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