The Bohr Model; Wave Mechanics and Orbitals
Attempt to explain H line emission spectrum Why lines? Why the particular pattern of lines? Emission lines suggest quantized E states…
Bohr’s Quantum Model of the Atom
e- occupies only certain quantized energy states
e- orbits the nucleus in a fixed radius circular path
Ee- in the nth state
depends on Coulombic attraction of nucleus(+) and e-(-)
always negative
Bohr’s Model of the H Atom
En = -2.18 x 10-18 J ( )1n2 n = 1,2,3,…
nucleus
First Four e- Energy Levels in Bohr Model
n=1
n=2n=3
n=4
nucleusn=3
n=2
n=1
E
ground state
excited states
n=4
E Levels are spaced increasingly closer together as n
En = -2.18 x 10-18 J ( )1n2
n = 1,2,3,…
Energy of H atom e- in n=1 state?
In J/atom: 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
n=1
n=2n=3
n=4
n=3
n=2
n=1
E
-2.42 x 10-19 J/atom
-5.45 x 10-19 J/atom
-2.18 x 10-18 J/atom
n=4 -1.36 x 10-19 J/atom
First Four e- Energy Levels in Bohr Model
the more - , the lower the En
n=1
n=2n=3
n=4
n=3
n=2
n=1
E
-2.42 x 10-19 J/atom
-5.45 x 10-19 J/atom
-2.18 x 10-18 J/atom
n=4 -1.36 x 10-19 J/atom
What is E for e- transition from n=4 to n=1? (Problem 1)
E = En=1 - En=4 = -2.18 x 10-18J/atom - (-1.36 x 10-19J/atom) = -2.04 x 10-18J/atom
What is of photon released when e- moves from n=4 to n=1? (Problem 1)
Ephoton = |E| = hc/
2.04 x 10-18J/atom = (6.63 x 10-34 J•s/photon)(3.00 x 108 m/s)/
= 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
En calculated by Bohr’s eqn predicts all ’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
h/mu
u = velocity m = mass
Wave-Particle Duality Mathematical expression (deBroglie)
Any particle has a 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
x . p > h/4 x = uncertainty in position p = uncertainty in momentum
p = mu, so p E
Heisenberg’s Uncertainty Principle
As p 0, x becomes large
In other words, If E (or p) of e- is specified, there is large
uncertainty in its position Unlike Bohr Model
x . p > h/4
Wave Mechanics(Schrodinger)
Wave mechanics = deBroglie + Heisenberg + wave eqns from physics
Leads to series of solutions (wavefunctions, ) describing allowed En of the e-
n corresponds to specific En Defines shape/volume (orbital) where e- with En is likely to be
n gives probability of finding e- in a particular space
probability density falls off rapidlyas distance from nucleus increases
Where 90% of thee- density is foundfor the 1s orbital
Ways to Represent Orbitals (1s)
1s
Quantum Numbers
Q# = conditions under which ncan 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)
l = 0 (s orbitals)
l = 1 (p orbitals)
these have different ml values
Orbital Shapes
these have different ml values
l = 2 (d orbitals)
Orbital Shapes
Energy of orbitals in a 1 e- atom
Three quantum numbers (n, l, ml) fully describe each orbital.
The ml distinguishes orbitals of the same type.
n=1
n=2
n=3
E
1s
2s 2p
3p 3d3s
orbitall = 0 l = 1 l = 2
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
ms = -1/2ms = +1/2
Spin Quantum Number, ms
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
E
1s
2s 2p
3p 3d3s
orbital
Filling Order of Orbitals in Multielectron Atoms
The Quantum Periodic Table
l = 0 l = 2l = 1
l = 3
n
1
2
3
4
5
6
7
67
s blockd block
p block
f block
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=2n=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 orbitall = 1: p orbitall = 2: d orbitall = 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?
n(l)1s one of these
2s one2p three
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
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)
l = 0, s orbital l = 1, p orbital l = 2, d orbital
ml 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