7 Quantum+Theory+II

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Quantum Theory IIChapter 7



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



Bohr Model for Hydrogen

n ± quantum

number that

designates orbit

Energy increases

as n gets bigger



Bohr Model of H Atoms



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



Absorption and Emission in

theB

ohr Model



Energy Levels in Hydrogen

¹ º

¸©ª

¨v!2

18-

nn

1J10-2.18E



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



Hydrogen Transitions



Find the energy change for the transition from

the n = 6 to n = 3 state of a hydrogen atom.



¹¹ º

¸©©ª

¨!

2

i

2

f

tn

1

n

1RE



nf = 3

ni = 6



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



nf = 3

ni = 6



What are the frequency and wavelength for this transition?




(Eatom = Ephoton

Ephoton = hR

PRc




(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



Problems with the Bohr Model

The Bohr Model only works for Hydrogen

Electrons travelling in orbits should lose

energy and be unstable



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



if electrons behave like particles, there should only

be two bright spots on the target

Expected Behavior for particles



It is observed that electrons present an interference

pattern, demonstrating the behave like waves

Electron Diffraction



What is the wavelength of a ball that has a mass of

100 g and is traveling at 100 mph?



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.



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



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



Trajectory vs. Probability



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



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



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



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



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



Probability Density Function

Graph of the first solution for the hydrogen atom



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



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



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



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



l = 0, the s orbital

1s orbital ± the lowest energy

orbital

spherical



As n gets larger, the size of

the orbital increases



2s and 3s orbitals



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



p orbitals



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



d orbitals



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



f Orbitals

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