CHEN 623-Chemical
Engineering Thermodynamics
The “internal” energy: Energies of atoms and molecules
Fall 2013
Instructor: Dr. Perla B. Balbuena
JEB 240
balbuena@tamu.edu
http://research.che.tamu.edu/groups/Balbuena/courses.htm
Thermodynamics
• Equilibrium
• Developed in the 1800’s as experimental
science (classical thermodynamics)
• Applications to many fields
• In the 1900’s statistical thermodynamics
• Statistical thermodynamics uses a few
concepts of quantum mechanics
• Our course focuses on “molecular
thermodynamics”
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Today we are going to talk about
contributions to the “internal”
energy
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Electronic energy of the H atom
Energy levels are quantized:
Ground state, 1st excited state, etc
What happens when n is very large?
What happens when an electron
makes a transition from one state
to another?
Electromagnetic radiation consists
of packets of energy called photons,
their energy is hn
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Emission or absorption of
electromagnetic radiation
e2 > e 1
energy is absorbed
e2 = e1 +h n1-2
how about the energy for the
transition 2-> 1?
hn = D e
D e is the absolute energy
difference (Bohr frequency
condition)
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Emission spectrum hydrogen atom
 1 1 
hn n2  e n  e 2  2.17869 x10 18 J  2  2 
2 n 
c=ln
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The Schrödinger equation
for a single particle in one dimension
potential energy
allowed energy
 d  ( x)

 V ( x) ( x)  e ( x)
2
2m dx
2
2
wave function
mass
position
Solving this eigenvalue equation for a particle in a given potential,
we obtain wave eigenfunctions and
corresponding allowed energies
For the H atom, the allowed energies have the form found experimentally and
the wave functions are the hydrogen atomic orbitals 1s, 2s, 2p, etc…
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what is the wave function?
 ( x)dx
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probability to find the particle between x and x+dx
specifies the state of the particle
for electrons, atoms, molecules, this probability is
less sharp than in everyday objects
each wave function has an associated energy (en)
more than one wave function may be associated with the
same energy (degeneracy of the level, gn)
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Translational Energy
So far we discussed only electronic energies
Other types of energy: kinetic (translational)
Example: a particle constrained to 0 < x < a
2
2
nh
en 
8ma 2
en n n
x y z
Allowed translational energies, n =1, 2, 3, …
levels are non-degenerate
2
2
2 

n
h  nx
nz 
y

 2  2
2
8m  a
b
c 
2
Particle in a 3-dimensional volume
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Electronic energies of atoms
other than H
• Can be calculated by numerical
solutions of the Schrodinger equation
• Can be measured (Moore Tables,
Atomic Energy Levels, see Tables 1.2,
1.3)
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First few states of atomic sodium
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Vibrational energies
Let’s consider the diatomic molecule
f  k ( R  Rc )  kx
Two masses connected by a spring
Hooke’s law
k is the force constant
if the spring is extended or compressed and then let it undergo its natural motion
x(t )  A cos 2nt
1
n
2
1/ 2
k
 

classical harmonic oscillator
A: amplitude of the motion
: reduced mass
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Internuclear potential of a
diatomic molecule
harmonic potential
is a good
approximation at
small
displacements
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Energy levels of a quantum
mechanical harmonic oscillator
QM solution of the Schrodinger equation
for the one-dimensional
Harmonic oscillator :
h
ev 
2
1/ 2
k 
1
1

   v    hn  v  
2
2

 
v = 0, 1, 2,…
levels are non-degenerate
The first level (v =0) is not zero
zero point energy
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Infrared (IR) spectrum of a
diatomic molecule
a transition for one vibrational state
to another one : absorbing or emitting
EM radiation
De  hn observed
only transitions between adjacent states are allowed Dn =
h
De  en 1  en 
2
+1
1/ 2
k
 

 hn
c=ln
1/ 2
n~
obs
1 k
 

2c   
observed frequency in wavenumbers, cm-1
fundamental vibrational frequency
for diatomic molecules, these frequencies appear in the IR region
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IR spectrum of HCl
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Dissociation energy (Do)and ground
state electronic energy (De)
Anharmonic oscillator: HCl
hn
Do  De 
2
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Rigid rotator
distance between the masses is fixed (Re)
note: an approximation since there is also vibration
CoM given by m1R1 = m2R2
The molecule rotates around the CoM
at a frequency nrot; define angular velocity w  2nrot
> u1  2R1nrot  R1w; u2  2R2nrot  R2w
Kinetic energy of the rigid rotator
K
1
1
1
1
2
2
2
2
m1n 1  m2n 2  (m1 R1  m2 R2 )w 2  Iw 2
2
2
2
2
I is the moment of inertia
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Solving the Schrödinger
equation for a rigid rotator
Allowed energies:
Degeneracy:
2
e J  J ( J  1)
2I
J = 0, 1, 2, …
g J  2J 1
Transitions are allowed only for adjacent states, so DJ = +1
De 
n
h2
4 I
2
h
4 I
2
( J  1)
J = 0, 1, 2, …
( J  1)
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Microwave spectroscopy
n
h
4 I
2
( J  1)  2 B ( J  1)
B is the rotational constant of the molecule
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Vibrations of polyatomic molecules
A molecule has n nuclei; complete specification requires 3n coordinates
(3n degrees of freedom):
3 for the center of mass (translational degrees of freedom)
For a linear molecule, 2 coordinates needed to define orientation about its
C of M (2 degrees of rotational freedom); remaining coordinates: 3n-5 specify
the relative positions of the n nuclei (vibrational degrees of freedom).
For a nonlinear molecule, 3 degrees of rotational freedom needed;
remaining coordinates: 3n-6 specify the relative positions of the n nuclei
(vibrational degrees of freedom).
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Normal modes CO2
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Molecules that are IR active
• Their dipole moment changes during
the motion of the normal mode
(condition to be IR active)
• What happens with CO2? What modes
are active?
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Rotational spectrum of a
polyatomic molecule
• Same energy levels as in the diatomic,
but the moment of inertia depends on
the configuration of the molecule (see
section 1.10)
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The energy of a molecule in the
rigid-rotator harmonic-oscillator
approximation
• Atom: translational + electronic
• Diatomic Molecule: translational
+rotational + vibrational + electronic
• Polyatomic Molecule: translational
+rotational + vibrational + electronic.
However rotational contribution
depends on I, and vibrational sums
over all modes:
1
nvib
e vib   hn j (u j  )
j 1
2
u j  0,1,2...
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1st homework-Due Friday, Sept. 6th
• Problems 1.24; 1.41; 1.42; 1.44; 1.46;
3.1; 3.5; 3.6; 3.9; 3.10
• For practice try other problems as
well, although they won’t be graded
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