the arrangement of electrons in atomschemistry study guide 4: arrangement of electrons in atoms p. 7...

The Arrangement of

Electrons in Atoms

CHEMISTRY STUDY GUIDE 4: ARRANGEMENT OF ELECTRONS IN ATOMS P. 2

OVERVIEW

I. THE BOHR MODEL OF THE ATOM ................................................ 3

II. THE QUANTUM MODEL OF THE ATOM ....................................... 7

A. THE NATURE OF ELECTRONS ........................................................................................... 7 B. QUANTUM NUMBERS .......................................................................................................... 8

III. ELECTRON CONFIGURATION & ORBITAL NOTATION ... 11

A. RULES GOVERNING ELECTRON CONFIGURATION .................................................... 11

B. ELECTRONIC NOTATION .................................................................................................. 13

C. ELECTRON CONFIGURATION .......................................................................................... 16


I. THE BOHR MODEL OF THE ATOM

Background.

Person Contribution

Democritus

(~400 B.C.E.)

Atomos: like grains of sand make up the desert, indivisible particles (atoms) make up

matter

Dalton (1808) Modern atomic theory based on five postulates. The atom remains an indivisible particle.

Thomson

(1897)

Electron. “Plum pudding” model of atom in which small, negative charges are dispersed

throughout a positive sphere.

Rutherford

(1911)

Nucleus. Small, positive nucleus orbited by small electrons. Most of atom is empty

space (electron cloud).

A major problem confronted Rutherford’s model of the atom: how were the electrons arranged in the

nucleus. If the electrons rotated around the nucleus, what kept them from collapsing into the nucleus? As

the physical laws were understood, then it would be centrifugal force: the same force that keeps the

planets in orbit around the sun. However, negatively-charged electrons circling around a positive nucleus

would eventually lose energy and should then spiral into the nucleus, collapsing the atom, and destroying

matter as we know it. However, this doesn’t happen. This perplexed scientists at the turn of the century.

A new atomic model began to emerge in the early twentieth century, due in a large part, to the study of

light and the relationship between light and the atom’s electrons. This required the expansion of classical

physics to quantum mechanics. To understand this development, we need some background information

about light.

Visible light is but one type of electromagnetic radiation. All forms of EM radiation comprise the

electromagnetic spectrum1 (Figure 1) and travel in transverse waves

2 (Figure 2).

Figure 1. Electromagnetic (EM) spectrum with selected radiation. Note: Visible light makes up only a

small part of the spectrum. (UV = ultraviolet; IR = infrared)

1 U.S. Frequency allocation chart: http://www.ntia.doc.gov/osmhome/allochrt.pdf

2 There are two types of waves: Transverse, like waves of water, where the direction of the particle movement is

perpendicular to the direction of the waves, and longitudinal, like sound or automobile traffic, where the direction

of the particles is parallel to the direction of the waves.


o Wavelength (; the Greek letter ‘l’ or

lambda) = distance between two

corresponding points on consecutive

waves.

o Frequency (; the Greek letter ‘n’ or nu) =

number of waves that pass a given point in

a specific unit of time, usually second

(units: cycles per second = cps = 1/second

= 1/s = Hertz = Hz)

o The speed of a wave (u) = *

for EM radiation, u = c = 3.00x108 m/s

Figure 2. Parts of a Wave.

The product of the frequency and wavelength of EM radiation is the speed of light:

c = (EQ. 1)

where c = speed of light (3.00 x 108 m/s)

= wavelength

= frequency

Note that frequency and wavelength are inversely proportional: as the frequency increases, the

wavelength decreases. For calculations, commonly used units for 1/second for frequency and nanometers

(nm = 10–9

m) for wavelength.

Problem

1. The frequencies of FM (frequency modulation) radio stations are in megahertz (MHz or 106 Hz)

and are in kilohertz (kHz or 103 Hz) for AM (amplitude modulation) radio stations. What is the

wavelength of the signal broadcast by 99.1 FM radio station?

Black-Body Radiation. When solid objects are heated, they emit radiation, such as the

red glow from the burner of an electric stove and the white light of an tungsten filament of

an incandescent light bulb. This is called ‘black-body radiation’ and the relationship

between the temperature and intensity and wavelength of the emitted light is not fully

explained by classical physics. In 1900, Max Plank explained this phenomenon by

proposed the idea of that energy is either released or emitted only in discrete packets, or

quanta (singular = quantum). He proposed that the energy (E) of a single quantum equals a constant

times the frequency of the radiation:

E = h (EQ. 2)

where E = energy of the radiation (Joules, J)

h = Plank’s constant (6.626 x 10–34

J-s)

= frequency

According the Plank’s theory, matter is only allowed to absorb or emit energy in whole-number multiples

of h (e.g., h, 2h, 3h). For example, if the energy emitted is 3h, then three quanta are emitted.


e-

Photoelectric Effect. Under certain circumstances, when light is shined on

a clean metal surface, electrons may be emitted from the metal (read:

produce electricity). But the pairing of the energy (frequency) of light with

the metal is specific. For example, light with a frequency of 4.60 x 1014

Hz

can cause cesium metal to emit electrons. Cesium will not emit electrons

when a light with a lower frequency is used even if the intensity is increased.

In 1905, Albert Einstein explained this photoelectric effect by explaining

that each photon has energy equal to Plank’s constant times the frequency:

Ephoton = E = h (EQ. 3)

Under the right situation, the energy of the photon striking the metal surface is sufficient to overcome the

attractive force holding the electron in the metal. The electron is then emitted from the metal surface. If

the energy of the photon is greater than the minimum energy, the excess energy is translated into kinetic

energy of the emitted electron.

Problem

2. The wavelength of the green light at a traffic signal is centered at 522 nm. What is the frequency

of this radiation ?

Bohr Model of the Atom. When electricity is applied to a tube containing an element, much like a

cathode-ray tube, and the light is passed through a prism, the element’s line

spectrum is revealed (Figure 3). Instead of the continual rainbow we associate

with light from the sun, lines of light with discrete wavelengths are displayed,

and the spectrum of each element is unique.

Hydrogen

Helium

Carbon

Figure 3. Line-Emission Spectra for Hydrogen, Helium, and Carbon. (Note that the background, which

should be black, is inverted to white for ease of visual disply on the paper.)


The Danish physicist, Niels Bohr, in 1913, adapted Plank’s idea of quantized energy and based a new

model of the atom on three postulates3:

1. Only orbits of certain radii, corresponding to certain definite energies, are permitted for the

electron in a hydrogen atom.

2. An electron in a permitted orbit has a specific energy and is in an

“allowed” energy state. An electron in an allowed energy state will

not radiate energy and therefore will not spiral into the nucleus.

3. Energy is emitted or absorbed by the electron only as the electron

changes from one allowed energy state to another. This energy is

emitted or absorbed as a photon, E = h.

In other words, the electron is only in certain allowable orbits (Figure 4). Other possible orbits are not

allowed. An analogy frequently used for the allowable energy levels are the steps on a staircase.

Figure 4. Bohr model of the atom. Only certain energy levels are allowed (B, C, D). Other energy

levels (e.g., A and E) are prohibited. This model is analogous to the steps on a staircase with the electron

being the ball and the different energy levels being the steps.

For hydrogen, the orbit closest to the nucleus is called the ground state (n = 1, or B in the above figure).

Each orbit further from the nucleus is an excited state: the second orbit, or n = 2, is the first excited state

(C in the above figure), the third orbit, or n = 3, is the second excited state (D in the above figure), and so

forth.

Using hydrogen as an example, let’s see how the line-spectrum is produced. First, the atom is at the

ground state (Figure 5).

A.

B.

C.

D.

Figure 5. Bohr model of the atom used to explain the line-emission spectrum of hydrogen. A. In this

model, the hydrogen is shown initially at the ground state (electron is at n=1). B. Energy (e.g., photon of

light) hits the electron with sufficient energy to excite it to the third excited state (n=4). C. However, the

attraction between the negatively-charged electron and the positive nucleus causes the electron to move

back towards the nucleus. In this case, it only returns to the first excited state (n=2) whereby it emits

energy in order to do this. D. Eventually, the electron does return to the ground state (n=1) by further

emitting energy.

3 Brown, Theordore, H. Eugene LeMay Jr., and Bruce E. Burnsten. (2006). Chemistry: The Central Science 10

th

Ed. Pearson Prentice-Hall, Upper Saddle River, NJ. p. 226.


Bohr’s model has the electrons in discrete energy levels or shells. The further the electron is from the

nucleus, the more potential energy it has. If an atom becomes excited by absorbing energy - from a flame

or electricity – the electron jumps away from the nucleus to a higher energy level. The energies, therefore

the frequencies and the colors of light, is characteristic for each fall in energy. When the electron drops

back down in energy, closer toward the nucleus, it releases energy with a specific amount of energy that

we perceive, if the wavelength is in the visible range, as a characteristic color. The amount of energy

contained by the quantum is proportional to the frequency of the emitted energy:

E = Ehigher energy level – Elower energy level = h (EQ. 4)

Problem

3. When the electron in a hydrogen atom falls from the sixth shell (n=6; 5th excited shell) to the

second shell (n=2; 1st excited shell), line-green light is emitted having a wavelength of 486 nm.

What is the energy difference between these two shells?

II. THE QUANTUM MODEL OF THE ATOM

A. THE NATURE OF ELECTRONS

de Broglie. Investigations about the electron yielded conflicting results – in some experiments, the

electron acted with particle-behavior. In other experiments, the results could only be explained by

modeling the electron as a wave. In 1924, Louis de Broglie solved this conundrum when he showed that

electrons be considered as waves confined in space, and that they could exist only at specific frequencies.

We now consider electrons to have dual wave-particle behavior.

Heisenberg. If electrons are both waves and particles, where are they located in the

atom? Electrons are detected by their interaction with photons – the way that we see are

the photons that bounced off the object and hit our eyes. Locating an electron depends

on its interaction with a photon. In 1927, Werner Heisenberg theorized that because

photons have about the same energy as electrons, any attempt to locate an electron with a

photon will knock the photon out of its course. Hence, the mere act of trying to locate an

electron will change its location:

4

)(h

mvx (EQ. 5)

where x = uncertainty of the position of the electron

mv = uncertainty of the momentum of the electron

h = Plank’s constant

In other words, according to Heisenberg’s uncertainty principle, one cannot know

simultaneously the location and momentum of an electron.


Schrödinger. So, if we can’t know where an electron is, how do we find one? In 1926, the Austrian

physicist Erwin Schrödinger developed an equation to treat electrons in atoms as waves. Together with

Heisenberg’s uncertainty principle, Schrödinger’s wave equation forms the basis for our modern

atomic theory. This theory is called the quantum theory and it describes mathematically the wave

properties of electrons and other very small particles. 2

2( , , , )( , , , ) ( , , ) ( , , , )

2

x y z ti x y z t V x y z x y z t

t m

The solution to the Schrödinger wave equation (above equation) for any given atom is called

a wave function and it gives the statistical probability of finding an electron in a given

location. According to the quantum theory, electrons do not travel around the nucleus in neat orbits

around the nucleus like Bohr postulated but rather are certain regions called orbtials, which are three-

dimensional regions of space around the nucleus that indicate the probable location of an electron.

It is noteworthy that because the Schrödinger wave equation is so complex, it is only solved for the

hydrogen atom. However, assumptions and estimations have proven extremely accurate for atoms and

even large molecules.

B. QUANTUM NUMBERS

In addition to energy levels, quantum numbers are used to describe atomic orbitals. One can think of

the four quantum numbers as the unique “energy address” where a given electron resides. There are four

quantum numbers:

n = principle quantum number

l = angular quantum number4

m = magnetic quantum number

s = spin quantum number

1. Principle Quantum Number (n).

This is the main energy level (or shell) that the electron occupies. The values are whole numbers

above zero, i.e., n = 1, 2, 3, etc. The electrons with the lowest ‘n’ are closest to the nucleus –

electrons in the 1st shell (n = 1) are closest to the nucleus, electrons in the 2

nd shell (n = 2) are further

away, electrons in the 3rd

shell (n = 3) are further still, etc.

The maximum number of electrons in any given shell is equal to 2n2. Thus, for the first shell there

are a maximum of 2(1)2 or 2 electrons. In the third shell, the maximum number of electrons is 2(3)

2

[=2(9)] or 18 electrons. The periods (horizontal rows) of the periodic table relate to the main energy

levels.

4 If you have trouble reading the font in this text, the angular quantum number is designated with the lower case ‘l’

(the 12th

letter of the alphabet).


Periodic Table Showing Principal Energy Levels (n)5

1 H He

2 Li Be B C N O F Ne

3 Na Mg Al Si P S Cl Ar

4 K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr

5 Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe

6 Cs Ba La Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn

7 Fr Ra Ac

Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu

Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lr

2. Angular Quantum Number (l).

The angular quantum number refers to the shape of the orbital. Unlike the solar-system model of the

atom, not all orbitals are spherical. From the rules for the quantum number (Table 1; l = 0 to n-1), the

first energy level can have only one shape of orbital (i.e., sub-energy level). The second energy level

can have two shapes of orbitals; the 3rd

, 3; and, the 4th , 4.

Table 1. Rules for Quantum Numbers

Quantum Number Rules Examples

n whole number n = 1, 2, 3, ...

l 0 to n–1 when n = l, l = 0

when n = 2, l = 0 or +1

m - l to + l when n = 0, l = 0, m = 0

when n = 1, l = 0, m = 0

l = –1, 0, or +1

s either +½ or –½ +½ or –½

When l = 0 (the first orbital for any given energy level), the shape is spherical and generally referred

to as the s-orbital (Table 2). When l = 1, the shape is dumbbell and generally referred to as the p-

orbital. When l = 2, the shape is referred to as the d-orbital, and when l = 3, the shape is referred to as

the f-orbital. The order of the orbitals is: s, p, d, f. These are the ground-state orbitals for the

elements.

5 N.B. The periodic table in the room should have La(57) and Ac(89) on the main table, and the insert beginning

with Ce(58) and Th(90)


Table 2. Orbital Letter Designation For Selected Angular Quantum Numbers (l).

l Letter Shape

0 s

1 p

2 d

3 f

3. Magnetic Quantum Number (m).

For the s-orbital (l = 0), the orbital can have only one orientation: it is the same in any direction (e.g.,

using the x-, y- and z-coordinate system) (Table 2). However, the p-orbital can have three different

orientations: one directed along the x-axis (px), one directed along the y-axis (py), and the third

directed along the z-axis (pz). This is a direct result of the rule for m = -l to + l. This may become

clearer in Table 3.

4. Spin Quantum Number (s). Every orbital can hold two electrons – one with a spin denoted + ½ and the other with a spin of –½.

Table 3. Quantum Numbers for The First Ten Elements.

Principal

Quantum Number

Angular

Quantum Number

Magnetic

Quantum Number

Spin

Quantum Number

Element

Z Symbol

n = 1 l = 0 m = 0 s = +½

s = -½

1

2

H

He

n = 2 l = 0 m = 0 s = +½

s = -½

3

4

Li

Be

l = 1 m = -1 s = +½ 5 B

m = 0 s = +½ 6 C

m = +1 s = +½ 7 N

m = -1 s = -½ 8 O

m = 0 s = -½ 9 F

m = +1 s = -½ 10 Ne

Each of the shapes of orbitals, determined by the second quantum number (= angular quantum number,

), is typically referred to by a letter – s, p, d, or f (Table 4).


Table 4. Types of Orbitals (Determined by the Angular Quantum Number)

Second Quantum

Number (l)

Letter Designating

Orbital

Number of

Orbitals

Maximum Number of

Electrons

0 s 1 2

1 p 3 6

2 d 5 10

3 f 7 14

III. ELECTRON CONFIGURATION & ORBITAL NOTATION

A. RULES GOVERNING ELECTRON CONFIGURATION

Electron configuration refers to the arrangement of electrons in an atom. The lowest energy level for an

atom is called the ground state electron configuration. It would be simple if the electrons entered the

orbitals in the order s p d f. However, it isn’t this simple because there is a balance between the

electron’s attraction to the positive nucleus and repulsion away from other electrons already in the atom.

So how do we fill the orbitals? There are three rules: (1) the Aufbau principle6, the Pauli exclusion

principle, and Hund’s rule.

1. Aufbau Principle: An electron occupies the lowest-energy orbital that can receive it.

Orbitals are filled from the lowest energy level up. However, due to balancing the forces of attraction

between the electron and the nucleus and the forces of repulsion between electrons, the order is not

simply 1s through 4f, etc. The principal energy levels are subdivided into sublevels. Electrons do not

always fill the energy sublevels in an s p d f order (Figure 6).

Principal energy levels

Diagram showing the energy of each sublevel.

Figure 6. Orbital energies.

The order of electrons filling the orbitals (see Figure 6) is as follows:

1s 2s 2p 3s 3p 4s 3d 4p 5s 4d 5p 6s etc.

To help remember this order, build a checkbox (Table 5). Fill the rows with the principal energy

levels (n = 1, 2, 3, etc) and the columns with the sub-energy levels (s, p, d, f). Then, connect the

corners of the boxes.

6 “building-up” principle


Table 5. Order of Electrons Entering An Atom.

s p d f

1 1s

2 2s 2p

3 3s 3p 3d

4 4s 4p 4d 4f

5 5s 5p 5d 5f

6 6s 6p 6d 6f

7 7s 7p 7d 7f

2. Pauli Exclusion Principle: No two electrons in the same atom can have the same set of four

quantum numbers.

The first three quantum numbers (n, l, and m) specify the energy, shape and orientation of an orbital.

The two possible spin quantum numbers (+½ and –½) allow two electrons of opposite spins to occupy

the one orbital: (spin +½ =; spin –½ =).

3. Hund’s Rule: orbitals of equal energy (e.g., px, py, and pz) are each occupied by one electron

before any orbital is occupied by a second electron, and all electrons in singly-occupied orbitals

must have the same spin.

In other words, an electron will enter orbital 2px with a +½ spin, the next electron will enter 2py (also

having a spin +½), the third electron will enter 2py (also having a spin +½). The fourth electron to

enter this sublevel will then enter 2px with a spin of –½.

The following gives examples of how these three rules are applied:

According to Aufbau principle, shell 1-s is filled before an

electron can enter shell 2-s

1s 2s 2p

According to the Pauli exclusion principle, shell 1-s is filled

with two electrons, one having spin +½ and the other having

spin –½, before an electron enters shell 2-s

1s 2s 2p

According to Hund’s rule, shell 2px, 2py, and 2pz must each

have one electron (+½ spin) before the second electron enters

shell 2-px.

1s 2s x y z

2p


B. ELECTRONIC NOTATION

Electron notation is a shorthand way to show the electrons as they fill the orbitals of atoms. Let’s start

locating the electrons in atoms by applying the three above rules.

The first energy level (H and He):

Hydrogen has only one electron. Therefore it goes into the lowest energy level with a spin of +½:

H:

1s 2s 2px 2py 2pz 3s 3px 3py 3pz

Helium has two electrons. According to the Aufbau principle, the second electron has to fill the 1s

orbitial before electrons can begin to enter 2s. According to the Pauli exclusion principle, this second

electron, however, has a spin of –½

He:


The ‘periodic table’ for the first energy level (n = 1) with H and He would look like this: 1 2

1 H He Hydrogen Helium

The second energy level (Li, Be, B, C, N, O, F, and Ne):

Lithium. With the first energy level filled (1s), we move on with the third electron, lithium, to the

second energy level which has both ‘s’ and ‘p’ sublevels. This third electron enters the 2s orbital with a

spin of +½:

Li:


Beryllium. Like helium, beryllium’s fourth electron fills the s-orbital with a spin of –½:

Be:


Boron. Boron’s fifth electron enters the 2px orbital:

B:


Carbon. According to Hund’s rule, before an electron can fill the px orbital, all of the p-sublevels must

be filled. This means that carbon’s sixth electron enters the 2py orbital with a spin of +½:

C:

1s 2s 2px 2py 2pz 3x 3px 3py 3pz


Nitrogen. As with carbon’s electron in the py orbital, nitrogen’s seventh electron enters the 2pz orbtial

with a spin of +½:

N:


Oxygen. Now, all of the p-sublevels have one electron. Before an electron can enter the next energy

level (n = 3; Aufbau principal), all of the p-sublevels must be filled. Thus, oxygen’s eighth electron

enters the px orbital, but with a spin of –½:

O:


Fluorine and Neon. These elements follow the pattern:

F:


Ne:


The ‘periodic table’ for the first and second energy levels (n = 1 & 2) would look like this: 1 2


3 4 5 6 7 8 9 10

2 Li Be B C N O F Ne Hydrogen Helium Boron Carbon Nitrogen Oxygen Fluorine Neon

The reason that helium is put above neon is because both have filled shells and similar chemical and

physical characteristics (i.e., noble gases).

The third energy level (Na, Mg, Al, Si, P, S, Cl, Ar):

Sodium and Magnesium. Fill the orbitals like Li and Be only for the 3s orbital:

Na:


Mg:


Fill in the orbital notation for Al, Si, P, S, Cl and Ar:

Al:


Si:


P:


S:



Cl:


Ar:


The ‘periodic table’ for the first three energy levels (n = 1, 2, 3) would appear to look like this: 1 2


3 4 5 6 7 8 9 10

2 Li Be B C N O F Ne Hydrogen Helium Boron Carbon Nitrogen Oxyge

n

Fluorine Neon

11 12 13 14 15 16 17 18

3 Na Mg Al Si P S Cl Ar Sodium Magnesium Aluminum Silicon Phosphorus Sulfur Chlorine Argon

However, the third energy level can have 3 sublevels: s-, p- and d-. Yet, according to the way the energy

levels fall, balancing the repulsive forces between electrons and the attractive force between the electrons

and the nucleus, 4s comes before 3d (see Aufbau principal, p. 11).

So, when filling orbitals, place the 4s before the 3d. The location atoms having unfilled orbitals is as

follows:

s s

s s p p p p p p

s s p p p p p p

s s d d d d d d d d d d p p p p p p

s s d d d d d d d d d d p p p p p p

If the f-orbitals were inserted into the periodic table, it would look like this:


s s

s s p p p p p p

s s p p p p p p

s s d f f f f f f f f f f f f f f d d d d d d d d d p p p p p p

s s d f f f f f f f f f f f f f f d d d d d d d d d p p p p p p

C. ELECTRON CONFIGURATION Once one understands orbital notation, it is relatively simple to write the electron configuration:

H: 1s1 (pronounced: one s one)

1s 2s 2px 2py 2pz

He: 1s2 (pronounced: one s two)

1s 2s 2px 2py 2pz

Li: 1s2 2s

1

1s 2s 2px 2py 2pz

And, it gets even easier. Noble-gas configuration. Lithium is written [He] 2s1; neon is [He] 2s

2 2p

6;

sodium is [Ne] 3s1: write the preceding noble gas in brackets, with the subsequent electron configuration

following.

D. TRENDS IN ELECTRON CONFIGURATIONS

Recall that elements in the same family or group on the periodic table have similar chemical and physical

properties. This is an outcome of the elements in the same group have the same number of valence

electrons. These are the outermost electrons – the ones specifically written with the noble-gas

configuration. For example, the noble-gas configuration for oxygen (Group 16) is [He] 2s2 2p

4; sulfur:

[Ne] 3s2 3p

4; sulfur: [Ar] 4s

2 4p

4. They combine with hydrogen to form H2O, H2S, and H2Se. The

electrons that are not in the highest energy level (e.g., 1s2 in lithium) are called inner-shell electrons.


ANSWERS TO PROBLEMS:

1.

m

s

s

m

c 03.31

10*1.99

10*00.3c

6

8

2.

Jxs

xsJxhE

sx

m

s

m

c

191434

14

9

8

1081.3)1

1075.5)(10626.6(

11075.5

10*522

10*00.3c

3.

Jxs

xsJxh

sx

mx

smx

x

191434

14

9

8

-9

1009.4)1

1017.6)(10626.6(E

11017.6

10486

/1000.3c ,c Because

m 10486 nm 486

the arrangement of electrons in atomschemistry study guide 4: arrangement of electrons in atoms p. 7...

Documents