8/6/2019 7 Quantum+Theory+II
http://slidepdf.com/reader/full/7-quantumtheoryii 1/44
Quantum Theory IIChapter 7
8/6/2019 7 Quantum+Theory+II
http://slidepdf.com/reader/full/7-quantumtheoryii 2/44
Bohr¶s First Quantum Model
Bohr developed a model for the hydrogen atom that
correctly predicted the hydrogen line spectrum
Bohr proposed that the electrons in atoms couldonly have very specific amounts of energy
In the Bohr model, electrons travel in circular orbits
Energy of the electron increased as the distancefrom the nucleus increased
8/6/2019 7 Quantum+Theory+II
http://slidepdf.com/reader/full/7-quantumtheoryii 3/44
Bohr Model for Hydrogen
n ± quantum
number that
designates orbit
Energy increases
as n gets bigger
8/6/2019 7 Quantum+Theory+II
http://slidepdf.com/reader/full/7-quantumtheoryii 4/44
Bohr Model of H Atoms
8/6/2019 7 Quantum+Theory+II
http://slidepdf.com/reader/full/7-quantumtheoryii 5/44
Electron TransitionsAn electron must gain or lose the exact amount of
energy corresponding to the difference in energy
between the final and initial orbits
Each line in the emission spectrum corresponds to
the difference in energy between two energy states
8/6/2019 7 Quantum+Theory+II
http://slidepdf.com/reader/full/7-quantumtheoryii 6/44
Absorption and Emission in
theB
ohr Model
8/6/2019 7 Quantum+Theory+II
http://slidepdf.com/reader/full/7-quantumtheoryii 7/44
Energy Levels in Hydrogen
¹ º
¸©ª
¨v!2
18-
nn
1J10-2.18E
8/6/2019 7 Quantum+Theory+II
http://slidepdf.com/reader/full/7-quantumtheoryii 8/44
Predicting the Spectrum of
HydrogenThe wavelengths of lines in the emission spectrum
of hydrogen can be predicted by calculating the
difference in energy between any two states
The Bohr Model predicts these lines very accurately
¹¹ º ¸
©©ª¨ (
2
i
2
f
Hatomn
1
n
1RE
(Eatom = Ephoton = hR
8/6/2019 7 Quantum+Theory+II
http://slidepdf.com/reader/full/7-quantumtheoryii 9/44
Hydrogen Transitions
8/6/2019 7 Quantum+Theory+II
http://slidepdf.com/reader/full/7-quantumtheoryii 10/44
Find the energy change for the transition from
the n = 6 to n = 3 state of a hydrogen atom.
8/6/2019 7 Quantum+Theory+II
http://slidepdf.com/reader/full/7-quantumtheoryii 11/44
¹¹ º
¸©©ª
¨!
2
i
2
f
tn
1
n
1RE
Find the energy change for the transition from
the n = 6 to n = 3 state of a hydrogen atom.
nf = 3
ni = 6
8/6/2019 7 Quantum+Theory+II
http://slidepdf.com/reader/full/7-quantumtheoryii 12/44
¹ º
¸©ª
¨v!
¹ º ¸©
ª¨v!
¹¹ º
¸©©ª
¨!
36
1
9
1J1018.2
61
31J1018.2
n
1
n
1RE
18
22
18
2
i
2
f
t
E = - (2.18 v 10-18 J)( 0.1111 - 0.0278)
= - 0.181 v 10-18 J
= -1.81 v 10-19 J
Find the energy change for the transition from
the n = 6 to n = 3 state of a hydrogen atom.
nf = 3
ni = 6
8/6/2019 7 Quantum+Theory+II
http://slidepdf.com/reader/full/7-quantumtheoryii 13/44
What are the frequency and wavelength for this transition?
8/6/2019 7 Quantum+Theory+II
http://slidepdf.com/reader/full/7-quantumtheoryii 14/44
What are the frequency and wavelength for this transition?
(Eatom = Ephoton
Ephoton = hR
PRc
8/6/2019 7 Quantum+Theory+II
http://slidepdf.com/reader/full/7-quantumtheoryii 15/44
What are the frequency and wavelength for this transition?
(Eatom = Ephoton Ephoton = hR
114
34
19
atomphotons1073.2
sJ1063.6
J1081.1
h
E
h
E
v!
v
v
!
(!!R
To find the wavelength we rearrange the equation,
PR!c
1010.1s1073.2
s1000.3c 6
114
18
v!vv
!R
!P
Convert the wavelength to nm.
nm1010.1m
nm10m1010.1 3
96
v!¹¹ º
¸©©ª
¨v
8/6/2019 7 Quantum+Theory+II
http://slidepdf.com/reader/full/7-quantumtheoryii 16/44
Problems with the Bohr Model
The Bohr Model only works for Hydrogen
Electrons travelling in orbits should lose
energy and be unstable
8/6/2019 7 Quantum+Theory+II
http://slidepdf.com/reader/full/7-quantumtheoryii 17/44
8/6/2019 7 Quantum+Theory+II
http://slidepdf.com/reader/full/7-quantumtheoryii 18/44
Wave Behavior of Particles
de Broglie proposed that particles could have
wave-like properties
mv
h!P
P = wavelength of a particle
h = Planck¶s constant
m = mass
v = velocity
8/6/2019 7 Quantum+Theory+II
http://slidepdf.com/reader/full/7-quantumtheoryii 19/44
if electrons behave like particles, there should only
be two bright spots on the target
Expected Behavior for particles
8/6/2019 7 Quantum+Theory+II
http://slidepdf.com/reader/full/7-quantumtheoryii 20/44
It is observed that electrons present an interference
pattern, demonstrating the behave like waves
Electron Diffraction
8/6/2019 7 Quantum+Theory+II
http://slidepdf.com/reader/full/7-quantumtheoryii 21/44
What is the wavelength of a ball that has a mass of
100 g and is traveling at 100 mph?
8/6/2019 7 Quantum+Theory+II
http://slidepdf.com/reader/full/7-quantumtheoryii 22/44
mv
hparticle !P
s
m4.44
km
m10
mi
km60.1
sec60
min
min60
hr
hr
mi100v
3
!¹¹ º
¸©©ª
¨¹ º
¸©ª
¨¹ º
¸©ª
¨¹ º
¸©ª
¨¹ º
¸©ª
¨!
The velocity must be converted from mph to m/s and the mass from g to kg
kg100.0g1000
kgg100m !¹¹
º
¸©©ª
¨!
m1049.1J
smkg
s
m4.44kg100.0
sJ1063.6
vm
h 342234
v!
¹¹¹¹
º
¸
©©©©
ª
¨
¹¹ º
¸
©©ª
¨
¹ º
¸©ª
¨
�v
!¹ º
¸©ª
¨�!P
Note: the definition, J = kg m2 s-2, was used.
8/6/2019 7 Quantum+Theory+II
http://slidepdf.com/reader/full/7-quantumtheoryii 23/44
Uncertainty Principle
¹ º ¸©
ª¨
Tuv
m
1
4
hvx
Heisenberg stated that was a limit on the accuracy
that could be achieved in measuring position andvelocity
x = position, (x = uncertainty in position
v = velocity, (v = uncertainty in velocityThe more accurately you know the position of a
small particle, the less you know about its speed
8/6/2019 7 Quantum+Theory+II
http://slidepdf.com/reader/full/7-quantumtheoryii 24/44
Determinacy vs. Indeterminacy
In classical physics, particles move in a predictatble
path determined by the particle¶s velocity, position,
and forces acting on it
In Quantum Theory it is impossible to know both theposition and velocity of a particle. Its exact path
cannot be predicted
The best we can do is to describe the probability an
electron will be found in a particular region usingstatistical functions
8/6/2019 7 Quantum+Theory+II
http://slidepdf.com/reader/full/7-quantumtheoryii 25/44
Trajectory vs. Probability
8/6/2019 7 Quantum+Theory+II
http://slidepdf.com/reader/full/7-quantumtheoryii 26/44
Modern Quantum Mechanics
Schrodinger and Heisenberg developed new
versions of quantum mechanics that give correct
results for all atoms and molecules
Energy is quantized as in the Bohr Model
In the new quantum mechanics, it is not possible
to determine the exact location and velocity of the
electrons in an atom
The math of the new quantum mechanics is very
complex and we will only be looking at the results
8/6/2019 7 Quantum+Theory+II
http://slidepdf.com/reader/full/7-quantumtheoryii 27/44
Wave Function, ]
The solutions to the Schroedinger equation is a
function called the wave function, ]
] contains all information that it is possible to
obtain about the electron in the atom
=!= E Schroedinger Equation
8/6/2019 7 Quantum+Theory+II
http://slidepdf.com/reader/full/7-quantumtheoryii 28/44
Quantum Numbers
The solutions to the Schroedinger equation is a function
called the wave function, ]
] is indexed by integers called quantum numbersn - principal quantum number
l - angular momentum quantum number
ml - magnetic quantum number
8/6/2019 7 Quantum+Theory+II
http://slidepdf.com/reader/full/7-quantumtheoryii 29/44
Hydrogen Atom Equation and
Solution
0r 4
ZeE
2
sinr
1sin
sinr
1
r r
r r
1
0
2
22
2
2222
2
2!]¹¹
º
¸©©ª
¨TI
QxN
]xU
¹ º
¸©ª
¨xUx]U
xUx
U¹
º
¸©ª
¨xx]
xx
J
Schrodinger Equation For The Hydrogen Atom
N
V
TU
±À
±¿¾
±°
±¯®
V V
¼¼½
»
¬¬«
¹¹ º
¸©©ª
¨!NU] imm
l
2
1
1l2
1n
l2
2
13
0
nlm e2
1c s
!ml2
!ml1l2Le
!ml2
!1ln
na
Z2),,r (
Solution
Don¶t Panic ± you won¶t see this again
8/6/2019 7 Quantum+Theory+II
http://slidepdf.com/reader/full/7-quantumtheoryii 30/44
Probability & Radial Distribution
Functions
]2 is the probability density - the probability of
finding an electron at a particular point in space
]decreases as you move away from the nucleus
the Radial Distribution function represents the
total probability at a certain distance from the
nucleus
8/6/2019 7 Quantum+Theory+II
http://slidepdf.com/reader/full/7-quantumtheoryii 31/44
Probability Density Function
Graph of the first solution for the hydrogen atom
8/6/2019 7 Quantum+Theory+II
http://slidepdf.com/reader/full/7-quantumtheoryii 32/44
Orbitals
The solutions of the Schrodinger Equation
define shapes that indicate where the electron
is likely to be. These are called orbitals. Unlike
orbits, orbitals are three dimensional
1s orbital 2p orbital
8/6/2019 7 Quantum+Theory+II
http://slidepdf.com/reader/full/7-quantumtheoryii 33/44
Quantum Numbers
n ± integer values > 0 1, 2, 3, 4 «
l - integer values from 0 to (n ± 1)
ml ± integer -l , -l +1, -l + 2, «. +l
If n = 3, l may be 2, 1 or 0
if l = 2 ml may be -2, -1, 0, 1, or 2
8/6/2019 7 Quantum+Theory+II
http://slidepdf.com/reader/full/7-quantumtheoryii 34/44
Principal Quantum Number, n
Characterizes the energy of the electron in a
particular orbital
Corresponds to Bohr¶s energy level
n can be any integer greater than 1
the larger the value of n, the larger the orbital and
the greater the energy
the number in an orbital name corresponds to the n
quantum number
8/6/2019 7 Quantum+Theory+II
http://slidepdf.com/reader/full/7-quantumtheoryii 35/44
The Shapes of Atomic Orbitals
l quantum number determines the shape of the orbital
l = 0 spherical - s orbitals
l = 1 dumbell shape - p orbital
l = 2 cross shape - d orbital
l = 3 8 lobes - f orbital
8/6/2019 7 Quantum+Theory+II
http://slidepdf.com/reader/full/7-quantumtheoryii 36/44
l = 0, the s orbital
1s orbital ± the lowest energy
orbital
spherical
8/6/2019 7 Quantum+Theory+II
http://slidepdf.com/reader/full/7-quantumtheoryii 37/44
As n gets larger, the size of
the orbital increases
8/6/2019 7 Quantum+Theory+II
http://slidepdf.com/reader/full/7-quantumtheoryii 38/44
2s and 3s orbitals
8/6/2019 7 Quantum+Theory+II
http://slidepdf.com/reader/full/7-quantumtheoryii 39/44
l = 1, p orbitals
For each value of n > 1 there are three p orbitals,
one for each of the possible values of ml
ml = -1, 0, +1
each of the three p orbitals point along a differentaxis
px, py, pz
8/6/2019 7 Quantum+Theory+II
http://slidepdf.com/reader/full/7-quantumtheoryii 40/44
p orbitals
8/6/2019 7 Quantum+Theory+II
http://slidepdf.com/reader/full/7-quantumtheoryii 41/44
l = 2, d orbitals
For each value of n > 2 there are five d orbitals
ml = -2, -1, 0, +1, +2
The five d orbitals differ either in shape or
orientation
8/6/2019 7 Quantum+Theory+II
http://slidepdf.com/reader/full/7-quantumtheoryii 42/44
d orbitals
8/6/2019 7 Quantum+Theory+II
http://slidepdf.com/reader/full/7-quantumtheoryii 43/44
l = 3, f orbitals
For each value of n > 3 there are seven f orbitals
ml = -3, -2, -1, 0, +1, +2, +3
The seven f orbitals differ either in shape or
orientation
8/6/2019 7 Quantum+Theory+II
http://slidepdf.com/reader/full/7-quantumtheoryii 44/44
f Orbitals
Top Related