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Chapter41
All 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 Atom
The Mass of the Atom
The absolute atomic mass matom can therefore be obtained by
measuring Avogadro’s number:
Mass of 1 moleof thesubstance
Mass of an atom
NA
The present best value for NA is
N A  (6.022045 0.000005) 1023 mole1
With the value of NA ,we can write:
matom 
Arel , 12 c
NA
gram
The Size of the Atom
Determining the Atomic Size from the Covolume
The Van der Waals equation for one mole of a real gas states
( P  a / V 2 )(V  b)  RT
The quantity b is equal to the fourfold volume of the particles
b  4
4 3
r  NA
3
Determining the Atomic Size from the Gases movement
1

4 2r 2 N
is the distance,the N is again the particle number density
41-2 Some Properties of Atoms
Basic 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 Emit and Absorb Light
The light is emitted or absorbed as a photon with energy:
hf  Ehigh  Elow
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)
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 Spin
To 41-9
Table 41-1 Electron States for an Atom
Quantum
Number
symbol
Principal
n
Orbital
Orbital magnetic
Spin
Allowed
Values
1,2,3……
0,1,2……(n-1)
0,±1,±2, ……±
s
Spin magnetic
Related to
Distance from the nucleus
Orbital angular momentum
Orbital angular momentum(z component)
Spin angular momentum
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.
There are 2n2 states in a shell.
There are 2(2 +1)states in a subshell
All states in a subshell have the same energy.
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
Magnetic Dipole Moments
Orbital Angular Momentum and Magnetism
The magnitude L of the orbital angular momentum
of an electron in an atom is quantized;it can have
only certain values:
L  l (l  1)
an orbital magnetic dipole moment

 orb
e 

L
2m
The “ - ” in this relation means that
is directed opposite
The magnitude of
 orb
must also be quantized and give by
e

l (l  1)
2m
The components
of the orbital magnetic dipole
moment are quantized and give by
orb , z  ml  B
is the Bohr magneton:
B 
e
e
 24

 9.274 10 J / T
4 πm 2m
The components Lz of the angular
momentum are also quantized,and
they are given by
LZ  ml 
we can extend that visual aide by
saying that makes a certain
angle with the z axis:
Lz
cos  
L
We can call the semi-classical angle
between vector and the z axis.
Spin Angular Momentum and Spin
Magnetic Dipole Moment
The magnitude S of the spin angular momentum of any
electron,whether free or trapped,has the single value given
by
1 1
S  s( s  1) 
  0.866
(  1)
2 2
The spin magnetic dipole moment
the spin angular momentum
,which is related to

e 
s   S
m
The magnitude of
also be quantized and give by
e
s 
m
s ( s  1)
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
S z  ms 
The electron is said to be spin up
The electron is said to be spin down
The components
are also quantized
of the spin magnetic dipole moment
 s , z  2msuB
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







J  ( L1  L 2   Lz )  ( S1  S 2   S z )
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 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
atoms. The beam passes between the poles of an electromagnet and
then lands on a glass detector plate where it forms a silver deposit.
The Experimental surprise
Stern 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.
When the field was
Fig.41-9
turned off,the beam
Results of a
was,of course,
modern
repetition of
undeflected and the
the SternGerlach
detector recorded the
experiment.
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.
The Meaning of the Results
It is not the magnetic deflecting force



F  q v B
The potential energy U of the dipole in the magnetic field
 
U    B
In Fig.41-8,the positive direction of the z axis and the
direction of
are vertically upward.
U   z B
Using (F=-dU/dx) for the z axis
dU
dB
FZ  
 z
dz
dz
According to
 s , z  2ms  B
The component are for quantum numbers ms=±1/2.
substituting into Eq.41-13 gives us
 s,z
1
 2( )  B    B
2
and
 s,z
1
 2( )  B    B
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
dB
Fz    B ( ) and
dz
dB
Fz    B ( )
dz
which result in the two spots of silver on the glass.
Sample Problem 41-1
FZ  B(dB / dz)
Fz  B (dB / dz )
az 

M
M
1 2
1  B (dB / dz ) 2
d  v0 z t  a z t  0  (
)t
2
2
M
1  B (dB / dz) w 2
d (
)( )
2
M
v
 (9.27 1024 J / T )(1.4 103 T / m)
(3.5 102 m) 2

2  (1.8 1025 kg )(750m / s) 2
 7.8 105 m  0.08m m
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 Resonance
The f required for the spin-flipping
hf  2 z B
a condition called magnetic resonance.
hf  2 z ( Bext  Blocal )
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 41-2
2 z B 2  (1.411026 J / T )(1.80T )
f 

h
6.631034 J  s
 7.66107 Hz  76.6MHz

c 3.0010 m / s
B
 
 3.92m
7
f
7.6610 Hz
8
41-7 The Pauli Exclusion Principle
Pauli 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
Rectangular Traps
1.One –dimensional trap.
width L
quantum number n
quantum number
2.Rectangular corral.
widths Lx,Ly
quantum numbers nx,ny
quantum number
3.Rectangular box.
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 7electron system to its first excited state.
(f) The system’s energy-level diagram,for the lowest three
total energies of the system.
Table 41-2 Ground-State Configuration
nx
2
2
2
1
1
1
1
ny
2
1
1
2
2
1
1
ms
Energy*
8
5
5
5
5
2
2
Total 32
*In multiples of
41-9 Building the Periodic Table
The values of l are represented by letters:
l = 0 1 2 3 4 5 ……
s p d f g h ……
To table 41-1
Guided by the Pauli exclusion principle
41-10 X Rays and the Numbering of
the 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
K 0  hf 
hc
min
hc
min 
K0
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
me4 1
13.6eV
En   2 2  
n2
8 0 h n
for n = 1,2,3,……
We can approximate the effective
energy of the atom by energy of the
atom by replacing the factor
in Eq.41-24 with
or
, That gives us
(13.6eV )(Z  1) 2
En  
n2
We may write the energy change as
E  E2  E1
 (13.6eV )(Z  1) 2  (13.6eV )(Z  1) 2


2
2
12
 (10.2eV )(Z  1) 2
Then the frequency f of the
line is
E
(10.2eV )(Z  1) 2
f 

15
h
(4.1410 eV  s)
 (2.4610 Hz)(Z  1)
15
2
Taking the square root of both sides yields
f  CZ  C
C is a constant (
)
Sample Problem 41-4
hc (4.141015 eV  s)(3.00108 m / s)
11
min  

3
.
55

10
m
3
k0
35.0 10 eV
Sample Problem 41-5
c
c
 CZ CO  C
and
0
C
Z X 1

X
ZC 1
O
O
Z X  30.0
c
X
 CZ X  C
178.9 pm Z X  1

143.5 pm 27  1
41-11 Lasers and Laser Light
Laser light special characteristics:
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.
41-12 How Lasers Work
Here are three processes by which
the atom can move from one of these
states to the other :
1.Absorption Fig.19 (a)
hf  Ex  E0
DEMO
2.Spontaneous emission.
3.Stimulated emission.
N x  N0 e
Ex  E0
In Fig.41-19(b)
In Fig.41-19(c)
( Ex  E0 ) / kT
N x  N0
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 heliumneon 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 41-6
(a)
E x  E0  hf 
hc

(6.631034 J  s )(3.00108 m / s )

(550109 m)(1.601019 J / eV )
 2.26eV
kT  (8.62105 eV / K )  300K  0.0259eV
N x / N 0  e ( N x  N0 ) / kT  e ( 2.26eV ) /(0.0259 eV )
 1.3 1038
(b)
T
Ex  E0
2.26eV

 38000K
5
k (ln 2) (8.6210 eV / K )(ln 2)
REVIEW & SUMMARY
Some Properties of Atoms
hf  Ehigh  Elow
Angular Momenta and Magnetic Dipole Moments
L  l (l  1)
LZ  ml 
an orbital magnetic dipole moment

 orb
e 

L
2m
orb , z  ml  B
is the Bohr magneton:
e
e
 24
B 

 9.274 10 J / T
4 πm 2m
  0.866
1 1
S  s( s  1) 
(  1)
2 2
S z  ms 

s
e 
 
S
m
 s , z  2ms  B
Spin and Magnetic Resonance
hf  2uz ( Bext  Blocal )
Pauli 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.
Building the Periodic Table
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
X Rays and the Numbering of the Elements
hc
min 
K0
Lasers and Laser Light
hf  Ex  E0