2/7/2013. Chapter 3. Quantum Theory and the Electronic Structure of Atoms. Chemistry: Atoms First Julia Burdge & Jason Overby

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1 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. 3 Quantum Theory and the Electronic Structure of Atoms 3.7 Principal Quantum Number (n) Angular Momentum Quantum Number (l) Magnetic Quantum Number (m l ) Electron Spin Quantum Number (m s ) 3.8 Atomic Orbitals s Orbitals p Orbitals d Orbitals and other High- 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 3.10 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. 1

2 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. 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 (m l ) specifies orientation 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. 2

3 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. l Orbital designation s p d f A collection of orbitals with the same value of n and l is referred to as a subshell. 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 designate shells, subshells, and orbitals. 3

4 Worked Example 3.8 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? Strategy Recall that the possible values of m l depend on the value of l, not on the value of n. Setup The possible values of m l are l, 0, +l. Solution The possible values of m l 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 m l. 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. A beam of atoms is split by a magnetic field. Statistically, half of the electrons spin clockwise, the other half spin counterclockwise. 4

5 To summarize quantum numbers: principal (n) size angular (l) shape Required to describe an atomic orbital magnetic (m l ) orientation principal (n = 2) 2p x related to the magnetic quantum number (m l ) angular momentum (l = 1) electron spin (m s ) direction of spin Required to describe an electron in an atomic orbital 3.8 Atomic Orbitals All s orbitals are spherical in shape but differ in size: < < 3s principal quantum number (n = 2) angular momentum quantum number (l = 0) m l = 0; only 1 orientation possible Atomic Orbitals The p orbitals: Three orientations: l = 1 (as required for a p orbital) m l = 1, 0, +1 5

6 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. 3s subshell 3 rd shell (n = 3p 3; 3) subshell l = 0) (n 3d = subshell 3; l = 1) (n = 3; l = 2) 2 nd subshell shell (n = 2) 2p subshell (n = 2; l = 1) (n = 2; l = 0) Worked Example 3.9 List the Think values of About n, l, and It Consult m l for each the of following the orbitals figure in a to 4dverify subshell. your answers. 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. 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. Solution 4d principal quantum number, n = 4 angular momentum quantum number, l = 2 Possible m l are -2, -1, 0, +1, +2. 6

7 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 orbital. principal (n = 1) Ground state electron configuration of hydrogen 1 angular momentum (l = 0) number of electrons in the orbital or subshell The use of an up arrow indicates an electron with m s = + ½ 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 1 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) 7

8 According to the Pauli exclusion principle, no two electrons in an atom can have the same four quantum numbers. configuration of helium 2 Quantum number describes the orbital describes the electrons in the orbital Principal (n) Angular moment (l) Magnetic (m l ) Electron spin (m s ) ½ ½ 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 configuration of Li The third electron must go in the next available orbital with the lowest possible energy. 2 1 The orbital can only accommodate 2 electrons (Pauli exclusion principle) 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 configuration of Be 2 2 8

9 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 configuration of B 2 2 2p 1 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 configuration of C 2 2 2p 2 The 2p orbitals are of equal energy, or degenerate. Put 1 electron in each before pairing (Hund s rule). 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 configuration of N 2 2 2p 3 The 2p orbitals are of equal energy, or degenerate. Put 1 electron in each before pairing (Hund s rule). 9

10 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 configuration of O 2 2 2p 4 Once all the 2p orbitals are singly occupied, additional electrons will have to pair with those already in the orbitals. 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 configuration of F 2 2 2p 5 When there are one or more unpaired electrons, as in the case of oxygen and fluorine, the atom is called paramagnetic. 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 configuration of Ne 2 2 2p 6 When all of the electrons in an atom are paired, as in neon, it is called diamagnetic. 10

11 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). 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 2 2 2p 6 3s 2 3p 6 4s p 6 3s 2 3p 6 4s 2 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 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 2 2 2p 6 3s 2 3p 6 4s 1. Because 2 2 2p 6 3s 2 3p 6 is the electron configuration of argon, we can simplify potassium s to [Ar]4s 1. configuration of K: 2 2 2p 6 3s 2 3p 6 4s 1 [Ar] [Ar]4s 1 11

12 and the Periodic Table Elements in Group 3B through Group 1B are the transition metals. and the Periodic Table Following lanthanum (La), there is a gap where the lanthanide (rare earth) series belongs. and the Periodic Table After actinum (Ac) comes the actinide series. 12

13 and the Periodic Table 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 ). Cr [Ar] 4s 3d 3d 3d 3d 3d Greater stability with half-filled 3d subshell 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 ). Cu [Ar] 4s 3d 3d 3d 3d 3d Greater stability with filled 3d subshell 13

14 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 = 15 electrons in As beyond its noble gas core Solution As [Ar]4s 2 3d 10 4p 3 Think About It Arsenic is a p-block element; therefore, we should expect its outermost electrons to reside in a p subshell. 14