PENETRATION EFFECTS, SLATER’s RULES


Example Problem #18 Determination of electron configuration of iron in its fundamental state and in one of its ionization states Using the diagonal rule, determine the electron configuration for iron (𝑍 = 26). Then determine the electron configuration of the 𝐹𝑒 ion. Solution: Iron has 26 protons; therefore, it has 26 electrons in its ground state. Using a diagonal diagram chart (as illustrated in Figure 1-39) we see that the first two electrons go into the 1𝑠 sub-shell and fill it and then the next two electrons go into the 2𝑠 sub-shell and fill it. That leaves twenty-two more electrons to place in higher orbitals. The next six go into the 2𝑝 sub-shell, filling it and leaving sixteen more. Two of them go into the 3𝑠 sub-shell, six electrons go into the 3𝑝 sub-shell and two go into the 4𝑠 sub-shell. That leaves six that will go into the 3𝑑 orbital. So the complete electron configuration of iron is 1𝑠 2𝑠 2𝑝 3𝑠 3𝑝 4𝑠 3𝑑 , abbreviated as [𝐴𝑟]4𝑠 3𝑑 , where [𝐴𝑟] denotes the electron configuration of argon (the preceding noble gas). An iron 𝐹𝑒 ion has 3 electrons less than the number of protons; therefore the iron 𝐹𝑒 ion has twenty three electron.

Figure 1-39. Orbital diagram for iron (𝑍 = 26) obtained with the diagonal rule chart.

Now we have to remove electrons successively from shells having the largest principal quantum number. The electron structure by shell placement for a neutral iron atom in its ground state is [𝐴𝑟]3𝑑 4𝑠 , where we placed “3𝑑 ” before “4𝑠 ” in the above notation. Therefore, we have the following electron configurations for iron ions: 𝐹𝑒 : [𝐴𝑟]3𝑑 4𝑠 , where one 4𝑠 electron has been removed 𝐹𝑒 : [𝐴𝑟]3𝑑 , where a second 4𝑠 electron has been removed 𝐹𝑒 : [𝐴𝑟]3𝑑 , where one 3𝑑 electron has been removed.

So the electron configuration for 𝐹𝑒 ion is [𝐴𝑟]3𝑑 , where [𝐴𝑟] denotes the electron configuration of argon (the preceding noble gas).

7. SCHIELDING and PENETRATION EFFECTS, SLATER’s RULES

The shielding or screening effect describes the decrease in magnitude of the electrostatic attraction between an electron and the positively charged nucleus in any atom with more than one shell. The effective nuclear charge is the positive charge that an electron experiences from the nucleus. When an atom has multiple shells (𝑛-shells) each electron in any of the shells feels the electrostatic attraction from the positively charged nucleus, but also electrostatic repulsion from other electrons found between the nucleus and the shell where that electron resides. This causes a net force on electrons - especially on the outer electrons – to be significantly smaller in magnitude. The shielding effect explains why the valence electrons (outermost electrons) are easily removed from the atom because they are as not strongly bonded to the nucleus as the electrons closer to the nucleus. In Bohr’s model of the atom we learned that for hydrogen like atoms (𝐻, 𝐻𝑒 , 𝐿𝑖 , 𝐵𝑒 , all of which have only one electron ) the ionization energy can be calculated using the following formula: 𝐸 = −13.606 ∙ 𝑍𝑛 𝑒𝑉 where 𝑍 stands for atomic number and 𝑛 is the principal quantum number.

SUMMARY


Example Problem #19 Use of Slater’s rules to calculate the effective nuclear charge seen by electrons in various orbitals Using Slater’s rules, calculate the effective nuclear charge for an 2𝑝 electron in nitrogen. Solution: Electronic configuration: (1𝑠 ) (2𝑠 , 2𝑝 ) Screening constant: 𝜎 = (0.35 x 6) + (0.85 x 2) = 3.8 Effective nuclear charge: 𝑍 = 𝑍 − 𝜎 = 9 − 3.8 = 5.2 So the effective nuclear charge seen by a 2𝑝 electron in nitrogen is 5.2 × 10 𝐶.

Example Problem #20 Using Slater’s rules, calculate the effective nuclear charge for an 3𝑝 electron in aluminum. Solution: Electronic configuration: (1𝑠 ) (2𝑠 , 2𝑝 ) (3𝑠 , 3𝑝 ) Screening constant: 𝜎 = (0.35 x 2) + (0.85 x 8) + (1.00 x 2) = 9.5 Effective nuclear charge: 𝑍 = 𝑍 − 𝜎 = 13 − 9.5 = 3.5 So the effective nuclear charge seen by a 3𝑝 electron in aluminum is 3.5 × 10 𝐶.

Example Problem #21 Using Slater’s rules, calculate the effective nuclear charge for an 4𝑠 electron and a 3𝑑 electron in iron. Solution: Electronic configuration: (1𝑠 ) (2𝑠 , 2𝑝 ) (3𝑠 , 3𝑝 ) (3𝑑 ) (4𝑠 )

For a 4𝑠 electronwe have: Screening constant: 𝜎 = (0.35 x 1) + (0.85 x 14) + (1.00 x 10) = 22.25 Effective nuclear charge: 𝑍 = 𝑍 − 𝜎 = 26 − 22.25 = 3.75 So the effective nuclear charge seen by a 3𝑝 electron in aluminum is 3.75 × 10 𝐶.

For a 3𝑑 electronwe have Screening constant: 𝜎 = (0.35 𝑥 5) + (1.00 𝑥 18) = 19.75 Effective nuclear charge: 𝑍 = 𝑍 − 𝜎 = 26 − 19.75 = 6.25 So the effective nuclear charge seen by a 3𝑝 electron in aluminum is 6.25 × 10 𝐶.

SUMMARY The atom is the smallest unit of matter that still maintains the identity and retains all chemical and physical properties of the element. An atom consists of a central, very small and dense nucleus that is surrounded by one or more negatively charged subatomic particles called electrons. Nucleus represents the core of the atom and consists of protons and neutrons, both called also nucleons. Quarks are considered the fundamental constituents of matter. Quarks combine to form particles called hadrons, the most stable of which are protons and neutrons, the constituents of atomic nuclei. Gluons are considered elementary particles that act as the exchange particles for the strong force between quarks, analogous to the exchange of photons in the electromagnetic force between two charged particles. Electrons are subatomic particles, extremely lightweight, that “orbit” the nucleus in a cloud having a radius that is about four-to-five orders of magnitude greater than the diameter of atom’s nucleus. Hydrogen is the most abundant element, making up for almost 75% of the mass of the universe. A hydrogen atom is around one hundred thousand (10 ) times larger than the proton that forms its nucleus. The diameter of hydrogen atom is roughly equal to 1.1 Angstroms (1 𝐴𝑛𝑔𝑠𝑡𝑟𝑜𝑚 = 10 𝑚), which is about twice the Bohr radius of hydrogen. The number of nucleons - protons and neutrons - in an atom is specified by the mass number 𝐴. If 𝑁 is the number of neutrons in an atom then the mass number is indicated as follows: 𝐴 = 𝑍 + 𝑁. The symbol of a particular element in the Periodic Table of the Elements contains the atomic number and mass number written as a superscript and subscript preceding the chemical symbol of the atom as 𝑆𝑦𝑚𝑏𝑜𝑙.

M. & C. Rizescu, Copyright © 2018 Shutter Waves, All Rights Reserved 57

Part 2: The Periodic Table of the Elements

1. INTRODUCTION to the PERIODIC TABLE of the ELEMENTS The Periodic Table of the Elements is a table structured in horizontal rows and vertical columns in such manner that elements are listed in order of increasing atomic number Z by following a pattern based on periodic trends of their electronic configuration and chemical properties (chemical periodicity). The periodic table of the elements is one of the most important references used by chemists in explaining properties of substances.

The standard form of the table includes seven rows called periods in the main body of the table with two additional rows in a separate block structured for inner-transition elements as well as eighteen columns (called groups) where eight of them belong to the main-block elements and ten belong to the transition and inner-transition elements (see Figure 2-1).

1.1. Electron Configuration and Block Classification on the Periodic Table All elements are arranged from left to right and top to bottom in order of increasing atomic number 𝑍, which generally coincides with increasing atomic mass. The period number of an element signifies the highest energy level an electron in that element occupies (in an unexcited state). The number of electrons in a period increases as we move down the periodic table; therefore, as the energy level of an atom increases, the number of energy sublevels (sub-shells) per energy level increases.

Figure 2-1. Standard form of the periodic table.

3. ELEMENTAL DATA on the PERIODIC TABLE

M. & C. Rizescu, Copyright © 2018 Shutter Waves, All Rights Reserved 64

Copper 𝐶𝑢 and silver 𝐴𝑔 have completely filled 𝑑-orbitals (3𝑑 and 4𝑑 , respectively) in their ground states. Explain why they are still considered transition metals. Solution: Copper (𝑍 = 29) and silver (𝑍 = 47) can exhibit a “+2” oxidation state (𝐶𝑢 and 𝐴𝑔 cations) where they will have incompletely filled 𝑑-orbitals, hence transition elements.

Example Problem #26 The four elements that form about 97-99 % of the living organisms List the four elements that form about 97 to 99 percent of the living organisms. Solution: Hydrogen, carbon, oxygen, and nitrogen.

3. ELEMENTAL DATA on the PERIODIC TABLE The Periodic Table of the Elements is one of the most recognizable powerful tools of chemistry. It is mostly displayed in almost every chemistry classroom and laboratory, physics or science laboratories as well. The periodic table (illustrated in Figure 2-4) contains significant amount of information, which – along with the way it was put together – helps the scientists to extract pertinent information concerning individual elements. For instance, a nuclear physicist can use uranium’s atomic mass to determine how many uranium atoms have not gone yet under disintegration out of a specified original amount of substance during a certain time interval. The periods in the table are numbered 1 through 7 sequentially from top to bottom and the groups are numbered 1 through 18 sequentially from left to right. Important data such as atomic weight, melting points, density in solid state at room temperature or near melting point, electron configuration, electronegativity (Pauling negativity number), type of crystalline structure along with lattice parameters in the most stable solid state, as well as atomic radius in metallic structures or in most stable solid state (at room temperature, r.t., or melting point, m.p.), covalent radius, van der Waals radius and ionic radius are data compiled for almost every single element in our periodic table, as illustrated below in Figure 2-3.

Figure 2-3. Standard elemental data on the Periodic Table. Several physical and chemical properties are tabulated in Table 2-1 and illustrated in our Periodic Table of the Elements in Figure 2-4. Next sub-chapters contain description of elemental properties and their periodic trends. Among them, ionization energy, electronegativity, and sizes of atoms, just to name a few. In latter chapters we will present in broader detail topics like chemical bonding, lattice structures and basic fundamental laws of chemical reactions.

3. ELEMENTAL DATA on the PERIODIC TABLE

M. & C. Rizescu, Copyright © 2018 Shutter Waves All rights reserved 65

TABLE 2-1. Physical, chemical and crystallographic data of the elements on the Periodic Table La

ttice

ang

les

𝛼,𝛽,𝛾 𝛼=12

0,𝛽=

=90 𝛼=𝛽

==90

𝛼=𝛽==9

0 𝛼=12

0,𝛽=

=90 𝛼=𝛽

==58.06

𝛼=𝛽==9

0 𝛼=12

0,𝛽=

=90 𝛼==

90 𝛽=13

2.53 𝛼=𝛽

==90

𝛼=𝛽==9

0 𝛼=𝛽

==90

𝛼=120,𝛽=

𝛾=90

𝛼=𝛽==9

0 𝛼=𝛽

==90

𝛼=71.84,

𝛽=90.37,

=71.56

𝛼=𝛽==9

0 𝛼=𝛽

==90

𝛼=𝛽==9

0 𝛼=𝛽

==90

𝛼=𝛽==9

0

Lattice constant 𝑐 [𝑝𝑚] 614 583 350.8 358.2 506 670.4 626.5 508.6 728 436.85

428.86

522.24

404.95

543.03

1126.1

2436.9

817.85 526 533.32

557.05

Lattice constant 𝑏 [𝑝𝑚] 376 357 350.8 228.43 506 246.2 386.1 342.9 328 436.85

428.86 320 404.95

543.03 550.3 1284.5

445.61 526 533.32

557.05

Lattice constant 𝑎 [𝑝𝑚] 376 357 350.8 228.43 506 246.2 386.1 540.3 550 436.85

428.86 320 404.95

543.03 1145 1043.7

622.35 526 533.32

557.05

Type

of l

attic

e or

crys

tal

syst

em a

t st

anda

rd

tem

pera

ture

an

d pr

essu

re

(STP

) or a

t m

eltin

g po

int

(a) HCP HCP BCC HCP Rhomb

ohedral

DHCP HCP Monoclinic

Monoclinic FCC BCC HCP FCC Diamon

d Cubic

Triclinic

Orthorhombic

Orthor

hombic

FCC BCC FCC

Ionic radius ***, [𝑝𝑚] - - 76(1+) 45(2+) 27(3+) 16(4+) 146(3-) 140(2-) 133(1-) - 102(1+) 72(2+) 53.5(3+) 40(4+) 38(5+) 184(2-) 181(1-) - 138(1+)

100(2+)

Van der Waals radius,[𝑝𝑚] 120 140 182 153 192 170 155 152 147 154 227 173 184 210 180 180 175 188 275 231

Covalent radius **, [𝑝𝑚] 31 28 128 96 84 76 71 66 57 58 166 141 121 111 107 105 102 106 203 176

Single-bond covalent radius *, [𝑝𝑚] 32 46 133 102 85 77 71 63 64 67 155 139 126 116 111 103 99 96 196 171

Atomic radius in solid state form, [𝑝𝑚]

- - 151.9 114.2 - 77.2 - - - 154.4 185.7 160 143.2 117.6 - - - 186 230.9 196.9

Pauling electronegativity number 2.2 - 0.98 1.57 2.04 2.55 3.04 3.44 3.98 - 0.93 1.31 1.61 1.9 2.19 2.58 3.16 - 0.82 1.0

First ionization energy, [𝑘𝐽/𝑚𝑜𝑙] 1312 2372.3

520.2 899.5 800.6 1086.5

1402.3

1319.9 1681 2080.7 495.8 737.7 577.5 786.5 1011.8 999.6 1251.2

1520.6 418.8 589.8

Melting point, [𝐾] 14.01 0.95 453.85

1560.15

2573.15 3915 63.29 50.5 53.63 24.703

371.15

923.15 933.4 1683.1

5 317.25

388.51

172.31 83.96 336.5 1112.1

5 Density near r.t. or m.p.,

[𝑔𝑟𝑎𝑚𝑠/𝑐𝑚 ] 0.088 (b) 0.214

0.534 1.848 2.34 3.52 1.026 1.149 1.108 1.444 0.968 1.738 2.6989 2.33 1.823 2.08 1.56 1.623 0.856 1.54

Standard atomic weight [𝑎. 𝑚. 𝑢.] 1.0079

47 4.0026

022 6.9412

9.0121

823 10.811

7 12.010

78 14.006

72 15.999

43 18.998

4032 20.179

76 22.989

7692 24.305

06 26.981

5386 28.085

53 30.973

7622 32.065

5 35.452

7 39.948

1 39.098

31 40.078

4

Atomic number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Symbol H He Li Be B C N O F Ne Na Mg Al Si P S Cl Ar K Ca

Nam

e of

th

e

Elem

ent

Hydrogen

Helium

Lithium

Beryllium

Boron (f)Carbon

Nitroge

n Oxygen

Fluorin

e Neon Sodium

Magne

sium Alumin

um Silicon

Phosph

orus Sulfur Chlorin

e Argon Potassi

um Calcium

4. PERIODIC TRENDS


The ionization energy increases from left to right across a period of the periodic table as a result of increasing nuclear charge. From left to right across a period, more protons are being added to the nucleus, but the number of core electrons in lower completely filled energy shells remains the same, so the outermost electrons feel a higher effective nuclear charge. Consequently, the outermost electrons are held more tightly, the radius of the atom decreases, and it becomes more difficult to remove an electron so the first ionization energy increases. There are some exceptions to this trend where we see a small decrease in ionization energy; i.e. from beryllium (𝑍 = 4) to boron (𝑍 = 5) and from nitrogen (𝑍 = 7) to oxygen (𝑍 = 8) in the second period and from magnesium (𝑍 = 12) to aluminum (𝑍 = 13) and from phosphorus (𝑍 = 15) to sulfur (𝑍 =16) in the third period. To explain this let us analyze the change in electron configuration from neutral atoms to positive ions (cations) of these elements. Table 2-2 contains the electronic configurations of those elements along with their first ionization energies.

TABLE 2-2. First ionization energies, electron configurations and ions for 𝐵𝑒, 𝐵, 𝑁, 𝑂, 𝑀𝑔, 𝐴𝑙, 𝑃, and 𝑆.

𝑩𝒆 𝑩𝒆 𝑩 𝑩 𝑵 𝑵 𝑶 𝑶

Electron configuration [𝐻𝑒]2𝑠 [𝐻𝑒]2𝑠 [𝐻𝑒]2𝑠 2𝑝 [𝐻𝑒]2𝑠 [𝐻𝑒]2𝑠 2𝑝 [𝐻𝑒]2𝑠 2𝑝 [𝐻𝑒]2𝑠 2𝑝 [𝐻𝑒]2𝑠 2𝑝

First ionization energy (kJ/mol) 899.5 800.6 1402.3 1319.9

𝑴𝒈 𝑴𝒈 𝑨𝒍 𝑨𝒍 𝑷 𝑷 𝑺 𝑺

Electron configuration [𝑁𝑒]3𝑠 [𝑁𝑒]3𝑠 [𝑁𝑒]3𝑠 3𝑝 [𝑁𝑒]3𝑠 [𝑁𝑒]3𝑠 3𝑝 [𝑁𝑒]3𝑠 3𝑝 [𝑁𝑒]3𝑠 3𝑝 [𝑁𝑒]3𝑠 3𝑝

First ionization energy (kJ/mol) 737.7 577.5 1011.8 999.6

Outcome Removing an

electron from a full 𝑠 sub-shell

Removing an electron from a 𝑝 sub-shell with

one electron

Removing an electron from a half-full 𝑝 sub-shell (three unpaired electrons)

Removing an electron from a 𝑝 sub-shell with

four electrons

From Table 2-2 we notice that for beryllium and magnesium, the first ionization energy is determined by removing an 𝑠 -electron from a full 𝑠 sub-shell. For boron and aluminum, the first ionization energy is determined by removing the only electron of the 𝑝 sub-shell (see Figure 2-6 for more clarity).

Figure 2-7. A more detailed plot of the ionization energies of the first eighteen elements. The irregularities from the general trend in increasing the ionization energy as a result of the increasing nuclear charge can be related to the slightly lower energies of electrons, thus greater stabilities, in half-filled (spin-unpaired) and completely-filled sub-shells.

4. PERIODIC TRENDS


energy levels. The total effect of this electrostatic attractive-repulsive combination is a net force on the valence electron to be significantly smaller in magnitude; therefore the valence electron will experience a net positive charge (𝑍 ) that is smaller than the nuclear charge (𝑍).

4. Paired or unpaired electron in its orbital; an electron belonging to a full or half-full subshell or it is on its own orbital requires additional energy to be removed. Two paired electrons in an orbital experience electrostatic repulsion from each other, which in turn, offsets the attraction from the nucleus and, consequently, the electron requires less energy to be removed.

Metals typically have low ionization energy while nonmetals typically have high ionization energy.

4.2 Electronegativity Electronegativity is a measure of the ability of an atom to attract a bonding-pair of electrons in a chemical bond. The higher the electronegativity of an atom is, the greater its attraction for bonding electrons is. Electronegativity – a concept introduced in 1932 by the American chemist Linus Pauling (1901-1993) – shows the tendency of a bonding-pair of electrons to be localized around a particular atom. The higher the electronegativity difference between two bonded atoms in a covalent bond, the more we expect that the bonding-pair of electrons will be localized around the more electronegative of the two atoms. Electronegativity of an element provides a general guide to its chemical behavior rather than an exact specification of its behavior in a particular compound; the same atom may exhibit different electronegativities in different chemical bonds.

Figure 2-8. Pauling electronegativity number on the periodic table of the elements. Darker shading indicates higher Pauling electronegativity number and lower shading indicates lower Pauling electronegativity number.

4. PERIODIC TRENDS


Figure 2-11. Illustration of schematic representations of ionic radii of sodium and chlorine ions in sodium chloride (left), metallic radii in metallic sodium (middle, upper side), van der Waals radii of krypton atoms in solid state at very low temperature (middle, lower side), and covalent radii in chlorine molecules, 𝐶𝑙 , part of a solid crystal at very low temperature along with its covalent and van der Waals radii (at right).

Because of these different ways in which atoms bond together, solids fall into different classes and several different types of radii can be defined, i.e. metallic radius, covalent radius, ionic radius, and van der Waals radius, as exemplified in Figure 2-11. For example, in molecules we speak of the covalent radii of the atoms, in metals we speak of the metallic radii of the atoms and, in ionic compounds, the sizes of the atoms (ions) are expressed in terms of ionic radii.

4.3.1 Atomic Radius in Crystalline Solid – Metallic Radius in Crystals When similar atoms combine in a solid element or compound the internuclear distance between the centers of the atoms can be determined by measuring the distances between adjacent rows of atoms in crystal lattice through X-ray diffraction analysis of crystals. Thus, the effective atomic radius can be calculated as half of the measured internuclear distance. Therefore, the (effective) atomic radius in solid state is half of the internuclear distance between two identical neighboring atoms in the crystalline form of the element.

When elements form solids in the crystalline state, they tend to pack together to form three-dimensional periodic and ordered arrays of atoms, ions, or molecules in what we call the crystal lattice. Atoms of majority of the elements - almost all metals at room temperature as well as noble gases (𝐻𝑒, 𝑁𝑒, 𝐴𝑟, 𝐾𝑟, and 𝑋𝑒) at very low temperature - can be thought of as identical perfect spheres that crystallize in cubic or hexagonal arrangements of the simplest and most symmetric of all the lattice types, as illustrated in Figure 2-12. A space lattice is a three-dimensional array of regularly spaced points coinciding with the atoms, ions or molecules positions in a crystalline structure. A space lattice is illustrated as points in space, representing the atoms, or ions or molecules of the crystalline solid, which can be connected with geometrical lines to form repeating shapes. Conversely, the collection of the lattice points that describe the crystalline solid defines a crystal lattice. A unit cell is the smallest structural unit or building block that possesses the highest symmetry present in the lattice that can describe completely the crystal structure. More than 90% of naturally occurring and artificially prepared solid elements crystallize predominantly in one of the three crystal lattices: body-centered cubic (BCC), face-centered cubic (FCC) or cubic-closed packed (ccp), and hexagonal (HEX) or hexagonal closed-packed (hcp) structures.

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4. PERIODIC TR

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4. PERIODIC TR

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RENDS

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4. PERIODIC TR

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RENDS

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SUMMARY


Figure 2-43. Structure of a sulfur molecule in the ball

and stick representation. Figure 2-44. Structure of a phosphorus molecule in the

ball and stick representation.

The nonmetal elements from phosphorus, sulfur, chlorine to argon crystallize in separate molecules (except for argon, which is monatomic) that are held together by weak van der Waals forces. The strength of the van der Waals forces depends on the size of the electron cloud of the molecules. The size of the electron cloud in each molecule depends on the size of the individual atoms as well as the number of atoms making up that molecule. Comparing the melting points of these elements we notice that for sulfur the melting point is a bit higher than that for phosphorus because sulfur molecules are larger than phosphorus, consisting of eight atoms per molecule for sulfur (a 𝑆 molecule is illustrated in Figure 2-43) as opposed to four atoms per molecule for phosphorus (as illustrated in Figure 2-44). Crystalline chlorine consists of diatomic molecules (𝐶𝑙 ) bounded together by even weaker van der Waals forces because the molecule is smaller, thus its melting point is smaller than those for the preceding elements, phosphorus and sulfur. Lastly, argon crystallizes in a monatomic structure where the van der Waals forces are much weaker than those for molecules, thus its melting point is the lowest in the third period. Noticed in Figure 2-42, the melting points for transition elements rise to a maximum around the middle of the series, then descend with increase in atomic number. The variation in melting point indicates that the strongest metallic bonds occur when the 𝑑-subshell is about half-filled; this is also the point at which the largest number of electrons occupy the bonding molecular orbitals in the metal. The presence of unpaired-electrons (up to six for the sixth group) contributes to higher interatomic forces; hence chromium, molybdenum and tungsten have very high melting points. When no un-paired electrons are present, as it is the case for 𝑍𝑛, 𝐶𝑑 and 𝐻𝑔, the melting points are lower than other 𝑑-block elements. However, manganese and technetium have abnormally low melting points, even if they have enough un-paired electrons in their electronic configurations. Nevertheless, the complex crystalline structure of these elements is responsible for this anomaly. As a general explanation for the relatively high melting points for transition metals is that the greater the number of unpaired 𝑑-electrons, the greater the number of bonds and therefore, the greater the strength of these bonds is.

SUMMARY The Periodic Table of the Elements is a table structured in horizontal rows and vertical columns in such manner that elements are listed in order of increasing atomic number by following a pattern based on periodic trends of their electronic configuration and chemical properties (chemical periodicity). The periodic table of the elements is structured in orbital blocks that are named according to the “last” sub-shell in which the valence electrons reside (𝑠- and 𝑝-blocks) or to the orbitals being filled during Aufbau process (for 𝑑- and 𝑓-block elements). The pattern of ordering elements in the periodic table is exactly the order of increasing energy according to the energy diagram for multi-electron atoms. There are 118 confirmed chemical elements included in the periodic table (as of 2015). The electrons that occupy the outermost (highest) energy level within an atom or ion are called valence electrons.

Selected References


39. A.V. Nikolaev, A.V. Tsvyashchenko, Puzzle of the → and other phase transitions in cerium, Lomonosov Moscow State University, online article 40. Chapter 3: Chemistry and Physics of Radon, Studies in Environmental Science, Volume 40, 1990, pp. 25-28, online article, Science Direct (Elsevier) 41. R.G. Haire, R.D. Baybarz, Studies of Einsteinium metal, Oak Ridge National Laboratory, Internal raport 42. Y. Endoh, G. Schirane, J.Jr. Skalyo, Lattice dynamics of solid neon at 6.5 and 23.7 K, Physical Review B 11(4): 1681-1688, February 1975 43. J.A. Leake, W.B. Daniels, J. Skalyo, B.C. Frazer, G. Schirane, Lattice dynamics of neon at two densities form coherent inelastic neutron scattering, Physical Review, Vol. 181, Issue 3, pp. 1251-1260 44. W. Van Witsenburg, Density of solid argon at the triple point and concentration of vacancies, Physics Letter A, Vol. 26, Issue 4, 28 August 1967, pp. 293-294 45. D.N. Batchelder, D.L. Losee, R.O. Simmons, Measurements of lattice constant, thermal expansion, and isothermal compressibility of neon single crystal, Physical Review, Vol. 162, Number 3, 15 October 1967, pp. 767-778 46. D.G. Henshaw, Atomic distribution in liquid and solid neon and solid argon by neutron diffraction, Physical Review, Vol. 111, Number 6, (1958), pp. 1470-1476 47. K.A. Gschneider, V.K. Pecharsky, Jaephil Cho, S.W. Martin, The to transformation in cerium – a twenty year study, Scripta Materials, Vol. 34, No. 11, pp. 1717-1722, 1996 48. Schiferl D., Barrett C.S., The crystal structure of arsenic at 4.2, 78 and 299 K, Journal of Applied Crystallography 2 (1969) 30-36 (4.3084, 4.3084, 11.274, 90, 90, 120, R-3m.

Above – Partial Solar Eclipse seen at its maximum on October 23rd, 2014, 05:54 PM (CST) above Dallas, TX. The giant dark sunspot 2192 located at the center of the solar disk was the largest seen in 24 years. Relative sizes of the planets and Sun are shown on the left. Sun has a diameter of 1,392,000 km, while the equatorial diameter of the Earth is 12,756 km. Compared to the Earth the Sun is huge!

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