Natural Sciences Ilecture 14: Structure of the Atom

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Now for something really different (sort of)...

CHEMISTRYPrelude

Today we leave the realm of what most scientists would describe as straight Physics and make our first foray into Chemistry. Even though we're giving the field a different name, the transition probably won't seem very abrupt, because there is no clear demarcation between what is Physics and what is Chemistry. The term "chemistry" is generally applied to science dealing with phenomena that involve associations of different kinds of atoms – formation of compounds, reactions between elements, properties of complex molecules, etc. However, because chemistry generally begins with atoms, and because studies of the structure and "internal workings" of the atom are usually classified as physics, we're presently in the never-never land between the two. In any event, we'll begin with a discussion of the structure of atoms...

STRUCTURE OF THE ATOM

The text opens the discussion of the atom (chapter 8) with a historical summary of observable properties and the relationship between atoms and elements. We'll take a complementary approach, beginning with a review of atomic structure and subatomic particles, emphasizing electrons because it is these specific subatomic particles that largely determine chemical behavior.

proton

neutron

electron

Z = atomic number= no. of protons

N = no. of neutronsA = mass no. = N + Z

Questions:

What is wrong with this picture?

How big is the nucleus?

How do the electrons really behave and why?

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The size of the atom and the nucleus...

Even before we could actually "see" atoms with the aid of an electron microscope, it was possible to calculate their sizes in various ways. The most definitive of these involved x-ray diffraction studies of crystalline materials. However, we can also obtain crude estimates simply by determining how many atoms are present in a given volume of material.

L

37 = 343 atoms (assume hard spheres)

3V = L / 343atom

1/3atom radius ~ (V )atom

The typical radius of an atom turns out-10

to be ~ 10 m (0.1 nm)

Determining the size of the nucleus was quite another matter, because it is "buried" somewhere within the atom – in a sense shielded by the electrons. Electron-electron repulsions justify the assumption of "hard-sphere" behavior of the atoms; no volumetric measurements can reveal the size of the nucleus. Early in the 20th century, British physicist Ernest Rutherford devised a way to figure out how big the nucleus is. His approach works something like this...

plywoodtarget

cannonballhidden behindtarget

How can the marksman determine the sizeof the cannonball using only his rifle?

Answer: By firing a lot of shotsand accumulating statistics onricochets off the cannonball

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Rutherford's experiment a little more sophisticated, but the principle isthe same:

lead containerwith source ofalpha particles

neercs gnitcdete

alpha beam" "

thin gold foil

)SnZ htiw detao(cflashes of lightreveal alpha "hits"

nucleus

atoms ofmetal foil

alpha particles fromradioactive source

(5 MeV)

Most of the alpha particles pass right through the foil, but some are deflected in

He++

Note: The thick-ness of the foil wasmuch greater than3 atom layers!

Two models had been put forth prior to Rutherford's experiments. Which do you now think is "correct"?

e

"saturanian"

large, massive(+) nucleus

(+)e

"plum pudding"

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On the basis of his alpha-scattering results, Rutherford concluded that the nucleus is tiny in comparison with the overall size of the atom: the size difference is about the same as that between the thickness of a dime and a football field (a factor of 100,000)!...So, in a way, the atom is mostly empty space. It has a tiny but very massive nucleus at its center in which almost all of the mass of the atom is concentrated.

Just for fun...Early in this course you had some practice with calculating the density

of materials ( ). Try computing the density of a material consis-ting solely of atomic nuclei using only the information on this page (what assumptions would you have to make in order to do this?).

r = m / V

Resseearn

l

nela

e ser

R s

atom

A few years after determining the size of the nucleus of a gold atom, Rutherford had another remarkable success: he broke up the nucleus of a nitrogen atom using alpha-particle "projectiles", and was able to identify the individual positive charges we now call protons. Rutherford believed the nucleus might also contain neutral particles of proton-like mass, but the existence of neutrons was not documented conclusively until the 1930s by James Chadwick.

What about those tiny, distant-orbiting electrons?...

The next major breakthrough in our understanding of the atom was made by Danish physicist Niels Bohr, who sought answers to questions about the orbiting electrons. According to Maxwell's electromagnetic wave model of light, an orbiting charged particle (being constantly accelerated) should give off electromagnetic radiation – but electrons did not seem to do this.

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Before proceeding to Bohr's model of the atom, let's briefly recap the state of knowledge about light in the early 20th century. Einstein had concluded that light consists of discrete packets of energy – now called . The energy of a photon is proportional to the vibrational frequency of the light through Planck's constant:

(this also means that E is inversely proportional to wavelength l). Bohr also knew that interactions between light and matter are "quantized": an incident photon is either totally absorbed by "target" matter or not absorbed at all.

The of the hydrogen atom was known to have certain regular characteristics:

PHOTONS

E = h f

Atomic Spectrum

slit

hothydrogen

prism

A Line Spectrum

visible wavelengths(Balmer series)

infrared(Paschen series)

ultraviolet(Lyman series)

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The Balmer series (see bottom of page 5) gets its name from a Swiss mathematician who figured out some regularities in the spacing of the lines:

Any hot gas at low pressure will produce a line spectrum, although the spectra of other gases are more complex than that of hydrogen. In the early 1900s, it was not realized that the continuous spectra of solids, liquids and dense gases – such as those Planck struggled to understand – result from "close-range" interactions among the atoms. Such interactions are absent in low-density gases, so we see the characteristics of the individual atoms. Bohr believed that the line spectra were telling him something simple and fundamental about the electronic structure of hydrogen, and he understood that any model he developed would have to account for these line spectra.

The model Bohr developed for the electronic structure of the hydrogen atom can be thought of as a "solar-system" model. It involves several important assumptions:

an electron can assume only specific "allowed" orbits

the allowed orbits are described by

(the justification for this assumption is only that it fits the Balmer series)

BOHR'S Model

= R1

l1

n2

12

2

wavelengtha constant

an integer

line color

violetvioletblue-greenred

n

6543

m v r = n h

2pPlanck's constant

integer (= orbit quantum no.)angular momentum(r = radius)

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Bohr knew about Newton and Coulomb, of course...

Since he viewed his problem as one involving an orbiting charged particle, Bohr set these two forces equal to one another and substituted for by using his relation at the bottom of page 6. This allowed him to

calculate the value of for any allowed orbit (i.e., any value of ).

vr n

F =2

m v r

F =k q q21

2r

(Newton's centripetal force)

(Coulomb's electrical force;q's are charges on electronand proton)

n

123

r (nm)

0.0530.2120.476

This is getting really interest-ing, because these electron orbit radii ( ) are pretty similar to known sizes of atoms – maybe Bohr was really onto something...

Bohr radiiproton

electron

r

Bohr's next obstacle concerned the expectation from Maxwell's theory that accelerating charged particles should give off electromag-netic radiation. Recall that a charge in a circular orbit is being contin-uously accelerated (the velocity vector is constantly changing), so radiation should be emitted. The problem was that no light was ob-served to be given off by hydrogen in its normal state. Bohr solved this dilemma simply by postulating an exception to Maxwell's theory:

.Electrons confined to their allowed orbits do not give off light

e

8

QUANTUM jumps

The next piece of Bohr's model concerned the energy states of electrons and the conditions under which they do give off light. Before we get going on that, a quick review of potential energy...

reference level; PE = 0

lowest PE(most negative)

objects at these levels have PE – work input

is needed to get to zeronegative

objects at these levels have PEpositive

In Bohr's electron energy level model, an electron has zero potential energy when it is completely removed from the atom. Like objects in the basement of the building above, electrons in orbit around a nucleus then have negative potential energies – it requires energy (work) input to remove them from the atom. An electron in the orbit closest to the nucleus has the lowest possible potential energy (this is referred to as the ).

From his equations, Bohr calculated the energies of the electrons for various values of , and discovered that

ground state

n

E =n

EL

n 2energy of electronof quantum no. n

energy of electronin ground state

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On page 8 we learned that:

the energy of each orbit is quantized

the energy of an electron is determined by which orbit it occupies

(there are no intermediate energy states)

The normal energy state for electrons in hydrogen atoms is the ground state, but they can be temporarily "excited" to higher orbits by high temperatures or electrical discharges. When electrons in higher orbits return to lower orbits, they give up energy in the form of light...

Transitions between energy states

A single photo is produced from each (energetically) downward electron transition, and that photo has an energy that is exactly equivalent to the difference in energy level of the two orbits:

tibro rehgih tibro rewol

p otonh

E = E - E = h fH Lphoton

The for the hydrogen atom looks like thisenergy-level diagram

1

2

3

4

5

6

n energy (eV)

-0.377

-0.544

-0.850

-1.51

-3.40

-13.6

7.3

E1

4 H

z

6.9

E1

4 H

z

6.2

E1

4 H

z

4.6

E1

4 H

z

Each of these transi-tions produces a pho-ton of specific energy – the discrete lines of the hydrogen spec-trum on page 5.

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Limitations of the Bohr model

f = (E - E ) / h H L

The Bohr model was a major breakthrough in thinking about the structure of the atom in general. It rationalized the line spectrum of the simplest of atoms (hydrogen) in terms of photons emitted as a consequence of specific inter-orbit transitions. Implicit in the model is the idea that all atoms (and molecules) have fundamental frequencies [ ] of light with which they will interact. The model successfully predicted spectra for other atoms having just one electron

+ ++(He and Li also worked), but it failed for more complicated atoms.

0.1

nm

n=54321

n=6

Paschen series

Ba

mr

ei

le

sre

s

Ln

rs

yma

se

ie

The scale of Bohr's hydrogen atom

Note: Photons of appropriate ener-gy are emitted during electron transitions with arrows as shown. Photons of the same energy can be absorbed dur-ing an "up-energy" transition as well.

electronorbitals

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QUANTUM MECHANICS

maybe electrons are also waves

About 10 years after the publication of Bohr's remarkable work, French physicist Louis de Broglie (then a graduate student!) put forth an even more remarkable idea. He believed that because light has the properties of both waves and particles, maybe it's too limiting to think of electrons only as particles – . He reasoned that if an electron is a standing wave at some average dis-tance from the nucleus, only specific wavelengths would be permitted at specific distances:

rnucleus

electron wave

This is allowed... This isn't...

rnucleus

The circumference of the circle is 2pr, so the wave can fit the circumference when

where is an integer.

De Broglie's challenge was to relate the wavelength in this equation to some property of the electron that was known – say, the mass...

(next page)

2 p r = nln

De Broglie Waves

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De Broglie knew the mass of the electron (thanks to Thomson and Millikan), but he needed a way to relate wavelength to mass. He borrowed from Einstein, who previously had related vibrational frequency to photon energy

but

so

Einstein had also concluded that mass and energy are related in a simple way

so

and

Einstein had thus deduced a relationship between wavelength and momentum for photons, and De Broglie assumed that the same relationship applies to electrons:

or

According to this equation, a moving particle has a wavelength associated with it that is inversely proportional to the momentum of the particle (p = m v). The equation implies that

(the wave-lengths are measurable only in the case of very small particles).

As wild as this idea might seem, it was soon supported by convincing evidence from research in the U.S. (at Bell Laboratories), where it was demonstrated that electron beams can be diffracted just as light beams can. So the intriguing conclusion is that

. This is the fundamen-tal basis of what soon came to be know as .

E = h f

f = c / l

E = (hc) / l

2E = m c

2(hc) / l = m c

h / l = mc

h / l = m v

l = h / mv

electrons and other particles are also waves when they are in motion

electrons exhibit the properties of waves and particles alike

Quantum Mechanics

This is mass times vel-ocity (i.e., momentum)

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...So, electrons in atoms are waves confined ("trapped") by a positively-charged nucleus.

The next big advances in building the theory of quantum mechanics were made using highly sophisticated mathematical approaches. There are two key components:

a 3-D description of vibrational states of electrons around the nucleus (remember the quantized vibrations of strings?)

exactly predicts the observed energy levels of electrons in the hydrogen atom

electrons are not orbiting particles as in the Bohr model; rather, their "position" is described in terms of regions of space around the nucleus where they might be found

(more on p. 14)

Applied to electrons in atoms, this principle states, in effect, that

. The principle arises from the dual wave/particle nature of light and matter, and the fact that all the techniques ("probes") we can use to observe particles and waves are themselves particles or waves. If you are using light (radiation) to observe a particle, the uncertainty in position of that particle is

Schrodinger wave equation

Heisenberg uncertainty principle

you cannot know both the location and the momentum of a particle (or, more accurately, the better you know one, the worse you know the other)

Dx > l

light "prbe"

o

particle of interest

diffract d or

eg

reflected li

th

..

Using the minimum possible wavelength seems like the way to go, but this gets you in trouble in another way: short wavelength

means high photon momentum ( ). This momentum is transferred to the particle you're trying to observe, so it no longer has the "intrinsic" momentum you were interested in. The uncertainty in momentum of the observed particle is

p = h / l

Dp > h / (4 p l)

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h tp o on beam

bpro e

momentum transferred toparticle of interest

The implications of the work of Schrodinger and Heisenberg are that we can describe the positions of electrons only in terms of proba-bilities – that is, at a given location in space and time near a nucleus, there is a certain probability of finding the electron of interest. The de Broglie waves associated with an electron are described in terms of a quantum-mechanical wave function that is in some ways analogous to equations for ordinary standing waves – e.g., guitar strings...

y = A sin nn2

xL

o360L

l = 2L1

l = L/24

l = 2L/33

l = L2

xy

This equation describes the position of any part-icle in the vibrating string in terms of x and y coordinates

n = no. of minima & maximano. nodes = n-1

l = 2L / nn

frequency a n

:

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The wave function ( ) is a 3-dimensional description of de Broglie waves that assigns a complex number to each point in space and time around the nucleus. The square of the value of the wave

function ( ) at point x and time t tells us not the actual location of the electron but the probability of finding it at a given place.

Y

2Y

Each wave function contains adesignation of position in terms of

, and and a designation of spin(up or down – more later)

Y

r q f

The guitar string on the previous page required only 1 quantum number, but we need 4 to describe the energy state of an electron:

l = 0,1,2,3

s,p,d,f

* s = "sharp" p = "principal"d = "diffuse" f = "fine" (from spectroscopy)

quantumno.

n

l

m

s

name

principal

angular momentum

magnetic

spin

determines

dist. from nucleus;main energy level

shape or symmetry

orientation

spin orientation

possible values

positive integers

integers to n-1

integers, -l to +l

+1/2, -1/2

*

nucleus

e

rq

f

x

y

z

16

The spin quantum number was an ad hoc addition to the first three. It was proposed by Wolfgang Pauli to circumvent a significant dilemma, namely that the Schrodinger wave equation does not preclude the possibility of all the electrons in an atom falling down into the lowest quantum level. Yet this was known not to happen – the energy states of the electrons in atoms remain highly structured. Pauli's "solves" this by stating that

. He further postulated that in order for two electrons to "co-occupy" the same state, there has to be something different about them. The different attribute was later discovered to be the intrinsic angular momentum of the electron itself – not of the orbital, which is given by m, but of the electron spinning on its own axis.

Let's look at an example, first using wave function notation

exclusion principle no two electrons can occupy the same quantum state at the same time

Yn,l,m

Y1,0,0

n = 1 l = 0m = 0

n = 1l = 0m = 0:

means the orbital is close to the nucleus means the shape is spherical

the orientation is irrelevant if the shape is spherical

[Note that l = 0 corresponds to an "s" orbital (see bottom of page 15),so you may recognize the above wave function as that of a 1s electron]

The radial probability curve looks like this...

This curve represents the probability of finding the 1s electron at some distance r from the nucleus

VISUALIZATION AIDS

r = Bohr radiusB

Y1,0,0

2

probabilityof findingthe electron

r

1rB 2rB

nucleus

17

The contour map approach:

1rB1rB 2rB 3rB32%

44%44%

18%

6%

The 3-D visualization:

z

y

x

Y1,0,0

n = 1 l = 0m = 0

1s electron Bohr radii

z

y

x

(See Figure 9-1 of your text (p. 222), which uses a baseball analogyto illustrate the principle shown on this page)

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Let's try another wave function...

Y2,0,0

n = 2 (further from nucleus than for n=1))

l = 0 (spherical shape)

m = 0 (no orientation aspect)

This is a 2s electron!

Y2,0,0

2

probabilityof findingelectron

r

probability curveextends further fromnucleus than for the1s electron

nucleus

z

y

x

2s electron

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Things get more complicated when the sphericalsymmetry is lost...

Y2,1, (0, +1, or -1)

n = 2 (energy level similar to 2s)

l = 1 (shape is not spherical)

m can have one of three possible values – i.e, three different orientations

This is a 2p electronZ

Y2,1,0

2

probabilityof findingelectron

rnucleus (along z axis)

z

y

x

z

y

x

All three 2p orbitalstogether

20

There are five possible 3d orbitals...(m can range from -2 to +2)

z

x

y

dXY dYZ dXZ

d 2 2x -y d 2

z

21

Partial summary table

n

1

2

3

4

l

0 (s)

0 (s)1 (p)

0 (s)1 (p)2 (d)

0 (s)1 (p)2 (d)3 (f)

m

0

0 1,0

0 1,0 2, 1,0

0 1,0 2, 1,0 3, 2, 1,0

s

½

½ ½, ½, ½

½ ½, ½, ½ ½, ½, ½, ½, ½

½ ½, ½, ½ ½, ½, ½, ½, ½ ½, ½, ½, ½, ½, ½, ½

no. of

electrons

2

26

26

10

26

1014

8

18

32

This diagram determines the order in which "orbitals" are filled with electrons in building up to high-Z atoms from H and He. Electrons are added in such a way as to preserve maximum spin alignment. Note that the principal quantum no. is not the sole determinant of energy level...

en

erg

y

4f

4d

3d

5p

4p

3p

2p

6s

5s

4s

3s

2s

1s

An energy-level diagram

Questions: Given theenergy-level occupancyshown, what is theidentity of the atom?

Is maximum spin align-ment maintained?