Physical Chemistry 2 nd Edition Thomas Engel, Philip Reid Chapter 16 The Particle in the Box and the Real World

Physical Chemistry 2Physical Chemistry 2ndnd Edition EditionThomas Engel, Philip Reid

Chapter 16 Chapter 16 The Particle in the Box and the Real WorldThe Particle in the Box and the Real World

© 2010 Pearson Education South Asia Pte Ltd

Physical Chemistry 2nd EditionChapter 16: The Particle in the Box and the Real World

ObjectivesObjectives

• Importance of the concept for particle in the box

• Understanding the tunneling of quantum mechanical particles



OutlineOutline

1. The Particle in the Finite Depth Box2. Differences in Overlap between Core

and Valence Electrons3. Pi Electrons in Conjugated Molecules

Can Be Treated as Moving Freely in a Box

4. Why Does Sodium Conduct Electricity and Why Is Diamond an Insulator?



OutlineOutline

5. Tunneling through a Barrier6. The Scanning Tunneling Microscope7. Tunneling in Chemical Reactions



16.1 The Particle in the Finite Depth Box16.1 The Particle in the Finite Depth Box

• For a box to be more realistic, we let the box to have a finite depth.

• The potential is defined by

• Outside the box,

2/,2/for ,

2/2/for ,0

0 axaxVxV

axaxV

2

02 where2/-for ''

2/for

h

EVmaxeBeAx

axBeAex

xx

xx



16.2 Differences in Overlap between Core and Valence 16.2 Differences in Overlap between Core and Valence ElectronsElectrons

• 16.1 Energy Eigenfunctions and Eigenvalues for a Finite Depth Box

• Strongly bound levels correspond to core electrons and weakly bound levels correspond to valence electrons.



16.3 Pi Electrons in Conjugated Molecules Can Be 16.3 Pi Electrons in Conjugated Molecules Can Be Treated as Treated as Moving Freely in a Box Moving Freely in a Box

• The absorption of light in UV of electromagnetic spectrum is due to excitation of electrons.

• If electrons are delocalized in an organic molecule with a π-bonded network, the absorption spectrum shifts from UV into visible range.

• Greater the degree of delocalization, the more absorption maximum shifts toward the red end of the visible spectrum.



16.4 Why Does Sodium Conduct Electricity and Why Is 16.4 Why Does Sodium Conduct Electricity and Why Is Diamond Diamond an Insulator? an Insulator?

• Valence electrons on adjacent atoms in a molecule or a solid can have an overlap.

• The energy required to remove an electron from the highest occupied state is the work function, ø.



16.5 Tunneling through a Barrier16.5 Tunneling through a Barrier

• Consider a particle with energy E confined to a very large box.

• A barrier of height V0 separates two regions in which E < V0.

• The particle can escape the barrier and go over the barrier, called tunneling.



16.5 Tunneling through a Barrier16.5 Tunneling through a Barrier

• To investigate tunneling, finite depth box is modified by having a finite thickness.

• The potential is now axxV

axVxV

xxV

for ,0

0for ,

0for ,0

0

where a = barrier width



16.6 The Scanning Tunneling Microscope16.6 The Scanning Tunneling Microscope

• 16.2 Tunneling through a Barrier

• Scanning Tunneling Microscope (STM) allows the imaging of solid surfaces with atomic resolution with a surprisingly minimal mechanical complexity.

• The STM is used to study the phenomena at near atomic resolution.




• Scanning Tunneling Microscope (STM)



ExampleExample

As was found for the finite depth well, the wave function amplitude decays in the barrier according to . This result will be used to calculate the sensitivity of the scanning tunneling microscope. Assume that the tunneling current through a barrier of width a is proportional ahEVmA 2

0

2/22exp

xhEVmAx 20 /2exp



ExampleExample

a. If is 4.50 eV, how much larger would the current be for a barrier width of 0.20 nm than for 0.30 nm?

b. A friend suggests to you that a proton tunneling microscope would be equally effective as an electron tunneling microscope. For a 0.20-nm barrier width, by what factor is the tunneling current changed if protons are used instead of electrons?

EV 0



SolutionSolution

a. Putting the numbers into the formula given, we obtain

Even a small distance change results in a substantial

change in the tunneling current.

78.8

100.110055.1

10602.150.41011.922exp

100.31022

2exp100.3

100.2

10234

1931

10102

010

10

h

EVm

maI

maI



SolutionSolution

b. We find that the tunneling current for protons is appreciably smaller than that for electrons.

This result does not make the proton tunneling microscope look very promising.

79

103127234

19

20

20

20

1023.1

100.21011.91067.110055.1

10602.150.422exp

22exp

22exp

22exp

ammh

EV

ah

EVm

ah

EVm

electronI

protonI

electronproton

electron

proton




• Most chemical reactions proceed faster as the temperature of the reaction mixture is increased.

• This is due to energy barrier which must be overcome in order to transform reactants into products.

• This barrier is referred to as the activation energy for the reaction.




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Physical Chemistry 2 nd Edition Thomas Engel, Philip Reid Chapter 16 The Particle in the Box and the Real World