V.Montgomery & R.Smith1 DEVELOPMENT of Quantum Mechanic.

V.Montgomery & R.Smith 1

DEVELOPMENTof

Quantum Mechanic


Wave Nature of Electrons

We know electrons behave as particles




de Broglie’s Equation

A free e- of mass (m) moving with a velocity (v) should have an associated wavelength: = h/mvLinked particle properties (m=mass, and v=velocity) with a wave property ()


Example of de Broglie’s Equation

Calculate the wavelength associated with an e- of mass 9.109x10-28 g traveling at 40.0% the speed of light.


Answer

C=(3.00x108m/s)(.40)=1.2x108m/s = h/mv

= (6.626 x 10-34 J•s) =6.06x10-

12m (9.11x10-31kg)(1.2x108m/s)

Remember 1J = 1(kg)(m)2/s2


Wave-Particle Duality

de Broglie’s suggested that e- has wave-like properties. Thomson’s experiments suggested that e- has particle-like properties measured charge-to-mass ratio

Davisson and Germer experiment

They were the first to prove experimentally that the electrons have both wave and particle nature .


Davisson and Germer experiment

They showed in their experiment that a beam of electrons are diffracted on a crystal, just like X-rays and could measure the wavelength of electrons.



In this experiment, electrons produced “the interference pattern” jut like in Young’s

experiment with light.

Conclusion

Since electrons produce the same pattern with the light, the electrons also have “the wave nature.”




Quantum mechanical model

SchrÖdingerHeisenbergPauliHund


Where are the e- in the atom?

e- have a dual wave-particle natureIf e- act like waves and particles at the same time, where are they in the atom?First consider a theory by German theoretical physicist, Werner Heisenberg.


Heisenberg’s Idea

e- are detected by their interactions with photonsPhotons have about the same energy as e-Any attempt to locate a specific e- with a photon knocks the e- off its courseALWAYS a basic uncertainty in trying to locate an e-


Heisenberg’s Uncertainty Principle

Impossible to determine both the position and the momentum of an e- in an atom simultaneously with great certainty.


SchrÖdinger’s Wave Equation

An equation that treated electrons in atoms as wavesOnly waves of specific energies, and therefore frequencies, provided solutions to the equationQuantization of e- energies was a natural outcome



Solutions are known as wave functionsWave functions give ONLY the probability of finding and e- at a given place around the nucleuse- not in neat orbits, but exist in regions called orbitals



Here is the equationDon’t memorize this or write it downIt is a differential equation, and we need calculus to solve it

-h (ә2 Ψ )+ (ә2Ψ )+( ә2Ψ ) +Vψ =Eψ8(π)2m (әx2) (әy2) (әz2 )

Scary???


Probability likelihoodOrbital wave function; region in space where the probability of finding an electron is highSchrÖdinger’s Wave Equation states that orbitals have quantized energiesBut there are other characteristics to describe orbitals besides energy

Definitions


Quantum Numbers

Definition: specify the properties of atomic orbitals and the properties of electrons in orbitalsThere are four quantum numbersThe first three are results from SchrÖdinger’s Wave Equation


Quantum Numbers (1)

Principal Quantum Number, n


Quantum Numbers

Principal Quantum Number, n Values of n = 1,2,3,… Positive integers only! Indicates the main energy level

occupied by the electron


Quantum Numbers

Principal Quantum Number, n Values of n = 1,2,3,… Describes the energy level, orbital

size As n increases, orbital size increases.


Principle Quantum Number

n = 1

n=2

n=3

n=4n=5n=6

Energy


Principle Quantum Number

More than one e- can have the same n valueThese e- are said to be in the same e- shellThe total number of orbitals that exist in a given shell = n2


Quantum Numbers (2)

Angular momentum quantum number, l


Quantum Numbers

Angular momentum quantum number, l Values of l = n-1, 0


Quantum Numbers

Angular momentum quantum number, l Values of l = n-1, 0 Describes the orbital shape


Quantum Numbers

Angular momentum quantum number, l Values of l = n-1, 0 Describes the orbital shape Indicates the number of sublevel

(subshells)(except for the 1st main energy level,

orbitals of different shapes are known as sublevels or subshells)


Orbital Shapes

For a specific main energy level, the number of orbital shapes possible is equal to n.

Values of l = n-1, 0 Ex. Orbital which n=2, can have one

of two shapes corresponding to l = 0 or l=1

Depending on its value of l, an orbital is assigned a letter.


Orbital Shapes

Angular magnetic quantum number, lIf l = 0, then the orbital is labeled s.s is spherical.


Orbital Shapes

If l = 1, then the orbital is labeled p.“dumbbell” shape


Orbital Shapes

If l = 2, the orbital is labeled d.“double dumbbell” or four-leaf clover


Orbital Shapes

If l = 3, then the orbital is labeled f.


Energy Level and Orbitals

n=1, only s orbitalsn=2, s and p orbitalsn=3, s, p, and d orbitalsn=4, s,p,d and f orbitals

Remember: l = n-1


Atomic Orbitals

Atomic Orbitals are designated by the principal quantum number followed by letter of their subshell Ex. 1s = s orbital in 1st main energy

level Ex. 4d = d sublevel in 4th main energy

level


Quantum Numbers (3)

Magnetic Quantum Number, ml


Quantum Numbers


Values of ml = +l…0…-l


Quantum Numbers


Values of ml = +l…0…-l Describes the orientation of the

orbital Atomic orbitals can have the same

shape but different orientations


Magnetic Quantum Number

s orbitals are spherical, only one orientation, so m=0p orbitals, 3-D orientation, so m= -1, 0 or 1 (x, y, z)d orbitals, 5 orientations, m= -2,-1, 0, 1 or 2


Quantum Numbers (4)

Electron Spin Quantum Number,ms


Quantum Numbers

Electron Spin Quantum Number,ms Values of ms = +1/2 or –1/2 e- spin in only 1 or 2 directions A single orbital can hold a maximum

of 2 e-, which must have opposite spins


Electron Configurations

Electron Configurations: arragenment of e- in an atomThere is a distinct electron configuration for each atom

There are 3 rules to writing electron configurations:


Pauli Exclusion Principle

No 2 e- in an atom can have the same set of four quantum numbers (n, l, ml, ms ). Therefore, no atomic orbital can contain more than 2 e-.


Aufbau Principle

Aufbau Principle: an e- occupies the lowest energy orbital that can receive it.Aufbau order:


Hund’s Rule

Hund’s Rule: orbitals of equal energy are each occupied by one e- before any orbital is occupied by a second e-, and all e- in singly occupied orbitals must have the same spin


Electron Configuration

The total of the superscripts must equal the atomic number (number of electrons) of that atom.The last symbol listed is the symbol for the differentiating electron.


Differentiating Electron

The differentiating electron is the electron that is added which makes the configuration different from that of the preceding element.The “last” electron.H 1s1

He 1s2

Li 1s2, 2s1

Be 1s2, 2s2

B 1s2, 2s2, 2p1


Orbital Diagrams

These diagrams are based on the electron configuration.In orbital diagrams: Each orbital (the space in an atom

that will hold a pair of electrons) is shown.

The opposite spins of the electron pair is indicated.


Orbital Diagram Rules1. Represent each electron by an arrow2. The direction of the arrow represents

the electron spin3. Draw an up arrow to show the first

electron in each orbital.4. Hund’s Rule: Distribute the electrons

among the orbitals within sublevels so as to give the most unshared pairs.

Put one electron in each orbital of a sublevel before the second electron appears.Half filled sublevels are more stable than partially full sublevels.


Orbital Diagram Examples

H _1s

Li _1s 2s

B __ __1s 2s 2p

N _1s 2s 2p


Dot Diagram of Valence Electrons

When two atom collide, and a reaction takes place, only the outer electrons interact.These outer electrons are referred to as the valence electrons.Because of the overlaying of the sublevels in the larger atoms, there are never more than eight valence electrons.


Rules for Dot Diagrams

:Xy:. .. .

S sublevel electrons

Px orbital

Py orbital

Pz orbital


Rules for Dot Diagrams

Remember: the maximum number of valence electrons is 8.Only s and p sublevel electrons will ever be valence electrons.Put the dots that represent the s and p electrons around the symbol.Use the same rule (Hund’s rule) as you fill the designated orbitals.


Examples of Dot Diagrams

H

He

Li

Be


Examples of Dot Diagrams

C

N

O

Xe


Summary

Both dot diagrams and orbital diagrams will be use full to use when we begin our study of atomic bonding.We have been dealing with valence electrons since our initial studies of the ions.The number of valence electrons can be determined by reading the column number. Al = 3 valence electrons Br = 7 valence electrons All transitions metals have 2 valence electrons.

V.Montgomery & R.Smith1 DEVELOPMENT of Quantum Mechanic.

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