Chemistry Classical Mechanics chem

8/13/2019 Chemistry Classical Mechanics chem

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QUANTUM THEORY

I. Review of Classical Mechanics

Re: Atkins (8th Ed.) Appendix 3

Atkins (9th

Ed.) Chapter 7 -Further information

Classical Mechanics

Cannot explain some well-knownphenomena observed experimentally nearthe end of the 19th Century/beginning of the20th Century:

Blackbody radiation

Photoelectric effect

Heat capacities of solids

Compton effect

Electron scattering

Atomic and molecular spectra


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Classical Mechanics

Cannot account for the behavior ofvery small particles (atoms and

molecules).

Quantum theory: a technique ofcalculation at the microscopic level,

for the study of the behavior ofindividual atoms and molecules.

Central Ideas in ClassicalMechanics

Some key expressions

Total energy:

Linear momentum:

Speed:

VEEEE KPK ++=

mvp =

m

p

dt

dxv ==

m

pvmEK

22

1 2

2=

( ) 2/12/1

)(22

=

m

VE

m

mEv K

As a function ofenergy:


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Newtons second law:

Acceleration:

amdt

xdm

dt

mvd

dt

dpF ===

2

2)(

2

2

dt

xda =

Classical Mechanics is based on Newtons

laws of motion, and predicts the responseof objects to FORCES.

Rotational analogy

angularvelocity

r radius

angle swept inradians

Length of the arc

r

r


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Translation Rotation

Mass Moment of inertia

Velocity Angular velocity

Linear momentum Angular momentum

Kinetic energy

Force Torque

2mrI=

rv/=

mvp =

v

m

IrmmvrprJ ==== 2

m

p

mvTEK 22

1

)(

22

== I

JIRE

K 22

1)(

22==

dt

dpF=

dt

dJT =

Free particle Total energy: E = EK+ EP EK+ V

For the free particle: V= const (say 0)

E = EK=const

1st order ODE:

Solution:

2/12

=

m

E

dt

dx Const

( )

00

00

02/1

)(

)(

/2)(

xtvtx

xtm

ptx

xtmEtx

+=

+=

+= Uniform

motion withconstant initialvelocity v0


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Harmonic Oscillator

Particle disturbedfrom its rest positionand experiences a

RESTORING FORCEwhose magnitude isdirectly proportionalto the displacement. x = r r

eq

req

r

xkF =

k force constant (in N/m)

Fr

Properties

Equation of motion:

Potential energy:

Total energy:

xkdt

rdm =

2

2

2

00 2

1. xkdxxkdxFEP

xx

===

22

2

1

2

1

..

xkvm

EPEKE

+=

+=

2

.

2)(

2

1

2

1eqrrkxkV ==


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Solution

2nd Order ODE

General solution:

Solution for HO: x =A sin t

A amplitude of the motion

angular frequency

02

2

=

+ x

m

k

dt

xd

2

tBtAx cossin +=

Properties of the oscillator

The position varies harmonically with time.

Momentum is least when the particle is atmaximum displacement.

Period: T= 2/ (s)

Frequency: s1 (Hz)

2

1==

T

2 . 5 5 7 . 5 1 0 1 2 . 5 1 5 1 7 . 5

-1

- 0 . 5

0. 5

1

T

T

x (t)

t


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Momentum: p= mv= m A cost

Potential energy:

Total energy:

Analogy with a vibrating chemical

bond:

2

2

1xkV =

2

2

1AkE=

Typically, k= 500-800 N/m

Properties of waves

Wavelength distancebetween two peaks ortroughs. ()

Frequency numberof cycles that passthrough a given pointin space per unit time.

()

Hz)/(

wavelength

speed

ycledistance/c

secondtravelled/distance

1ors

m

sm ==

=

v

Wavenumber:

1~=


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The electromagnetic field

A great deal of information on molecular structure isrevealed by the interaction of radiation with matter.

Radiation emission and transmission of energy inthe form of waves.

Light electromagnetic radiation (how energytravels through space).

The electromagnetic field is an oscillating electricand magnetic disturbance that spreads as aharmonic wave through vacuum (empty space).

Magnetic and Electric fields

For all electromagnetic radiation:

speed c = 2.998 108 m/s

The electromagnetic spectrum


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Features of electromagneticradiation

Oscillating electric field:

Oscillating magnetic field:

If = or , the peaks of one wavecoincide with the troughs of the other thewaves are out-of-phase: destructiveinterference.

If = 0, the peaks and troughs of the twowaves coincide together the waves go inphase: constructive interference.

])/2(2cos[),( 0 += xtEtxE

])/2(2cos[),( 0 += xtBtxB

We can show thatE(x,t) andB (x,t)satisfy the following two equations:

where (x,t) is eitherE(x,t) orB (x,t).

2

2

2

2 4=

x

222

2

4=

t


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Constructive and destructiveinterference

NO spot

In between

Partial cancellation

Produce Dull spot

Diffraction pattern of an NaCl crystal

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