Chapter41Chapter41

All About AtomsAll About Atoms

Atoms are the basic building blocks of matter that make up everyday objects. A desk, the air, even you are made up of atoms! There are 90 naturally occurring kinds of atoms. Scientists in labs have been able to make about 25 more.

41-1 The Mass and Size of the Atom41-1 The Mass and Size of the Atom

The Mass of the AtomThe absolute atomic mass matom can therefore be obtained by measuring Avogadro’s number:

AN

substance theof mole 1 of Massatoman of Mass

12310)000005.0022045.6( moleN A

gramN

cAm

A

relatom

12,

The present best value for NA is

With the value of NA ,we can write:

The Size of the Atom

Determining the Atomic Size from the Covolume

Nr 224

1

RTbVVaP ))(/( 2

The Van der Waals equation for one mole of a real gas states

ANrb 3

3

44

The quantity b is equal to the fourfold volume of the particles

Determining the Atomic Size from the Gases movement

is the distance,the N is again the particle number density

41-2 Some Properties of AtomsBasic properties of atoms:

1. Atoms are stable.

2. Atoms combine with each other.

Atoms Are Put Together Systematically

The numbers of elements in the six periods are:

2, 8, 8, 18, 18, 32.

Results of Mass Spectrometry

In atomic physics,mass spectrometers are primarily of interest as instruments for analysing the isotopic composition of chemical elements. An element often has several isotopes,for example chlorine:an isotope with mass number 35 occurs with an abundance of 75.4%;the other stable isotope with mass number A=37 has na abundance of 24.6%.The resulting relative atomic mass of the isotope mixture is Arel=35.457. There are elements with only one stable isotope,for example ; and others with two stable isotopes,

and finally there are elements with many stable isotopes.

Atoms Have Angular Momentum and Magnetism

In quantum physics,each quantum state of an election in an atom involves an angular momentum and magnetic dipole moment that have opposite directions.

(those vector quantities are said to be coupled)

lowhigh EEhf

Atoms Emit and Absorb LightThe light is emitted or absorbed as a photon with energy:

The Einstein-de Haas Experiment

This clever experiment designed to show that the angular momentum and magnetic moment of individual atoms are coupled.Observation of the cylinder’s rotation verified that the angular momentum and the magnetic dipole moment of an atom are couple in opposite directions. Moreover,it dramatically demonstrated that the angular momenta associated with quantum states of atoms can result in visible rotation of an object of everyday size.

41-3 Electron Spin41-3 Electron Spin

Quantum Number

symbol Allowed

Values Related to

Principal n 1,2,3…… Distance from the nucleus

Orbital 0,1,2……(n-1) Orbital angular momentum

Orbital magnetic0,±1,±2, ……±

Orbital angular momentum(z component)

Spin s Spin angular momentum

Spin magnetic Orbital angular momentum(z component)

All states with the same value of form a shell.

All states with the same values of n and l form a subshell.

All states in a subshell have the same energy.

There are 2n2 states in a shell. There are 2(2 +1)states in a subshell

Table 41-1 Electron States for an Atom

To 41-9

In quantum physics,spin angular momentum is best thought of as a measurable intrinsic property of the election.

Table 41-1,shows the four quantum numbers n,l, ml, and ms, that completely specify the quantum states of the electron in a hydrogen atom.The same quantum numbers also specify the allowed states of any single election in a multielectron atom.

41-4 Angular Momenta and Magneti41-4 Angular Momenta and Magnetic Dipole Momentsc Dipole Moments

Orbital Angular Momentum and Magnetism

)1( llL

The magnitude L of the orbital angular momentum of an electron in an atom is quantized;it can have only certain values:

Lm

eorb 2

The “ - ” in this relation means that is directed opposite

an orbital magnetic dipole moment

The magnitude of must also be quantized and give by

)1(2

llm

eorb

The components of the orbital magnetic dipole moment are quantized and give by

Blzorb m ,

is the Bohr magneton:

TJm

eeB /274.9

2πm4 1024

L

Lzcos

lZ mL

The components Lz of the angular momentum are also quantized,and they are given by

we can extend that visual aide by saying that makes a certain angle with the z axis:

We can call the semi-classical angle between vector and the z axis.

Spin Angular Momentum and Spin Magnetic Dipole Moment

The spin magnetic dipole moment ,which is related to the spin angular momentum

Sm

es

)1( ssm

es

The magnitude of also be quantized and give by

866.0)12

1(

2

1)1( ssS

The magnitude S of the spin angular momentum of any electron,whether free or trapped,has the single value given by

The electron is said to be spin up

The electron is said to be spin down

sz mS

Neither nor can be measured in any way.however,we can measure their components along any given axis---call it the z axis.the components of the spin angular momentum are quantized and given by

The components of the spin magnetic dipole moment are also quantized

Bszs um2,

Orbital and Spin Angular Momenta Combined

Define a total angular momentum which is the vector sum of the angular momenta of the individual electrons. the number of electrons in a neutral atom is the atomic number Z. For a neutral atom

)()( 2121

zz SSSLLLJ

Instead, the effective magnetic dipole moment for the atom is the component of the vector sum of the individual magnetic dipole moments in the direction of

41-5 The stern-Gerlanch Experiment41-5 The stern-Gerlanch Experiment In the Stern-Gerlanch experiment,as it is now know,silver is vaporized in an oven, and some of the atoms in that vapor escape through a narrow slit in the oven wall,into an evacuated tube.Some of those escaping atoms then pass through a second narrow slit,to form a narrow beam of

The Experimental surpriseStern and Gerlach found was that the atoms formed two distinct spots on the glass plate,one spot above the point where they would have landed with no deflection and the other spot just as far below that point.This two-spot result can be seen in the plots of Fig.41-9,which shows the outcome of a more recent version of the Stern-Gerlach experiment.

atoms. The beam passes between the poles of an electromagnet and then lands on a glass detector plate where it forms a silver deposit.

When the field was turned off,the beam was,of course, undeflected and the detector recorded the central-peak pattern shown in Fig.419.When the field was turned on,the original beam was split vertically by the magnetic field into two smaller beams,one beam higher than the previously undeflected beam and the other beam lower.As the detector moved vertically up through these two smaller beams,it recorded the two-peak pattern show in Fig.41-9.

Fig.41-9 Results of a modern repetition of the Stern-Gerlach experiment.

The Meaning of the ResultsIt is not the magnetic deflecting force

The potential energy U of the dipole in the magnetic field

BU z

BvqF

BU In Fig.41-8,the positive direction of the z axis and the direction of are vertically upward.

Using (F=-dU/dx) for the z axis

dz

dB

dz

dUF zZ

According to

Bszs m 2,

The component are for quantum numbers ms=±1/2. substituting into Eq.41-13 gives us

BBzsBBzs )2

1(2 and )

2

1(2 ,,

Then substituting these expressions for Uz in Eq.41-17,we find that the force component Fz deflecting the silver atoms as they pass through the magnetic field can have only the two values

)( and )(dz

dBF

dz

dBF BzBz

which result in the two spots of silver on the glass.

Sample Problem 41-1Sample Problem 41-1

220 )

)/((

2

10

2

1t

M

dzdBtatvd B

zz

mmm

smkg

m

mTTJ

v

w

M

dzdBd B

08.0108.7

)/750)(108.1(2

)105.3(

)/104.1)(/1027.9(

))()/(

(2

1

5

225

22

324

2

)/( dzdBF BZ M

dzdB

M

Fa Bz

z

)/(

The separation between the two subbeams is twice this,or 0.16 mm. This separation is not large but is easily measured.

41-6 Magnetic 41-6 Magnetic ResonanceResonance

Bhf z2a condition called magnetic resonance.

The f required for the spin-flipping

)(2 localextz BBhf

Nuclear magnetic resonance is a property that is the basis for a valuable analytical tool, particularly for the identification of unknown compounds. Figure 41-11 shows a nuclear magnetic resonance spectrum.

Sample Problem 41-2Sample Problem 41-2

Bm

Hz

sm

f

c92.3

1066.7

/1000.37

8

MHzHz

sJ

TTJ

h

Bf z

6.761066.7

1063.6

)80.1)(/1041.1(22

7

34

26

41-7 The Pauli Exclusion Principle41-7 The Pauli Exclusion PrinciplePauli exclusion principle : For elections,it states that

No two elections confined to the same trap can have the same set of values for its quantum numbers.

This principle means that no two elections in an atom can have the same four values for the quantum numbers n,l,ml, and ms. In other words,the quantum numbers of any two elections in an atom must differ in at least one quantum number.

41-8 Multiple Electrons in 41-8 Multiple Electrons in Rectangular TrapsRectangular Traps

1.One –dimensional trap.

2.Rectangular corral.

3.Rectangular box.

width L quantum number n quantum number

widths Lx,Ly quantum numbers nx,ny quantum number

widths Lx,Ly,Lz quantum numbers nx,ny,nz quantum number

Finding the Total Energy

(a) Energy-level diagram for one electron in a square corral of widths L.

(b) Two electrons occupy the lowest level of the one-electron energy-level diagram.

(c) A third electron occupies the next energy level.

(d) The system’s ground-state configuration,for all 7 electrons.

(e) Three transitions to consider as possibly taking the 7-electron system to its first excited state.

(f) The system’s energy-level diagram,for the lowest three total energies of the system.

nx ny msEnergy*

2 2 8

2 1 5

2 1 5

1 2 5

1 2 5

1 1 2

1 1 2

Total 32

*In multiples of

Table 41-2 Ground-State Configuration

41-9 Building the Periodic Table41-9 Building the Periodic Table

To table 41-1

The values of l are represented by letters:

l = 0 1 2 3 4 5 ……

s p d f g h …… Guided by the Pauli exclusion principle

41-10 X Rays and the Numbering of 41-10 X Rays and the Numbering of the Elementsthe Elements

The distribution by wavelength of the X rays produced when 35 kev electrons strike a molybdenum target.The sharp peaks and the continuous spectrum from which they rise are produced by different mechanisms.

The Continuous X-Ray Spectrum

min0

hchfK

0min K

hc

The Characteristic X-ray Spectrum

The peaks arise in a two-part process(1) An energetic electron strikes an atom in the target and,while it is being scattered,the incident electron knocks out one of the atom’s deep-lying (low n value) electrons.If the deep-lying election is in the shell defined by n=1,there remains a vacancy,or hole,in this shell.

(2)An electron in one of the shells with a higher energy jumps to the K shell,filling the hole in this shell.During this jump,the atom emits a characteristic x-ray photon.

Accounting for the Moseley Plot

The hydrogen atom is

for n = 1,2,3,……

2220

4 6.131

8 n

eV

nh

meEn

We can approximate the effective energy of the atom by energy of theatom by replacing the factor in Eq.41-24 with or , That gives us

2

2)1)(6.13(

n

ZeVEn

2

2

2

2

2

12

)1)(2.10(

1

)1)(6.13(

2

)1)(6.13(

ZeV

ZeVZeV

EEE

We may write the energy change as

Then the frequency f of the line is

215

15

2

)1)(1046.2(

)1014.4(

)1)(2.10(

ZHz

seV

ZeV

h

Ef

Taking the square root of both sides yields

CCZf

C is a constant ( )

Sample Problem 41-4Sample Problem 41-4

meV

smseV

k

hc 113

815

0min 1055.3

100.35

)/1000.3)(1014.4(

Sample Problem 41-5Sample Problem 41-5

CCZc

CCZc

XX

Cc

O

and

0

127

1

5.143

9.178

XZ

pm

pm

1

1

O

O

C

X

X

C

Z

Z

0.30XZ

41-11 Lasers and Laser Light41-11 Lasers and Laser Light

1. Laser light is highly monochromatic.2. Laser light is highly coherent.3. Laser light is highly directional.

4. Laser light can be sharply focused.

Laser light special characteristics:

41-12 How Lasers Work41-12 How Lasers WorkHere are three processes by which the atom can move from one of these states to the other :

1.Absorption Fig.19 (a)

0EEhf x

DEMO

2.Spontaneous emission. In Fig.41-19(b)

3.Stimulated emission. In Fig.41-19(c)kTEE

xxeNN /)(

00

0NN x 0EEx

To produce laser light,we must have more photons emitted than absorbed.So there requires more atoms in the excited state than in the ground state,as in Fig.41-20b. However,since such a population inversion is not consistent with thermal equilibrium, we must think up clever ways to set up and maintain one.

The Helium-Neon Gas Laser

Four essential energy levels for helium and neon atoms in a helium-neon gas laser.Laser action occurs between levels E2 and E1 of neon when more atoms are at the E2 level than at the E1 level.

metastable state metastable state

metastable state

Four energy systems

Sample Problem 41-6Sample Problem 41-6

(a)

eV

eVJm

smsJ

hchfEEx

26.2

)/1060.1)(10550(

)/1000.3)(1063.6(199

834

0

eVKKeVkT 0259.0300)/1062.8( 5

38

)0259.0/()26.2(/)(0

103.1

/ 0

eVeVkTNNx eeNN x

(b) KKeV

eV

k

EET x 38000

)2)(ln/1062.8(

26.2

)2(ln 50

REVIEW & SUMMARYREVIEW & SUMMARYSome Properties of Atoms

lowhigh EEhf

Angular Momenta and Magnetic Dipole Moments

)1( llL

lZ mL

Lm

eorb 2

an orbital magnetic dipole moment

is the Bohr magneton:

TJm

eeB /274.9

2πm4 1024

866.0)12

1(

2

1)1( ssS

Blzorb m ,

sz mS

Sm

es

Bszs m 2,

Spin and Magnetic Resonance

)(2 localextz BBuhf

No two elections confined to the same trap can have the same set of values for its quantum numbers.

Pauli exclusion principle : For elections,it states that

The values of l are represented by letters:

l = 0 1 2 3 4 5 ……

s p d f g h ……

Guided by the Pauli exclusion principle

Building the Periodic Table

X Rays and the Numbering of the Elements

0min K

hc

Lasers and Laser Light

0EEhf x