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

Quantum Theory and the Electronic Structure of Atoms 3 3.7 Quantum Numbers Principal Quantum Number (n) Angular Momentum Quantum Number (l) Magnetic Quantum Number (ml) Electron Spin Quantum Number (ms) 3.8 Atomic Orbitals s Orbitals p Orbitals d Orbitals and other High-Energy Orbitals Energies of Orbitals 3.9 Electron 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 ψ2 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 3.7 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 (ml) – specifies orientation

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.

The allowed values of l range from 0 to n – 1. 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 1 2 3 Orbital designation s p d f

Quantum Numbers The magnetic quantum number (ml) describes the orientation of the orbital in space. The values of ml 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 What are the possible values for the magnetic quantum number (ml) when the principal quantum number (n) is 3 and the angular quantum number (l) is 1? Strategy Recall that the possible values of ml depend on the value of l, not on the value of n. Setup The possible values of ml are – l,…0,…+l. Solution The possible values of ml are -1, 0, and +1. 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 ml.

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

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

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

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

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

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

Worked Example 3.9 Think About It Consult the following figure to verify your answers. List the values of n, l, and ml for each of the orbitals in a 4d subshell. 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 ml, which will have to be deduced from the value of l. 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 (ml) can have integral values of – l,…0,…+l. Solution 4d Possible ml are -2, -1, 0, +1, +2. principal quantum number, n = 4 angular momentum quantum number, l = 2

Electron Configurations 3.9 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. Ground state electron configuration of hydrogen principal (n = 1) 1s1 number of electrons in the orbital or subshell Energy 2s 2p 2p 2p angular momentum (l = 0) The use of an up arrow indicates an electron with ms = + ½ 1s

Electron Configurations If hydrogen’s electron is found in a higher energy orbital, the atom is in an excited state. A possible excited state electron configuration of hydrogen 2s1 Energy 2s 2p 2p 2p 1s

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. The ground state electron configuration of helium Energy 2p 2p 2p 1s2 2s Quantum number Principal (n) Angular moment (l) Magnetic (ml) Electron spin (ms) 1 1 1s describes the 1s orbital describes the electrons in the 1s orbital + ½ ‒ ½

Electron Configurations The Aufbau principle states that electrons are added to the lowest energy orbitals first before moving to higher energy orbitals. Li has a total of 3 electrons The ground state electron configuration of Li 1s22s1 Energy 2p 2p 2p 2s The third electron must go in the next available orbital with the lowest possible energy. 1s The 1s orbital can only accommodate 2 electrons (Pauli exclusion principle)

Electron Configurations The Aufbau principle states that electrons are added to the lowest energy orbitals first before moving to higher energy orbitals. Be has a total of 4 electrons The ground state electron configuration of Be 1s22s2 Energy 2p 2p 2p 2s 1s

Electron Configurations The Aufbau principle states that electrons are added to the lowest energy orbitals first before moving to higher energy orbitals. B has a total of 5 electrons The ground state electron configuration of B 1s22s22p1 Energy 2p 2p 2p 2s 1s

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. C has a total of 6 electrons The ground state electron configuration of C 1s22s22p2 Energy 2p 2p 2p 2s The 2p orbitals are of equal energy, or degenerate. Put 1 electron in each before pairing (Hund’s rule). 1s

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. N has a total of 7 electrons The ground state electron configuration of N 1s22s22p3 Energy 2p 2p 2p 2s The 2p orbitals are of equal energy, or degenerate. Put 1 electron in each before pairing (Hund’s rule). 1s

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. O has a total of 8 electrons The ground state electron configuration of O 1s22s22p4 Energy 2p 2p 2p 2s Once all the 2p orbitals are singly occupied, additional electrons will have to pair with those already in the orbitals. 1s

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. F has a total of 9 electrons The ground state electron configuration of F 1s22s22p5 Energy 2p 2p 2p 2s When there are one or more unpaired electrons, as in the case of oxygen and fluorine, the atom is called paramagnetic. 1s

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. Ne has a total of 10 electrons The ground state electron configuration of Ne 1s22s22p6 Energy 2p 2p 2p 2s When all of the electrons in an atom are paired, as in neon, it is called diamagnetic. 1s

Electron Configurations General rules for writing electron configurations: Electrons will reside in the available orbitals of the lowest possible energy. Each orbital can accommodate a maximum of two electrons. Electrons will not pair in degenerate orbitals if an empty orbital is available. 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). 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. Solution Ca 1s22s22p63s23p64s2 1s2 2s2 2p6 3s2 3p6 4s2 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 3.10 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 1s22s22p63s23p64s1. Because 1s22s22p63s23p6 is the electron configuration of argon, we can simplify potassium’s to [Ar]4s1. The ground state electron configuration of K: 1s22s22p63s23p64s1 1s22s22p63s23p64s1 [Ar] [Ar]4s1

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

Electron Configurations and the Periodic Table 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

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]4s13d5 and not [Ar]4s23d4 as expected. Copper (Z = 29) is [Ar]4s13d10 and not [Ar]4s23d9 as expected. The reason for these anomalies is the slightly greater stability of d subshells that are either half-filled (d5) or completely filled (d10). 4s 3d [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]4s13d5 and not [Ar]4s23d4 as expected. Copper (Z = 29) is [Ar]4s13d10 and not [Ar]4s23d9 as expected. The reason for these anomalies is the slightly greater stability of d subshells that are either half-filled (d5) or completely filled (d10). 4s 3d [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. 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. 2 2 6 2 6 10 2 3 Solution As [Ar]4s23d104p3 Think About It Arsenic is a p-block element; therefore, we should expect its outermost electrons to reside in a p subshell.