Transcript Energy is Quantized

Intro/Review of Quantum
QM-1
So you might be thinking… I thought I could avoid
Quantum Mechanics?!?
Well… we will focus on thermodynamics and
kinetics, but we will consider this topic with
reference to the molecular basis that underlies the
laws of thermodynamics. Since molecules behave
quantum mechanically, we will need to know a few
of the results that are provided from quantum
mechanics.
Those interested in more details should take CHE372 this spring!
Energy is Quantized
Energy
Energy
Microscopic
Small things, large relative
energy spacings, must
consider the energy levels to
be quantized
Time
Energy
Big things, small relative
energy spacings, energy
looks classical (i.e.,
continuous)
Energy
Macroscopic
QM-2
Time
Energy is Quantized by h
QM-3
Planck suggests that radiation (light, energy) can
only come in quantized packets that are of size hν.
Planck, 1900
E  h
Energy (J)
Planck’s constant
h = 6.626 × 10-34 J·s
Frequency (s-1)
Note that we can specify the energy by specifying any one of the following:
1. The frequency, n (units: Hz or s-1):
2. The wavelength, λ, (units: m or cm or mm):
Recall:
  c
3. The wavenumber,
EX-QM1
1
Recall: ~ 

~ (units: cm-1 or m-1)
E  h
E
hc

E  hc~
Where can I put energy?
QM-4
Connecting macroscopic thermodynamics to a molecular understanding
requires that we understand how energy is distributed on a molecular level.
ATOMS:
The electrons: Electronic energy. Increase the
energy of one (or more) electrons in the atom.
Nuclear motion: Translational energy. The atom
can move around (translate) in space.
MOLECULES:
The electrons: Electronic energy. Increase the energy of one (or more)
electrons in the molecule.
Nuclear motion:
Translational energy. The entire molecule can translate in space.
Vibrational energy. The nuclei can move relative to one another.
Rotational energy. The entire molecule can rotate in space.
Schrödinger Equation
QM-5
Erwin Schrödinger formulated an equation used in quantum
mechanics to solve for the energy of different systems:
Schrödinger
2 2

 ( x)  V ( x) ( x)   ( x)
2
2m x
Kinetic energy
Potential
energy

h
2
Total energy
 (x)
is the wavefunction. The wavefunction is the most complete possible
description of the system.
Solving the differential equation (S.E.) gives one set of wavefunctions,  (x )
and a set of associated eigenvalues (i.e., energies) E.
Interested in solving this problem for specific systems?!?! Take CHE 372
in the spring! Meanwhile, you are required such to be familiar with the
solutions for the systems we will encounter.
ATOMS I: H atom electronic levels
QM-6
Convert J to cm-1; Can you?
 2.17869 1018
 109680 1
J
cm
Electronic Energy Levels: el 
2
2
n
n
n must be an integer.
+
(r = ∞)
Series limit, n = ∞, the electron and
proton are infinitely separated, there
is no interaction.
-
Ground state, n = 1, the most probable
distance between the electron and
proton is rmp = 5.3 × 10-11 m.
-
EX-QM2
+
Wavefunctions and Degeneracy
QM-7
The wavefunctions are
the atomic orbitals.
3s
2s
1s
The number of wavefunctions,
or states, with the same
energy is called the
degeneracy, gn.
ATOMS II: Translational Energy
QM-8
In addition to electronic energy, atoms have translational energy.
To find the allowed translational energies we solve the Schrödinger
a
equation for a particle of mass, m.
0
x
In 1D, motion is along the x dimension
and the particle is constrained to the
interval 0 ≤ x ≤ a.
n 2 h 2 n  1,2,3,...
n 
8ma 2
z
In 3D…
n
2
2
2 

n
h  nx
n
y
z 



8m  a 2 b 2 c 2 
2
x ,n y ,nz
nx  1,2,3,...
n y  1,2,3,...
nz  1,2,3,...
c
b
a
These states can be degenerate. For example, if a=b=c, then
the two different states (nx=1, ny=1, nz=2) and (nx=2, ny=1,
nz=1) have the same energy.
x
Electronic Energy Levels, Generally
QM-9
As we have seen, the electronic energy levels of the hydrogen
atom are quantized. However, there is no simple formula for the
electronic energy levels of any atom beyond hydrogen. In this
case, we will rely on tabulated data.
For the electronic
energy levels, there
is a large gap from
the ground state to
the first excited state.
As a result, we
seldom need to
consider any states
above the ground
state at the typical
energies that we will
be working with.
MOLECULES I: Vibrational
QM-10
We model the vibrational motion as a harmonic
oscillator, two masses attached by a spring.
Solving the Schrödinger equation for
the harmonic oscillator you find the
following quantized energy levels:
The energy levels
nu and vee!
1
2
 v  h (v  )
v  0,1,2,...
The level are non-degenerate, that
is gv=1 for all values of v.
The energy levels are equally spaced by hn.
The energy of the lowest state is NOT
zero. This is called the zero-point energy.
Re
R
 0  12 h
MOLECULES II: Rotational
Moment of inertia:
QM-11
I  m1R12  m2 R22
6 2
3 
I
Treating a diatomic molecule as a rigid
rotor, and solving the Schrödinger
equation, you find the following quantized
energy levels…
2

 J  J ( J  1)
2I
J  0,1,2,...
The degeneracy of these energy levels is:
g J  2J 1
2 
2

1 
I
J  0
3
I
2
J=3
J=2
J=1
J=0
Rotational energy
10 2
4 
I J=4
Dissociation Energy
QM-12
The dissociation energy and the electronic energy of a
diatomic molecule are related by the zero point energy.
Energy
Negative of the
electronic energy
h
De  D0 
2
Dissociation energy
For H2…
De = 458 kJ·mol-1
D0 = 432 kJ·mol-1
~= 4401 cm-1 (=52 kJ·mol-1)
EX-QM3
Polyatomic Molecules I: Vibrations
QM-13
For polyatomic molecules we can consider each of the nvib vibrational
degrees of freedom as independent harmonic oscillators. We refer to
the characteristic independent vibrational modes as normal modes.
For example, water
has 3 normal modes:
Since the normal modes
are independent, the total
energy is just the sum:
 vib   h j v j  12 
Bending
Mode
nvib
j 1
EX-QM4
~  1595cm 1
~  3686cm 1
Symmetric Stretch
~  3725cm 1
Asymmetric Stretch
Polyatomic Molecules II: Rotations
QM-14
Linear molecules: The same as diatomics with the moment of inertia
defined for more than 2 nuclei:
2
 J  J ( J  1) J  0,1,2,...
2I
g J  2J 1
n
I   m j ( x j  xcm ) 2
j 1
Nonlinear molecules: There is one moment of inertia for each of
the 3 rotational axes. This leads to three ways to define
polyatomic rotors:
Spherical top (baseball, CH4): IA = IB = IC
Symmetric top (American football, NH3): IA = IB ≠ IC
Asymmetric top (Boomerang, H20): IA ≠ IB ≠ IC
Degrees of Freedom
QM-15
To specify the position of a molecule with n nuclei in space we
require 3n coordinates, this is 3 Cartesian coordinates for each
nucleus. We say there are 3n degrees of freedom.
We can divide these into translational, rotational, and vibrational
degrees of freedom:
Degrees of Freedom
(3n in total)
Translation:
Motion of the center of mass
3
Rotation (Orientation about COM):
Linear Molecule
Non-Linear Molecule
Vibration (position of n nuclei):
Linear Molecule
Non-Linear Molecule
EX-QM5
2
3
3n-5
3n-6
Total Energy
QM-16
The total energy is the energy of each degree of freedom:
   trans   rot   vib   elec
n
x ,n y ,nz
2
h 2  nx2 n y nz2 

 
8m  a 2 b 2 c 2 
nx  1,2,3,...
n y  1,2,3,...
nz  1,2,3,...
 v  h (v  12 )
For each vib. DOF
2
 J  J ( J  1)
2I
J  0,1,2,...
For linear molecules.
Look up values in
a table (i.e., De).
Relative Energy Spacings
QM-17
The general trend in energy spacing:
Electronic > Vibrations > Rotations > > Translations
J=2
EX-QM6
J=1
J=0
Population: Boltzmann Distribution
QM-18
The Boltzmann distribution determines the relative
population of quantum energy states.
Ej
pj 
Ludwig Boltzmann
Probability that a
randomly chosen system
will be in state j with Ej
e
e
i
k BT
 Ei
k BT
Partition function
This equation is the key equation in statistical mechanics,
the topic of the next few sections of this class. Statistical
mechanics is used to comprehend ‘macroscopic’
thermodynamics in terms of a ‘microscopic’ molecular basis.