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Chapter 43

Molecules and Solids

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Outline Molecular bonds Bonding in solids Energy states and spectra of

molecules Free electron theory of metals Band theory of solids Electrical conduction in

metals, insulators, and semiconductors

Semiconductor devices and superconductivity

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Molecular bonds The bonding mechanisms in a molecule

are fundamentally due to electric forces between atoms (or ions).

Potential energy function that can be used to model a molecule:

r: internuclear separation distance between the two atoms

A and B are parameters that can be determined by experiments.

system atom- twoafor mn r

B

r

ArU

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Plot of U(r) ~ r for a two-atom system

Equilibrium separation: U(r) is a minimum and the two atoms are in stable equilibrium.

At large separation distance: the slope is positive, corresponding to a net attractive force.

At small separation distance: the slope is negative, corresponding to a net repulsive force.

Binding energy: The additional energy the system has to be given to break up the diatomic molecule (so that r = ).

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Classification of molecular bonding mechanisms

Ionic bonds Example: Sodium Chloride (NaCl)

Covalent bonds Example: H2 molecule

Van der Waals bonds Example: Condensation of inert gas

atoms into the liquid phase Hydrogen bonds

Example: DNA molecules

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Example: Covalent bonding

Ground-state wave functions 1(r) and 2(r) for two hydrogen atoms making a covalent bond ( )

(a) The atoms are far apart and their wave functions overlap minimally.

(b) The atoms are close together, forming a composite wave function 1(r) + 2(r) for the system. The probability amplitude for an electron to be between the atoms is high.

0/

30

1

1)( ar

s ea

r

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Bonding in solids A Crystalline solid consists of a large

number of atoms (ions) arranged in a regular array, forming a periodic structure.

Classification of bonding in solids: Ionic solids. Example: Sodium Chloride

(NaCl crystal) Covalent solids. Example: Diamond, silicon,

germanium Metallic solids. Example: Copper, silver,

sodium, etc.

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Examples of bonding in solids

NaCl Diamond Metal

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Energy states and spectra of molecules

Total energy of a molecule:E = Eel + Etrans + Erot + Evib

Rotational motion of molecules Vibrational motion of molecules Molecular spectra

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Rotational motion of molecules

Erot = (1/2) I2. I = r2, = m1m2/(m1 + m2), the

reduced mass of the molecule Quantization of the magnitude of the

molecule’s angular momentum

J: rotational quantum number Allowed values of rotational energy:

Energy separation between adjacent rotational levels:

,..., , J JJIL 210 )1(

...,, J JJI

Erot 210 )1(2

2

...,,JJI

hEEE JJ 321

4 2

2

1

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Allowed Rotational transitions

Selection rule: J = 1 For most molecules, transitions

between adjacent rotational energy levels result in radiation that lies in the microwave range of frequencies (f ~ 1011 Hz).

Example 43.1: The J = 0 to J = 1 rotational transition of the CO molecule occurs at a frequency of 1.15 x 1011 Hz.

(a) Find the Moment of inertia of the molecule.

(b) Find the bond length of the molecule.

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Vibrational motion of molecules

Frequency of vibration for the system:

Allowed values of vibrational energy:

Energy separation between successive vibrational levels:

k

f2

1

...,,vkh

vhfvEvib 210 2

)2

1()

2

1(

hfkh

Evib 2

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Allowed vibrational transitions

Selection rule: v = 1 Transitions between

vibrational levels are caused by absorption of photons in the infrared region of the spectrum.

Example 43.2: The frequency of the photon that causes the v = 0 to v = 1 transition in the CO molecule is 6.42 x 1013 Hz. Find the force constant k for

this molecule.

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Molecular spectra Total energy of the molecule:

Each energy level is indexed by the two quantum numbers v and J.

Absorptive transitions between the v = 0 and v = 1 vibrational states: (1) J = +1 and (2) J = -1

Energies of the absorbed photons:(1) E = hf + (ħ2/I)(J+1), J= 0,1,2,…(2) E = hf - (ħ2/I)J, J=1,2,3…

)1(2

)2

1(

2

JJI

hfvE

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Absorption spectrum of the HCl molecule

Quick Quiz: There is a gap between the two sets of peaks. Why?

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Homework

Chapter 43, P. 1434, Problems: #3, 8, 9, 14, 17.