Structure of atom

M

ODULE

-I

I

NTRODUCTION

TO

A

TOMIC

S

TRUCTURE

Chemistry

CHY-101

By: Dr. Himanshu Arora

24 September 2012

24 September 2012

Sample footer

2

•

Structure of the Atom,

•

Introduction to Periodic Table,

•

Evolution of Atomic Theory,

•

Thomson’s plum pudding model, Rutherford’s and, Bohr’s models,

Rutherford-Geiger-Marsden Experiment,

•

Planck-Einstein Relationship, Black body radiation, Planck’s constant;

•

Bohr’s postulates;

•

Matter-Energy interactions involving hydrogen atom;

•

quantum states; electron orbital transitions; s, p, d, f, orbitals;

•

electronic configuration based on quantum states;

•

Bohr-Sommerfield Model, Quantum numbers;

•

Balmer and Pfund Series, Rydberg Equation;

•

Stern-Gerlach Experiment;

•

Aufbau Principle; Pauli’s Exclusion Principle; Hund’s Rule;

•

Heisenberg’s Uncertainty Principle;Wave- Particle duality;

•

Schrodinger Equation; Simple Harmonic Oscillator; Particle in a Box.

3

Structure of atom

In 1932 Chadwick discovered the presence of particles having no charge in the atom called neutrons

In 1886, E. Goldstein discovered new radiations in gas discharge and called them canal rays. These rays were positively charged. This later led to the discovery of the positively charged particles called protons in the atom.

In 1900, J.J. Thomson discovered the presence of the negatively charged particles called ELECTRONS in the atom.

3

Subatomic

particles Symbol

Unit

charge Unit mass

T

HE

P

ERIODIC

L

AW

Mendeleev understood the ‘Periodic Law’ which states:

When arranged by increasing atomic number, the chemical elements display

a regular and repeating pattern of chemical and physical properties.

Atoms with similar properties appear in groups or families (vertical columns)

on the periodic table.

They are similar because they all have the same number of valence (outer

shell) electrons, which governs their chemical behavior.

A D

IFFERENT

T

YPE

OF

G

ROUPING

Besides the 4 blocks of the table, there is another way of

classifying element:

Metals

Nonmetals

Metalloids or Semi-metals.

M

ETALS

, N

ONMETALS

, M

ETALLOIDS

There is a zig-zag or

staircase line that divides

the table.

Metals are on the left of

the line, in blue.

Nonmetals are on the

right of the line, in

%

Period Group

Alkali Metal _{Noble Gas}

Halogen

9

Sample footer

Dalton’s Atomic Theory

• Elements are made of extremely small particles

called atoms.

• Atoms of a given element are identical in size, mass,

and other properties; atoms of different elements differ

in size, mass, and other properties.

• Atoms cannot be subdivided, created, or destroyed.

• Atoms of different elements combine in simple

whole-number ratios to form chemical compounds.

• In chemical reactions, atoms are combined, separated,

or rearranged.

24 September 2012

10

Thomson model of an atom

According to Thomson an atom is similar to a Christmas pudding. The

pudding had positive charge and the electrons having negative charge were

like plums on the pudding.

He proposed that

i) An atom consists of a positively charged sphere and the electrons are

embedded in it.

24 September 2012

Sample footer

11

●

Passing an electric current makes a beam appear to move from the

negative to the positive end

Thomson’s Experiment

Voltage source

+

-11

24 September 2012

Sample footer

12

Voltage source

Thomson’s Experiment

●

By adding an electric field he found that the moving pieces were

negative

+

-12

24 September 2012

Sample footer

13

How does the atom emit radiation?

This model soon came into conflict with experiments by Rutherford

Rutherford-Geiger-Marsden gold foil experiment in 1909 disproved the

Plum Pudding Model and showed that instead of a soup of positive

charge, an atom consisted of a small nucleus of strong positive charge

paving the way to Rutherford’s Atomic Model.

24 September 2012

Sample footer

14

R

UTHERFORD

’

S

M

ODEL

Atom consist of two parts:

(a)

Nucleus: Almost the whole mass of the atom is concentrated in this small

region

24 September 2012

15

24 September 2012

Sample footer

16

Rutherford model of atoms

Rutherford’s alpha scattering experiment

Rutherford allowed a beam of fast moving alpha particles ( α –particles)

having positive charge to fall on a thin gold foil. He observed that

:-i) Most of the α – particles passed straight through the gold foil.

ii) Some of the α – particles were slightly deflected by small angles.

24 September 2012

Sample footer

17

Defects of Rutherford’s model of the atom

Any particle in a circular orbit would undergo acceleration and during

acceleration the charged particle would radiate energy. So the revolving

electrons would lose energy and fall into the nucleus and the atom would be

unstable. We know that atoms are stable.

Negatively charged

electron

Positively charged nucleus Very small positively

charged nucleus

Negatively charged electrons in orbits around the nucleus

-+

24 September 2012

Sample footer

18

Bohr’s model of an atom

i) An atom has a positively charged nucleus at its centre and most of the mass of the

atom is in the nucleus.

ii) The electrons revolve around the nucleus in special orbits called discrete orbits.

iii) These orbits are called shells or energy levels and are represented by the letters

K, L, M, N etc. or numbered as 1, 2, 3, 4, etc.

iv) While revolving in the discrete orbits the electrons do not radiate energy.

24 September 2012

19

Distribution of electrons in different shells

The distribution of electrons in the different shells was suggested by Bhor and

Bury. The following are the rules for filling electrons in the different shells.

i) The maximum number of electrons in a shell is given by the formula 2n

where n is the number of the shell 1, 2, 3 etc.

First shell or K shell can have = 2n

= 2 x 1

= 2x1x1 = 2 electrons

Second shell or L shell can have = 2n

= 2 x 2

= 2x2x2 = 8 electrons

Third shell or M shell can have = 2n

=

2 x 3

= 2x3x3 = 18 electrons

Fourth shell or N shell can have = 2n

=

2 x 4

= 2x4x4 = 32 electrons

and so on.

ii) The maximum number of electrons that can be filled in the outermost shell

is 8.

24 September 2012

20

B

’

P

Retained key

features of

Rutherford’s

model.

Concept of

stationary

circular orbits.

Quantization of

angular momentum.

24 September 2012

21

Drawbacks of Bohr’s Model

Explains spectrum of only hydrogen and hydrogen like species

( single electron species)

Does not explain Stark effect and Zeeman effect

Not in accordance with Dual nature and Heisenberg’s uncertainty principle

24 September 2012

Sample footer

22

Bohr-Sommerfeld

Model

Bohr-Sommerfield Model

•

Bohr’s model failed in case of

heavier

elements

where

the

spectral lines observed did not

corroborate

with

the

applied

magnetic field. It was found that

spectral lines are not homogenous

but consists of several convenient

lines.

•

Sommerfield proposed that not

only do electrons travel in certain

orbits but the orbits have different

shapes and the orbits could tilt in

the presence of a magnetic field.

This explained well the splitting of

spectral lines observed for heavier

elements

24 September 2012

Sample footer

23

A

D

B

-S

M

• Sommerfield’s Model predicted the splits in the spectrum. The electrons moving on the two orbits of the same n number but of different shape have a bit different energies which explained the splitting of spectral lines or very closely spaced spectral lines.

• Sommerfield’s Model also showed that orbits don’t have to lie on the same plane and could tilt in the presence of a magnetic field.

24 September 2012

Sample footer

24

Principles and rules to write the

electronic configuration

1. Aufbau principle

2. Pauli’s exclusion principle

A

UFBAU

PRINCIPLE

24 September 2012

25

Sample footer

• In German, "Aufbau" means "construction" (also Aufbau rule or building-up principle), is used to determine the electron configuration of an atom, molecule or ion.

• The principle postulates a hypothetical process in which an atom is "built up" by progressively adding electrons

• According to the principle, electrons fill orbitals starting at the lowest available (possible) energy states before filling higher states (e.g. 1s before 2s).

24 September 2012

26

Modern Physics 26

n+

=1 _{n+ =2}

n+ =3

24 September 2012

Sample footer

27

24 September 2012

Sample footer

28

P

AULI

’

S

E

XCLUSION

P

RINCIPLE

In an atom no two electrons can have the same set of

four quantum numbers.

24 September 2012

Sample footer

29

The Pauli Exclusion Principle

The maximum number of electrons and their orbital

diagrams are:

9/24/2012 29

Sub Shell No. Values Orbitals (-l to +l)

Max No. Electrons

s (l = 0) 1 (0) 2

p (l = 1) 3 (-1, 0, +1) 6

d (l =2) 5 (-2,-1,0,+1,+2) 10

24 September 2012

30

H

’

Among the orbitals of same energy, electrons do not start pairing, until all these orbitals are singly occupied”.

Hund’s rule is also called as the principle of minimum pairing and the principle of maximum multiplicity.

Case - 1

Case - 2

1s2 2s2 2px1 2py1 2pz1

24 September 2012

31

Quantum Numbers

• The principal quantum number

n

governs the electron's energy

and average distance from the nucleus.

• The orbital quantum number

l

determines the magnitude of an

atomic electron's angular momentum.

• The magnetic quantum number

m

specifies the direction of an

atomic electron's angular momentum.

24 September 2012

32

Quantum Numbers

principal quantum number

n

1,2,3,…..

orbital quantum number

l

0,1,2,….n-1

magnetic quantum number

m

-

l

to +

l

for n=2 -2,-1,0,1,2

spin magnetic quantum number +½ or –½ spin

T

-

24 September 2012

33

P

-

24 September 2012

24 September 2012

35

Sample footer

24 September 2012

36

Sample footer

24 September 2012

37

Sample footer

Planck-Einstein Relationship

h

= Planck’s Constant = 6.626 x 10

joule seconds (J s)

24 September 2012

38

Sample footer

Radiant energy is emitted or absorbed discontinuously in the form of

quanta.

24 September 2012

39

Sample footer

The ratio of the energy of a photon of 2000 Å wavelength radiation

to that of 6000 Å wavelength radiation is

(a)

¼ (b) 4

(b)

½ (d) 3

I

P

Hence, answer is (d).

24 September 2012

40

Sample footer

D

EFINITION

OF

A

BLACK

BODY

• A black body is an ideal body which allows the

whole of the incident radiation to pass into itself

(without reflecting the energy) and absorbs

within itself this whole incident radiation

(without passing on the energy).

• This propety is valid for radiation corresponding

to all wavelengths and to all angels of incidence.

Therefore, the black body is an ideal absorber of

incident radaition.

24 September 2012

41

Sample footer

B

LACKBODY

R

ADIATION

24 September 2012

42

•

All objects emit radiant energy.

•

Hotter objects emit more energy than colder objects. The amount of

energy radiated is proportional to the temperature of the object raised to

the fourth power.

•

This is the

Stefan Boltzmann Law

F = σ T

F = flux of energy (W/m

)

T = temperature (K)

σ = 5.67 x 10

W/m

K

(a constant)

•

The hotter the object, the shorter the wavelength (λ) of emitted energy.

This is

Wien’s Law

λ

= hc/5kT

43

Sample footer

43

B

LACK

-B

ODY

R

ADIATION

L

AWS

•

It agrees with experimental

measurements

for

long

wave-lengths.

•

It predicts an energy output

that diverges towards infinity

as

wavelengths

grow

smaller.

•

The failure has become

known as the ultraviolet

catastrophe.

24 September 2012

44

Sample footer

Planck assumed that the radiation in the cavity was emitted (and absorbed) by some sort of “oscillators” contained in the walls. He used Boltzman’s statistical methods to arrive at the following formula:

Planck made two modifications to the classical theory:

• The oscillators (of electromagnetic origin) can only have certain discrete energies determined by

• The oscillators can absorb or emit energy in discrete multiples of the fundamental quantum of energy given by Planck’s radiation law

Planck’s radiation law

24 September 2012

45

Sample footer

S

H

S

1

2

3

4

5

6

7

8

Lyman

Balmer

Paschen

Brackett

Pfund

λ

24 September 2012

46

Sample footer

S

L

H

S

Series n_i n_f Region

24 September 2012

47

Sample footer

W

W

T

R_H=Rydberg’s constant=1.097x 107m-1

where

Balmer Spectral Series

•

The Balmer series is particularly useful in astronomy

because the Balmer lines appear in numerous stellar objects

due to the abundance of hydrogen in the universe, and

therefore are commonly seen and relatively strong

compared to lines from other elements.

24 September 2012

49

Sample footer

I

LLUSTRATIVE

P

ROBLEM

A gas sample of hydrogen atoms are shot with electrons of 12.75 eV energy.What will be the energy of radiation emitted in Balmer series?

24 September 2012

50

Dual Nature

de Broglie

Matter has both wave

and particle nature.

According to Planck’s

quantum theory

According to Einstein’s equation

(1)

(2)

Equating (1) and (2)

H

EISENBERG

’

S

U

NCERTAINITY

P

RINCILE

•

It is impossible to know

both

the position and momentum

exactly, i.e.,

Δx

=0 and

Δp

=0

•

These uncertainties are inherent in the physical world and

have nothing to do with the skill of the observer

•

Because

h

is so small, these uncertainties are not

observable in normal everyday situations

A

NOTHER

C

ONSEQUENCE

OF

H

EISENBERG

’

S

U

NCERTAINTY

P

RINCIPLE

A quantum particle can never be in a state of rest, as this

would mean we know both its position and momentum

precisely

The more accurately we know the energy of a body, the

less accurately we know how long it possessed that

24 September 2012

53

Sample footer

H

’

U

P

The

position

and

momentum

of a subatomic particle cannot be determined

S

-G

E

XPERIMENT

24 September 2012

55

Sample footer

I

LLUSTRATIVE

E

XAMPLE

Heisenberg’s uncertainty principle

⇒ Δx × Δp =

Δx × mΔv =

Δx =

Δx = 1.95 cm Solution

24 September 2012

56

Sample footer

S

Describes the probability of finding an

electron in a given volume element.

Schrodinger’s equation can only be solved exactly for the hydrogen atom.

24 September 2012

57

Sample footer

The Schrodinger Equation

Solving this equation will give us

• the possible energy levels of a system (such as an atom)

• The probability of finding a particle in a particular region of space

24 September 2012

58

Sample footer

P

ARTICLE

IN

A

BOX

Rigid walls Newton’s view

24 September 2012

59

Sample footer

T

HE

PARTICLE

IN

A

BOX

IS

NOT

FREE

,

IT

IS

“

BOUND

”

BY

U(

X

)

Examples: An electron in a long molecule or in a straight wire

“Boundary conditions”:

ψ(x) = 0 at x=0, L and all values of x outside this box, where U(x) = infinite

To be a solution of the SE, ψ(x) has to be continuous everywhere, except where U(x) has an infinite

24 September 2012

60

Sample footer

S

OLUTIONS

TO

THE

S.E.

FOR

THE

PARTICLE

IN

A

BOX

dψ/dx also has to be continuous everywhere, (except where U(x) has an infinite discontinuity) because you need to find d2ψ/dx2

24 September 2012

61

Sample footer

From 0 < x < L, U(x) = 0, so in this region, ψ(x) must satisfy:

24 September 2012

62

Sample footer

You may be tempted to conclude that

, the solution for a free particle, is a possible solution for the bound one too.

WRONG!!!

!

Why not?

24 September 2012

63

Sample footer

S

O

WHAT

IS

THE

SOLUTION

THEN

?

Try the next simplest solution, a superposition of two waves

24 September 2012

64

Sample footer

W

AVE

FUNCTIONS

FOR

THE

PARTICLE

IN

A

BOX

24 September 2012

65

Sample footer

T

HE

ENERGY

OF

A

PARTICLE

IN

A

BOX

CANNOT

BE

ZERO

!

66

Wave function Probability distribution function

67

I

NFINITE

S

QUARE

-W

ELL

P

OTENTIAL

The simplest such system is that of a particle trapped in a box with infinitely

hard walls that the particle cannot penetrate. This potential is called an

infinite square well and is given by

Clearly the wave function must be zero where the potential is infinite.

Where the potential is zero inside the box, the Schrödinger wave

equation becomes

where

.

68

Q

UANTIZATION

Boundary conditions of the potential dictate that the wave function must be zero

at x = 0 and x = L. This yields valid solutions for integer values of n such that kL

= nπ.

The wave function is now

We normalize the wave function

The normalized wave function becomes

69

Q

UANTIZED

E

NERGY

The quantized wave number now becomes

Solving for the energy yields

Note that the energy depends on the integer values of

n

. Hence the energy is

quantized and nonzero.

70

S

IMPLE

H

ARMONIC

O

SCILLATOR

Simple harmonic oscillators describe many physical situations: springs, diatomic molecules and atomic lattices.

Consider the Taylor expansion of a potential function:

Redefining the minimum potential and the zero potential, we have

Substituting this into the wave equation:

71

P

ARABOLIC

P

OTENTIAL

W

ELL

If the lowest energy level is zero, this violates the uncertainty principle.

The wave function solutions are where H_n(x) are Hermite polynomials of order

n.

In contrast to the particle in a box, where the oscillatory wave function is a sinusoidal curve, in this case the oscillatory behavior is due to the polynomial, which dominates at

small x. The exponential tail is provided by the Gaussian function, which dominates at

72

A

NALYSIS

OF

THE

P

ARABOLIC

P

OTENTIAL

W

ELL

The energy levels are given by

The zero point energy is called the Heisenberg limit:

Classically, the probability of finding the mass is greatest at the ends of motion and smallest at the center (that is, proportional to the amount of time the mass spends at each position).