Chemistry: Atoms First Julia Burdge & Jason Overby Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Chapter 3 Quantum Theory and the Electronic Structure of Atoms Kent L. McCorkle Cosumnes River College Sacramento, CA

Quantum Theory and the Electronic Structure of Atoms 3 3.7Quantum Numbers Principal Quantum Number (n) Angular Momentum Quantum Number (l) Magnetic Quantum Number (m l ) Electron Spin Quantum Number (m s ) 3.8Atomic Orbitals s Orbitals p Orbitals d Orbitals and other High-Energy Orbitals Energies of Orbitals 3.9Electron Configuration Energies of Atomic Orbitals in Many-Electron Systems The Pauli Exclusion Principle Aufbau Principle Hund’s Rule General Rules for Writing Electron Configurations 3.10 Electron Configurations and the Periodic Table

Quantum Mechanics Erwin Schrödinger derived a complex mathematical formula to incorporate the wave and particle characteristics of electrons. Wave behavior is described with the wave function ψ. The probability of finding an electron in a certain area of space is proportional to ψ 2 and is called electron density.

Quantum Mechanics The Schrödinger equation specifies possible energy states an electron can occupy in a hydrogen atom. The energy states and wave functions are characterized by a set of quantum numbers. Instead of referring to orbits as in the Bohr model, quantum numbers and wave functions describe atomic orbitals.

Quantum Numbers Quantum numbers are required to describe the distribution of electron density in an atom. There are three quantum numbers necessary to describe an atomic orbital.  The principal quantum number (n) – designates size  The angular moment quantum number ( l ) – describes shape  The magnetic quantum number (m l ) – specifies orientation 3.7

Quantum Numbers The principal quantum number (n) designates the size of the orbital. Larger values of n correspond to larger orbitals. The allowed values of n are integral numbers: 1, 2, 3 and so forth. The value of n corresponds to the value of n in Bohr’s model of the hydrogen atom. A collection of orbitals with the same value of n is frequently called a shell.

Quantum Numbers The angular moment quantum number ( l ) describes the shape of the orbital. The values of l are integers that depend on the value of the principal quantum number The allowed values of l range from 0 to n – 1.  Example: If n = 2, l can be 0 or 1. A collection of orbitals with the same value of n and l is referred to as a subshell. l 0123 Orbital designationspdf

Quantum Numbers The magnetic quantum number (m l ) describes the orientation of the orbital in space. The values of m l are integers that depend on the value of the angular moment quantum number: – l,…0,…+ l

Quantum Numbers Quantum numbers designate shells, subshells, and orbitals.

Worked Example 3.8 Strategy Recall that the possible values of m l depend on the value of l, not on the value of n. What are the possible values for the magnetic quantum number (m l ) when the principal quantum number (n) is 3 and the angular quantum number ( l ) is 1? Solution The possible values of m l are -1, 0, and +1. Setup The possible values of m l are – l,…0,…+ l. Think About It Consult Table 3.2 to make sure your answer is correct. Table 3.2 confirms that it is the value of l, not the value of n, that determines the possible values of m l.

Quantum Numbers The electron spin quantum number (m s ) is used to specify an electron’s spin. There are two possible directions of spin. Allowed values of m s are +½ and −½.

Quantum Numbers A beam of atoms is split by a magnetic field. Statistically, half of the electrons spin clockwise, the other half spin counterclockwise.

Quantum Numbers To summarize quantum numbers: principal (n) – size angular ( l ) – shape magnetic (m l ) – orientation electron spin (m s ) direction of spin Required to describe an atomic orbital Required to describe an electron in an atomic orbital 2px2px principal (n = 2) angular momentum ( l = 1) related to the magnetic quantum number (m l )

Atomic Orbitals All s orbitals are spherical in shape but differ in size: 1s < 2s < 3s 2s2s angular momentum quantum number ( l = 0) m l = 0; only 1 orientation possible principal quantum number (n = 2) 3.8

Atomic Orbitals The p orbitals: Three orientations: l = 1 (as required for a p orbital) m l = –1, 0, +1

Atomic Orbitals The d orbitals: Five orientations: l = 2 (as required for a d orbital) m l = –2, –1, 0, +1, +2

Energies of Orbitals The energies of orbitals in the hydrogen atom depend only on the principal quantum number. 2 nd shell (n = 2) 3d subshell (n = 3; l = 2) 2p subshell (n = 2; l = 1) 3 rd shell (n = 3) 2s subshell (n = 2; l = 0) 3p subshell (n = 3; l = 1) 3s subshell (n = 3; l = 0)

Worked Example 3.9 Strategy Consider the significance of the number and the letter in the 4d designation and determine the values of n and l. There are multiple values for m l, which will have to be deduced from the value of l. List the values of n, l, and m l for each of the orbitals in a 4d subshell. Solution 4d Possible m l are -2, -1, 0, +1, +2. Setup The integer at the beginning of the orbital designation is the principal quantum number (n). The letter in an orbital designation gives the value of the angular momentum quantum number ( l ). The magnetic quantum number (m l ) can have integral values of – l,…0,…+ l. principal quantum number, n = 4 angular momentum quantum number, l = 2 Think About It Consult the following figure to verify your answers.

Electron Configurations The electron configuration describes how the electrons are distributed in the various atomic orbitals. In a ground state hydrogen atom, the electron is found in the 1s orbital. 1s11s1 principal (n = 1) angular momentum ( l = 0) number of electrons in the orbital or subshell 1s1s 2s2s2p2p2p2p2p2p Energy The use of an up arrow indicates an electron with m s = + ½ Ground state electron configuration of hydrogen 3.9

Electron Configurations If hydrogen’s electron is found in a higher energy orbital, the atom is in an excited state. 2s12s1 1s1s 2s2s2p2p2p2p2p2p Energy A possible excited state electron configuration of hydrogen

Electron Configurations In a multi-electron atoms, the energies of the atomic orbitals are split. Splitting of energy levels refers to the splitting of a shell (n=3) into subshells of different energies (3s, 3p, 3d)

Electron Configurations According to the Pauli exclusion principle, no two electrons in an atom can have the same four quantum numbers. 1s21s2 1s1s 2s2s 2p2p2p2p2p2p Energy The ground state electron configuration of helium Quantum number Principal (n) Angular moment ( l ) Magnetic (m l ) Electron spin (m s ) ½ ‒ ½ describes the 1s orbital describes the electrons in the 1s orbital

Electron Configurations The Aufbau principle states that electrons are added to the lowest energy orbitals first before moving to higher energy orbitals. 1s22s11s22s1 1s1s 2s2s 2p2p2p2p2p2p Energy The ground state electron configuration of Li The 1s orbital can only accommodate 2 electrons (Pauli exclusion principle) The third electron must go in the next available orbital with the lowest possible energy. Li has a total of 3 electrons

Electron Configurations The Aufbau principle states that electrons are added to the lowest energy orbitals first before moving to higher energy orbitals. 1s1s 2s2s 2p2p2p2p2p2p Energy 1s22s21s22s2 The ground state electron configuration of Be Be has a total of 4 electrons

Electron Configurations The Aufbau principle states that electrons are added to the lowest energy orbitals first before moving to higher energy orbitals. 1s1s 2s2s 2p2p2p2p2p2p Energy The ground state electron configuration of B 1s22s22p11s22s22p1 B has a total of 5 electrons

Electron Configurations According to Hund’s rule, the most stable arrangement of electrons is the one in which the number of electrons with the same spin is maximized. 1s22s22p21s22s22p2 1s1s 2s2s 2p2p2p2p2p2p Energy The ground state electron configuration of C The 2p orbitals are of equal energy, or degenerate. Put 1 electron in each before pairing (Hund’s rule). C has a total of 6 electrons

Electron Configurations According to Hund’s rule, the most stable arrangement of electrons is the one in which the number of electrons with the same spin is maximized. 1s22s22p31s22s22p3 1s1s 2s2s 2p2p2p2p2p2p Energy The ground state electron configuration of N The 2p orbitals are of equal energy, or degenerate. Put 1 electron in each before pairing (Hund’s rule). N has a total of 7 electrons

Electron Configurations According to Hund’s rule, the most stable arrangement of electrons is the one in which the number of electrons with the same spin is maximized. 1s22s22p41s22s22p4 1s1s 2s2s 2p2p2p2p2p2p Energy The ground state electron configuration of O O has a total of 8 electrons Once all the 2p orbitals are singly occupied, additional electrons will have to pair with those already in the orbitals.

Electron Configurations According to Hund’s rule, the most stable arrangement of electrons is the one in which the number of electrons with the same spin is maximized. 1s22s22p51s22s22p5 1s1s 2s2s 2p2p2p2p2p2p Energy The ground state electron configuration of F F has a total of 9 electrons When there are one or more unpaired electrons, as in the case of oxygen and fluorine, the atom is called paramagnetic.

Electron Configurations According to Hund’s rule, the most stable arrangement of electrons is the one in which the number of electrons with the same spin is maximized. 1s22s22p61s22s22p6 1s1s 2s2s 2p2p2p2p2p2p Energy The ground state electron configuration of Ne Ne has a total of 10 electrons When all of the electrons in an atom are paired, as in neon, it is called diamagnetic.

Electron Configurations General rules for writing electron configurations: 1) Electrons will reside in the available orbitals of the lowest possible energy. 2) Each orbital can accommodate a maximum of two electrons. 3) Electrons will not pair in degenerate orbitals if an empty orbital is available. 4) Orbitals will fill in the order indicated in the figure.

Worked Example 3.10 Write the electron configuration and give the orbital diagram of a calcium (Ca) atom (Z = 20). Solution Ca1s 2 2s 2 2p 6 3s 2 3p 6 4s 2 Setup Because Z = 20, Ca has 20 electrons. They will fill in according to the diagram at right. Each s subshell can contain a maximum of two electrons, whereas each p subshell can contain a maximum of six electrons. 1s21s2 2s22s2 2p62p6 3s23s2 3p63p6 4s24s2 Think About It Look at the figure again to make sure you have filled the orbitals in the right order and that the sum of electrons is 20. Remember that the 4s orbital fills before the 3d orbitals.

Electron Configurations and the Periodic Table The electron configurations of all elements except hydrogen and helium can be represented using a noble gas core. The electron configuration of potassium (Z = 19) is 1s 2 2s 2 2p 6 3s 2 3p 6 4s 1. Because 1s 2 2s 2 2p 6 3s 2 3p 6 is the electron configuration of argon, we can simplify potassium’s to [Ar]4s 1. 1s22s22p63s23p64s11s22s22p63s23p64s1 The ground state electron configuration of K: [Ar][Ar]4s s22s22p63s23p64s11s22s22p63s23p64s1

Electron Configurations and the Periodic Table Elements in Group 3B through Group 1B are the transition metals.

Following lanthanum (La), there is a gap where the lanthanide (rare earth) series belongs. Electron Configurations and the Periodic Table

After actinum (Ac) comes the actinide series. Electron Configurations and the Periodic Table

There are several notable exceptions to the order of electron filling for some of the transition metals.  Chromium (Z = 24) is [Ar]4s 1 3d 5 and not [Ar]4s 2 3d 4 as expected.  Copper (Z = 29) is [Ar]4s 1 3d 10 and not [Ar]4s 2 3d 9 as expected. The reason for these anomalies is the slightly greater stability of d subshells that are either half-filled (d 5 ) or completely filled (d 10 ). 4s4s3d3d3d3d3d3d3d3d3d3d [Ar]Cr Greater stability with half-filled 3d subshell Electron Configurations and the Periodic Table

There are several notable exceptions to the order of electron filling for some of the transition metals.  Chromium (Z = 24) is [Ar]4s 1 3d 5 and not [Ar]4s 2 3d 4 as expected.  Copper (Z = 29) is [Ar]4s 1 3d 10 and not [Ar]4s 2 3d 9 as expected. The reason for these anomalies is the slightly greater stability of d subshells that are either half-filled (d 5 ) or completely filled (d 10 ). Electron Configurations and the Periodic Table 4s4s3d3d3d3d3d3d3d3d3d3d [Ar]Cu Greater stability with filled 3d subshell

Worked Example 3.11 Write the electron configuration for an arsenic atom (Z = 33) in the ground state. Solution As[Ar]4s 2 3d 10 4p 3 Setup The noble gas core for As is [Ar], where Z = 18 for Ar. The order of filling beyond the noble gas core is 4s, 3d, and 4p. Fifteen electrons go into these subshells because there are 33 – 18 = 15 electrons in As beyond its noble gas core Think About It Arsenic is a p-block element; therefore, we should expect its outermost electrons to reside in a p subshell.