genchem_lec02

BITS Pilani, Pilani Campus

CHEM F111 : General Chemistry

Lecture 2

BITS Pilani Pilani Campus

1

Summary (Lecture 1)

2

2

Black body radiation (1860):

Planck’s Explanation (1901-1909):

Energy quantized.

Photoelectric effect (1887):

Einstein Explanation (1905):

Particle character of light.

Energy Quantized

Classical Mechanics: Failures to explain


3

Atomic Spectra: The phenomena

White light gets separated into its component colors after passing

through the prism.

When such a light is passed through hot gas/element, some

wavelength of light gets absorbed, while rest when analyzed

through spectroscope shows a series of characteristics lines at

definite wavelength: LINE SPECTRUM


•Each element has its own

characteristics spectrum.

•Complicated spectra of

elements

•Comparatively simple spectra

for hydrogen.

Positions of lines of light, wavenumber,

n = 3, 4, 5……. & R: some constant

4

Mathematical explanation of Line Spectra of

Hydrogen Atom

Balmer (1885): Explained the lines in the visible region of

Hydrogen.

22

1

2

11~

nR

Rydberg:Generalized Formula

n1 and n2 (n2>n1) are positive

integers. (RH = 109737.315 cm-1)

Pfund (5)

Lyman (1) Paschen (3)

Brackett (4)

2

2

2

1

111~

nnRH


Bohr’s Explanation: Model of Hydrogen-

like Atom

5

• Electron (mass m and charge –e) moving in circular orbit of radius r

about nucleus of charge Ze;

mev2/r = Ze2/4πε0r

2

• Only those orbits are stable for which the magnitude of

L (orbital angular momentum of the electron) is quantized.

L = mevr = nh/2π =nħ, n=1,2,3,…

quantization of angular momentum

2

22

0

em

nhr

e

Putting n = 1; r = a0 = 0.529 x 10-10 m;

First Bohr radius BITS Pilani, Pilani Campus

(Electron must radiate energy, fall into the nucleus) [Classical Concept]

• Bohr postulated that an electron can revolve around the nucleus only

in certain stable orbits without emitting any electromagnetic radiation.

Bohr’s Explanation on Line Spectra of

Hydrogen Atom

6

• Transition between orbits are

allowed, but only when an

electron absorbs or emit a

photon of energy equal to the

difference between the energy

of the orbits.

hc

EE

hE

121~

2

2

2

1

32

0

4

2

2

2

1

22

0

4

12

222

0

4

0

22

11

8

~

11

8

842

v1

nnch

emv

nnh

emEEE

nh

em

r

emVKEE

e

e

een

The equation corresponds with the experimentally observed result.


ch

emR e

H 32

0

4

8

Bohr’s Model of Hydrogen-like Atom

7

Bohr Model can make some approximate

predictions about the emission spectra for

atoms with a single outer-shell electron.

eVn

Z

nh

emZE e

n 2

2

222

0

42 6.13

8

For example:

Be3+: Z = 4: single electron system


Limitation: Bohr Model cannot be applied to elements with more than

one electron

Orbiting electrons existed in orbits

possessing discrete quantized energies

eVnnh

emHydrogenE e

n 2222

0

4 6.13

8)(

Wave Character of Particles

8

Davisson-Germer Experiment (1920-1925)

Scattering of electron beam

after interaction with single

crystal of nickel.

Series of bright and dark

fringes

• If light (radiation) can be viewed as a collection of particles,

then can entities (particles) also be seen as waves ?


9

de Broglie (1924): Wave-particle duality

• Just as light exhibits both ‘wave-like’ (diffraction), and

‘particle-like’ characteristics, so should all material objects.

• A particle with linear momentum p, has an associated

‘matter-wave’ of wave length ( and mathematical form

sin(2πx/).

E = pc (particle) = hc/ (wave) p = h/

•The Joint wave-particle character of matter and radiations is

called wave-particle duality.


For light (photon)

10

Wave-particle duality

• Macroscopic objects are so massive that the de Broglie

wavelengths are immeasurably small, even if they travelling

slow.

• The wavelength of the matter wave associated with a

particle should decrease as the particles speed increases

Lower p has a longer ; Higher p has a shorter

.


11

• A wave of wavelength

- corresponds to a particle with a constant p

-wave is spread through out the space.

-Precise location determination: Impossible.

Uncertainty: A natural consequence

of wave-particle duality

• A particle with fixed position (x)

-associated with wave of the form sin(2πx/).

-Localized wavefunction(≠0) at x and 0 elsewhere.


12

• A sharply localized

wavefunction can be generated by

superposition of large no. of

wavefunctions. (resultant wave:

spread of waves of different

wavelengths)

-Precise momentum determination: Impossible

Uncertainty or Indeterminacy!


• Superposition of many waves

corresponds to superposition of

many different linear momenta.

(which of these wavelength should

fit into de broglie equation to give

momentum)

Heisenberg Uncertainty Principle

13

• It is impossible to predict, measure, or know both the exact

position of an object and its exact momentum at the same

time. In fact, an object does not have an exact position and

momentum at the same time! (Contrary to classical Mechanics)

px x ħ/2 px : uncertainty in linear momentum

x : uncertainty in position


(a) (x: large) then (px : small)

(b) (x: small) then (px : large)

• Complementary variables, increase in the precision of one

possible only at the cost of a loss of precision in the other.

Trajectories not defined precisely.

14

Quantum Mechanics (QM): Discovery

Black Body Radiation

&

Line spectra of atoms

Atoms/molecules

exist with only

certain energies

Quantum Mechanics:

A theoretical Science:

Davisson-Germer Exp.

&

Photoelectric Effect

Wave exhibit particle like

properties &

Particle exhibit wave like

properties


De broglie duality

&

uncertainty principle

Particle moving in precise

paths cannot have fixed

speeds.

• A particle is spread through space like wave. To describe this

distribution, a mathematical equation is required.

Wavefunction

15

Wavefunction: /ψ (PSI/psi)

Describing wave behavior of matter (any system) using an

expression of a wave: wavefunction [ ]

In QM, a particle is not localized. Approximately, wavefunction is a

blurred version of path.


Wavefunction determines the probability distribution of finding

the particle at a point at any instance.

At any particular instance of time: should depend upon (x,y,z,t)

(state of a system): time dependent wavefunction

(x,y,z,t) = ψ(x,y,z) f(t) ψ is time-independent wavefunction, (wavefunction for a stationary

State)

16

Wavefunction: Born Interpretation

: Probability Amplitude 2: Probability Density

It is difficult to establish that a particular particle (electron) is in a

particular space at a particular time. Rather, over a period of time, the

particle has a certain probability (P) of being in a certain region

(between two points a & b).

• Probability that the particle is

located in the infinitesimal element

of volume dV about the given point,

at time t = 2dV

P = * dV = 2 dV


Total Probability between a & b =

Characteristics of Wavefunction:

17

• all space 2 dτ must be finite, ie., must be normalizable.

(Born Interpretation) all space

2 dτ = 1 or 2 dτ = 1


Only those wavefunctions which follow the following criteria

are considered acceptable wavefunctions:

• and d/dx must be continuous

• must be single valued & Bounded (Born Interpretation)

Normalization

18

Question: The wavefunction (x)= sin (πx/2) for a system exist.

If the region of interest if from x= 0 to x = 1. Normalize the

function.

Solution: (x)(unnormalizable)

Such that * = N2 = 1

= N2 [1/2] = 1 ; N =

Thus, (x)= sin (πx/2) is now Normalizable

N (x) (Normalizable)


If the given wavefunction is not normalize: we could make it

normalize by multiplying it by any constant and integrating

over its limits and determine the value of Normalization

constant.

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• A wavefunction should be defined in such a way that various

observable properties of the system can be determined.

Observables & Operators

Every observable in quantum mechanics is represented by an operator which is used to

obtain physical information about the observable from the state function. For an

observable that is represented in classical physics by a function Q(x,p), the corresponding

operator is ),( pxQ

.

Observable Operator

Position x

Momentum

xip

Energy )(

2)(

2 2

222

xVxm

xVm

pE

Each individual property (x, p, E): An observable.

For every observable: an Operator exist (Mathematical instruction)

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Multiplication operator: M (2,3) = 2 x 3 = 6

Differentiation operator: D [F(x)] = d/dx (9x2+8x) = 18x+8

Momentum of a system = p [F(x)] = [F(x)]

System Information from

observables

For determining values of observables: perform mathematical

operation (operator) on the wavefunction.

Observable operator acts on a function to give information

about the property

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Eigen value Function

21

Special type of operator-function combination:

Operator when operated on function gives some constant(s)

times the original function: Eigen function of operator

A Φ = K Φ

Φ is the Eigen function of the operator.

Let the function defining a system be: Φ Let the operator corresponding to a observable be : A

K = constant = Eigen Value

Not all functions are Eigen functions of all operators.

Only those physical properties are determined by function if

that wavefunction is an eigen function of the corresponding

operator.

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