l - Bryn Mawr College

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The Bohr Model; Wave
Mechanics and Orbitals
Bohr’s Quantum Model of the Atom

Attempt to explain H line emission spectrum



Why lines?
Why the particular pattern of lines?
Emission lines suggest quantized E states…
Bohr’s Model of the H Atom

e- occupies only certain quantized energy states

e- orbits the nucleus in a fixed radius circular path
nucleus

Ee- in the nth state
En = -2.18 x
10-18
1
J( 2
n
)
n = 1,2,3,…

depends on Coulombic attraction of nucleus(+) and e-(-)

always negative
First Four e- Energy Levels in
Bohr Model
n=4
n=3
nucleus
excited states
n=2
E
n=1
n=2
n=3
n=1 ground state
n=4
E Levels are spaced increasingly closer together as n
Energy of H atom e- in n=1 state?
En = -2.18 x

1
J( 2
n
)
n = 1,2,3,…
In J/atom:


10-18
En=1 = -2.18 x 10-18 J/(12) = -2.18 x 10-18 J/atom
In J/mole:

En=1 = -2.18 x 10-18 J/atom(6.02 x 1023 atoms/mol)(1kJ/1000J) = -1310kJ/mol
First Four e- Energy Levels in
Bohr Model
n=4
n=3
-1.36 x 10-19 J/atom
-2.42 x 10-19 J/atom
n=2
-5.45 x 10-19 J/atom
E
n=1
n=2
n=3
n=4
n=1
-2.18 x 10-18 J/atom
the more - , the lower the En
What is DE for e- transition from n=4
to n=1? (Problem 1)
n=4
n=3
-1.36 x 10-19 J/atom
-2.42 x 10-19 J/atom
n=2
-5.45 x 10-19 J/atom
E
n=1
n=2
n=3
n=1
-2.18 x 10-18 J/atom
n=4
DE = En=1 - En=4 = -2.18 x 10-18J/atom - (-1.36 x 10-19J/atom) = -2.04 x 10-18J/atom
What is l of photon released when emoves from n=4 to n=1? (Problem 1)
Ephoton = |DE| = hc/l
2.04 x 10-18J/atom =
(6.63 x 10-34 J•s/photon)(3.00 x 108 m/s)/ l
l = 9.75 x 10-8 m or 97.5 nm
A line at 97.5 nm (UV region) is
observed in H emission
spectrum.
Bohr Model Explains H Emission
Spectrum
DEn calculated by Bohr’s eqn predicts all l’s
(lines).
Quantum theory explains the behavior of e- in H.
But, the model fails when applied to any
multielectron atom or ion.
Wave Mechanics
Quantum, Part II
Wave Mechanics


Incorporates Planck’s quantum theory
 But very different from Bohr Model
Important ideas
 Wave-particle duality
 Heisenberg’s uncertainty principle
Wave-Particle Duality

e- can have both particle and wave properties
 Particle: e- has mass
 Wave: e- can be diffracted like light waves
e- or light wave
wave split into pattern
slit
Wave-Particle Duality

Mathematical expression (deBroglie)
l = h/mu
u = velocity m = mass

Any particle has a l but wavelike properties are
observed only for very small mass particles
Heisenberg’s Uncertainty Principle

Cannot simultaneously measure position (x) and
momentum (p) of a small particle
Dx . Dp > h/4p
Dx = uncertainty in position
Dp = uncertainty in momentum
p = mu, so p a E
Heisenberg’s Uncertainty Principle
Dx . Dp > h/4p


As Dp  0, Dx becomes large
In other words,
 If E (or p) of e- is specified, there is large
uncertainty in its position
 Unlike Bohr Model
Wave Mechanics(Schrodinger)

Wave mechanics =
deBroglie + Heisenberg + wave eqns from physics

Leads to series of solutions (wavefunctions, Y)
describing allowed En of the e
Yn corresponds to specific En


Defines shape/volume (orbital) where e- with En is likely to be
(Yn )2 gives probability of finding e- in a particular space
Ways to Represent Orbitals (1s)
Where 90% of the
e- density is found
for the 1s orbital
(Y1s )2
probability density falls off rapidly
as distance from nucleus increases
Quantum Numbers

Q# = conditions under which Yn can be
solved

Bohr Model uses a single Q# (n) to describe
an orbit

Wave mechanics uses three Q# (n, l, ml) to
describe an orbital
Three Q#s Act As Orbital ‘Zip Code’

n = e- shell (principal E level)

l = e- subshell or orbital type (shape)

ml = particular orbital within the subshell (orientation)
Orbital Shapes
l = 0 (s orbitals)
l = 1 (p orbitals)
these have different ml values
Orbital Shapes
l = 2 (d orbitals)
these have different ml values
Energy of orbitals in a 1 e- atom
l=0
l=1
l=2
orbital
n=3
3s
3p
3d
n=2
E
2s
2p
n=1
1s
Three quantum numbers (n, l, ml)
fully describe each orbital.
The ml distinguishes orbitals of
the same type.
Spin Quantum Number, ms

In any sample of atoms, some e- interact one way with magnetic
field and others interact another way.

Behavior explained by assuming e- is a spinning charge
Spin Quantum Number, ms
ms = +1/2
ms = -1/2
Each orbital (described by n, l, ml) can contain a maximum of
two e-, each with a different spin.
Each e- is described by four quantum numbers (n, l, ml , ms).
Energy of orbitals in a 1 e- atom
orbital
E
3s
3p
2s
2p
1s
3d
Filling Order of Orbitals in
Multielectron Atoms
The Quantum Periodic Table
s block
l =0
n
=1
p lblock
=2
dl block
1
2
3
4
5
6
7
6
7
f lblock
=3
More About Orbitals and
Quantum Numbers
n = principal Q#


n = 1,2,3,…
Two or more e- may have same n value



e- are in the same shell
n =1: e- in 1st shell; n = 2: e- in 2nd shell; ...
Defines orbital E and diameter
n=1
n=2
n=3
l = angular momentum or azimuthal Q#




l = 0, 1, 2, 3, … (n-1)
Defines orbital shape
# possible values determines how many orbital
types (subshells) are present
Values of l are usually coded
l = 0: s orbital
l = 1: p orbital
l = 2: d orbital
l = 3: f orbital
A subshell l = 1 is a ‘p subshell’
An orbital in that subshell is a ‘p orbital.’
ml = magnetic Q#

ml = +l to -l

Describes orbital orientation
# possible ml values for a particular l tells how
many orbitals of type l are in that subshell

If l = 2
then ml = +2, +1, 0, -1, -2
So there are five orbitals in the d (l=2) subshell
Problem: What orbitals are present
in n=1 level? In the n=2 level?

If n = 1




l = 0 (one orbital type, s orbital)
ml = 0 (one orbital of this type)
Orbital labeled 1s
If n = 2

l = 0 or 1 (two orbital types, s and p)



for l = 0, ml = 0 (one s orbital)
for l = 1, ml = -1, 0, +1 (three p orbitals)
Orbitals labeled 2s and 2p
n(l)
1s one of these
2s one
2p three
Problem: What orbitals are present
in n=3 level?

If n = 3

l = 0, 1, or 2 (three types of orbitals, s, p,and d)




ml




l = 0, s orbital
l = 1, p orbital
l = 2, d orbital
for l = 0, ml = 0 (one s orbital)
for l = 1, ml = -1, 0, +1 (three p orbitals)
for l = 2, ml = -2, -1, 0, +1, +2 (five d orbitals)
Orbitals labeled 3s, 3p, and 3d
n(l)
3s one of these
3p three
3d five
Problem: What orbitals are in the
n=4 level?

Solution




One s orbital
Three p orbitals
Five d orbitals
Seven f orbitals
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