Atoms first chapter 3.7 11

Chemistry: Atoms FirstJulia Burdge & Jason Overby

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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 Quantum Theory and the Electronic Structure of AtomsAtoms

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 MechanicsQuantum 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 anelectron in a certain area ofspace is proportional to ψ2 andis called electron density.

Quantum MechanicsQuantum 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 NumbersQuantum 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 (ml) – specifies orientation

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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 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 0 1 2 3

Orbital designation s p d f


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 designate shells, subshells, and orbitals.

Worked Example 3.8

Strategy Recall that the possible values of ml depend on the value of l, not on the value of n.

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?

Solution The possible values of ml are -1, 0, and +1.

Setup The possible values of ml 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 ml.


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.


A beam of atoms is split by a magnetic field.

Statistically, half of the electrons spin clockwise, the other half spin counterclockwise.


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

Required to describe an electron in an atomic orbital

2px

principal (n = 2)

angular momentum (l = 1)

related to the magnetic quantum number (ml )

Atomic OrbitalsAtomic Orbitals

All s orbitals are spherical in shape but differ in size:

1s < 2s < 3s

2sangular momentum quantum number (l = 0)

ml = 0; only 1 orientation possible

principal quantum number (n = 2)

3.8


The p orbitals:

Three orientations:l = 1 (as required for a p orbital)

ml = –1, 0, +1


The d orbitals:

Five orientations:l = 2 (as required for a d orbital)

ml = –2, –1, 0, +1, +2

Energies of OrbitalsEnergies of Orbitals

The energies of orbitals in the hydrogen atom depend only on the principal quantum number.

2nd shell (n = 2)

3d subshell (n = 3; l = 2)

2p subshell (n = 2; l = 1)

3rd 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 ml, which will have to be deduced from the value of l.

List the values of n, l, and ml for each of the orbitals in a 4d subshell.

Solution 4d

Possible ml 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 (ml) 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 ConfigurationsElectron 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.

1s1principal (n = 1)

angular momentum (l = 0)

number of electrons in the orbital or subshell

1s

2s 2p 2p 2p

The use of an up arrow indicates an electron with ms = + ½

Ground state electron configuration of

hydrogen

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If hydrogen’s electron is found in a higher energy orbital, the atom is in an excited state.

2s1

1s

2s 2p 2p 2p

A possible excited state electron configuration of hydrogen


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)


According to the Pauli exclusion principle, no two electrons in an atom can have the same four quantum numbers.

1s2

1s

2s

2p 2p 2p

The ground state electron configuration of helium

Quantum number

Principal (n)

Angular moment (l)

Magnetic (ml)

Electron spin (ms)

1

00

+ ½

1

00

‒ ½

describes the 1s orbital

describes the electrons in the 1s orbital


The Aufbau principle states that electrons are added to the lowest energy orbitals first before moving to higher energy orbitals.

1s22s1

1s

2s

2p 2p 2p

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



1s

2s

2p 2p 2p 1s22s2

The ground state electron configuration of Be

Be has a total of 4 electrons



1s

2s

2p 2p 2p

The ground state electron configuration of B

1s22s22p1

B has a total of 5 electrons


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.

1s22s22p2

1s

2s

2p 2p 2p

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



1s22s22p3

1s

2s

2p 2p 2p

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



1s22s22p4

1s

2s

2p 2p 2p

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.



1s22s22p5

1s

2s

2p 2p 2p

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.



1s22s22p6

1s

2s

2p 2p 2p

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.


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

Ca 1s22s22p63s23p64s2

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.

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 TableElectron 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 1s22s22p63s23p64s1.

Because 1s22s22p63s23p6 is the electron configuration of argon, we can simplify potassium’s to [Ar]4s1.

1s22s22p63s23p64s1

The ground state electron configuration of K:

[Ar] [Ar]4s1

3.10

1s22s22p63s23p64s1


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.


After actinum (Ac) comes the actinide series.


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 3d 3d 3d 3d[Ar]Cr

Greater stability with half-filled 3d subshell


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 3d 3d 3d 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.

Solution

As [Ar]4s23d104p3

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

2

2

6

6

3

10

Think About It Arsenic is a p-block element; therefore, we should expect its outermost electrons to reside in a p subshell.

Atoms first chapter 3.7 11

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