Chem 467
Chem 467 Review Sheet for Exam 2Review Sheet for Exam 2 Fall 2012Fall 2012
Postulates of Quantum Mechanics Postulates of Quantum Mechanics
•• The state of the system is described by aThe state of the system is described by awavefunctionwavefunction that must satisfy certain conditions;that must satisfy certain conditions; continuous, single valued, co
continuous, single valued, continuous first derivative, square integrable, finite.ntinuous first derivative, square integrable, finite.
•• The productThe product
∗∗
d d
gives the probability of finding the particle within a volumegives the probability of finding the particle within a volume d d
.. •• All physical observables have All physical observables have a corresponding mathematical operator; specifically linear a corresponding mathematical operator; specifically linearmomentum is given by momentum is given by p p
x x=
=
ℏℏ
ii∂
∂
∂
∂
x x and position is given byand position is given by x x== x x . . The onlThe only possy possibleible values that can be observed are eigenvalues of the given operator.values that can be observed are eigenvalues of the given operator.
•• The expectation value of an observableThe expectation value of an observable J J is given byis given by
〈〈
J J〉=
〉=
∫
∫
∗∗ J J
d d
.. Solutions to Model SystemsSolutions to Model Systems
The Schrödinger equation can only b
The Schrödinger equation can only be solved analytically for a handful of systems.e solved analytically for a handful of systems. Free Particle
Free Particle – A par– A particle that experiences no ticle that experiences no interactions in all space in 1 interactions in all space in 1 dimension.dimension. Potential:
Potential: V V = 0 for all= 0 for all x x Hamiltonian: Hamiltonian: H H
=
=
T T x x=
=
−ℏ
−ℏ
2 2 2 2 mm∂
∂
22∂
∂
x x22 Wavefunctions: Wavefunctions: x x==cc11ee ii kk xx cc22ee − −ii k xk x wherewhere k k
=
=
22m E m Eℏℏ
22
1 1//22
E
Enneerrggyy:: AAlll l nnoonn--nneeggaattiivve e eenneerrggiiees s aarre e aalllloowweedd The free particle
The free particle can be anywhere with equal probabilican be anywhere with equal probabilityty. . It has a well It has a well characterized momentum. characterized momentum. If If k k is positive, it is traveling to the
is positive, it is traveling to the right; if right; if k k is negative, it is negative, it is traveling to is traveling to the left. the left. (Or,(Or, cc22is zero for right-is zero for
right-moving and
moving and cc11is zero for left-moving.) is zero for left-moving.) If the particle is not trIf the particle is not traveling in a preferred direction,aveling in a preferred direction, cc11 == cc22..
Particle in a Box
Particle in a Box – W– Walls of infinite potential restrict the particle to being alls of infinite potential restrict the particle to being in a box in a box of lengthof length L L.. Potential:
Potential: V V
x x=
=
00 for for 00 x x L L VV x x=∞=∞ for for x x
00 ,, xx L L Hamiltonian (inside box):Hamiltonian (inside box): H H
=
=
T T x x=
=
−ℏ
−ℏ
2 2 2 2 mm
∂
∂
22∂
∂
x x22 WWavfunctions (inside avfunctions (inside box):box): x x==
22L Lsinsin n nxx L L nn = 1, 2, 3 ...= 1, 2, 3 ...
(outside box):
x=
0Energy: E n
=
h2
n2
8m L2 n = 1, 2, 3 ...
The wavefunction has n nodes (counting the two at the edges as a single node). As n increases, the probability of finding the particle becomes uniform throughout the box.
Particle in a Finite-Well – The walls are not infinite.
Potential: V
x=
0 for 0 x LV
x=
V for x
0 , x L Hamiltonian (outside box): H
=
−ℏ
2
2 m
∂
2∂
x2
VWavefunctions (outside box):
=
c e±k x where k=
2 m
V−
E
ℏ
The wavefunction decays exponentially. The + sign is for the left side, the – sign is for the right side. Hamiltonian (inside box): H
=
−ℏ
2
2 m
∂
2∂
x2 Wavfunctions (inside box):
x≈
N sin n
xL n = 1, 2, 3 ...
The wavefunction has to match at the edges of the box, which distorts the wavefunction. The particle can penetrate slightly into the barrier. This lowers the energy slightly from the infinite walled case.
Tunneling – A particle encounters a barrier of height V . In the region of the barrier, the wavefunction decays exponentially. If the barrier is thin enough, the particle can appear on the other side. The
probability of tunneling is given by P t
=
e−2 L where=
2 m
V−
E K
ℏ
Particle in a 3-D Box – We extend the problem to 3 dimensions.
Potential: V
x , y , z=
0 for 0 x L x , 0 y L y , 0 z L z V
x , y , z=∞
for x0 , y0 , z
0 , x
L x , y
L y , z
L z Hamiltonian: H =−ℏ 2 2m[
∂2 ∂ x2 ∂2 ∂ y2 ∂2 ∂ z 2]
Wavefunctions:
x , y , z=
8 L x L y L z
1/2 sinn x
x L x sin n y
y L y sin n z
z L z Energy: E n x , n y , n z=
h2 8m
n x2 L x2
n y2 L y2
n z 2 L z 2
If the box is cubic, meaning L x
=
L y=
L z=
L , then E n x , n y , n z=
h2 8m L2
n x2
n y2
n z 2
Now multiple states have the same energy; they aredegenerate.Particle on a Ring, or 2-D Rigid Rotor – A rotor restricted to motion in a plane, or particle on a ring.
Potential: V
=
0 at r , V=∞
otherwise Hamiltonian: H
rot=
−ℏ
2 2 I∂
2∂
2 where I=
r 2 Wavefunctions:
rot=
1
2
e i ml with ml =0,±1,±2,. ..Energy: E rot
=
ℏ
2
ml 2 2I
3-D Rigid Rotor, or Particle on a Sphere
Potential: V =0 at r , V =∞ otherwise Hamiltonian: H
rot=
−ℏ
2 2 I[
1 sin
∂
∂
sin ∂
∂
1 sin2
∂
2∂
2]
or: H
=
1 2 I
l 2where l 2 is the angular momentum operator
l 2=−ℏ
2[
1 sin
∂
∂
sin ∂
∂
1 sin2
∂
2∂
2]
Wavefunctions: Spherical Harmonics Y l ml
,
where l=
0,1,2,.. . There is a second quantum number, ml =l ,l −1 ,. . .−l , for z angular momentum.Energy: E rot
=
l
l
1
ℏ
2
2 I
Harmonic Oscillator Potential: V
=
1 2k x 2 Hamiltonian: H
=
−ℏ
2 2 m∂
2∂
x2
1 2k x 2 Wavefunctions: n x=
1/4
1 2nn!
1/2 H n ye− y 2 /2 where n=0,1,2,...H n are polynomials of order n.
=/ℏ
,=
k/
, and y=
1/2 x .Energy: E n
=
n
12
ℏ =
n
1 2
h
Solutions to Real SystemsHydrogenic Atom Potential: V
r=
−
Z e 2 4
0r Hamiltonian: H
=
1 2mr 2
l 2−
ℏ
2 2m r 2∂
∂
r
r 2∂
∂
r
−
e2 4
0r Wavefunctions:
r ,
,=
R
r
Y l ml
,
with radial functions Rn , l
r=−
[
4 Z3 n4a03
n−
l−
1
![
n
1
!]
3]
1/2
2 Z r n a0
l e− Z r /n a0 Ln2l l 1
2 Z r n a0
where Ln2 l 11 are polynomials and a0=
4
0ℏ
2
mee2 is the Bohr radius Principle quantum number n = 1, 2, 3, . . .
Orbital angular momentum l = 0, 1, 2, . . . , n – 1
Magnetic quantum number ml = -l , -(l – 1), . . . , 0, . . . , (l – 1), l
These functions are known asatomic orbitals and are labeled by n and l , where
l = 0 1 2 3 s p d f Energy: E n
=
−
Z 2
ee 4 8
02 h2n2=
−
Z 2e2 8
0a0n2 Energy only depends on n.Multielectron Atoms – Can't be solved analytically, because of electron-electron interactions. We use the hydrogenic atomic orbitals as our basis set for describing other atoms. The additional terms in the Hamiltonian cause the degenerate states to split.
Electron Spin – Electrons are described by another property known asspin. This has no classical analog. The two possible spin states are m s=±1/2
Pauli Exclusion Principle – No two electrons can have the same set of quantum numbers. Because of spin, each atomic orbital can contain at most 2 electrons.
Stability of Atoms – Atoms are stable because of a balance between kinetic and potential energy.
Hydrogen Molecular Ion (H2+)
Hamiltonian: H
=−
ℏ
2 2 m p∇
A 2−
ℏ
2 2 m p∇
B 2−
ℏ
2 2 me∇
e 2−
e 2 4
0r A−
e2 4
0r B
e2 4
0 R If we consider the nuclei as being at fixed positions, we can treat R as a parameter and only worry about the behavior of the electron. This is known as theBorn-Oppenheimer approximation. This is justified because nuclei are much more massive than electrons.Modified Hamiltonian: H
R=−
ℏ
2 2 me∇
e 2−
e 2 4
0r A−
e 2 4
0r B
e 2 4
0 R Wavefunctions:
±=
N
H 1 s A±
H 1 s B
We use the atomic orbitals as our basis set, one on each proton. This is alinear combination of atomic orbitals. When we add the two functions, constructive interference leads to an increase in probability of finding the electron between the nuclei. This is abonding orbital and has lower energy
than the separate atomic orbitals. When we subtract the two functions, destructive interference leads to a decrease of probability of finding the electron between the nuclei. This is anantibonding orbital. The energy is higher than the separate atomic orbitals.
Molecular Orbital Theory – We must have the same number of molecular orbitals as total atomic orbitals. For diatomic molecules, bonds that are symmetric about the axis of the bond are called
bonds. If the bond is formed above and below the axis of the bond, this is a
bond. For polyatomic molecules, the molecular orbitals are spread out over the whole molecule.Spectroscopy
We can use light to measure the difference between energy states of various systems. Two conditions must be met:
The energy of the light must equal the difference in energy of the states E = E j− E i=h
The transition dipole moment must not be zero
x ji=
∫
j∗
x
idx=
0∫
j∗ x
idx The second condition determines if a transition is allowed. It also defines theselection rules. Forbidden transitions are much less likely to occur, but high power lasers can cause them, or they occur over a long time (emission only).Particle in a Box – Allowed transitions
n=±
1(This can model electronic transitions in long, conjugated molecules.)
Rotational Spectroscopy – We define the rotational constant as B
=
ℏ
2
2 I
=
ℏ
22
r 02 The rotational states have energy E rot = J J 1 B Selection rules: molecule must have a permanent dipole, and
J=±
1Difference in energy levels:
E J J 1=
2 B
J
1
Spacing between spectroscopic peaks: E =2 B
Rotational transitions occur in the microwave region of the electromagnetic spectrum. Rotational spectroscopy can be used to determine equilibrium bond lengths since I
=
r 2Vibrational Spectroscopy – The frequency relates to force constant of the bond and the reduced mass
of the bonding partners 2
==
k/
Vibrational states have energy E n
=
n
12
h
Selection rules: motion must change the dipole moment, and
n=±
1Difference in energy levels: E nn1=h
Spacing between spectroscopy peaks: Only a single peak should be observed, however any transition must also obey the rotational selection rules. This gives a pattern of peaks with spacing 2 B.
Vibrational transitions occur in the infraredregion of the spectrum. Vibrational spectroscopy can be used to determine force constants of bonds, and bond lengths through the rotational structure.
Anharmonicity – Real molecules are not harmonic oscillators; the bond will break if stretched enough. To include this, a term is added to the potential, and to the energy of the states.
E n
=
n
1 2
h−
n
1 2
2 xeh
where xe= 4 De and Deis the dissociation energy. Electronic Spectroscopy – Transitions between the electronic states of atoms or molecules.
Atomic Spectroscopy
Atomic spectra are characterized by discrete lines. We won't formally discuss the selection rules.
Electronic transitions occur in the visible and ultraviolet (UV) regions of the spectrum. (X-rays can be used to probe core electrons.)
Molecular Spectroscopy
Under the Born-Oppenheimer approximation, we determine potential energy surfaces. This surface determines the vibrational motion of the nuclei. If an excited electronic state is not bound, excitation to that state causes the bond to break. If an excited state is bound, it has its own vibrational states. If the wavefunctions of the ground and excited states overlap well, the absorbance increases; this is the Franck-Condon principle. Electronic spectra therefore contain vibrational information.
Fluorescence and Phosphorescence
When a molecule is excited, that energy must go somewhere. If it is emitted from the excited state back to the ground state, we call thisfluorescence. If the molecule changes to a different excited state
with a longer lifetime, the emission takes place over a longer time and is known asphosphorescence. Quenching provides a non-radiative path back to the ground state; no light is emitted.
Lasers – Critical components are a gain medium, pump source, and optical cavity. Lasers operate on the principles of absorption, spontaneous emission, and stimulated emission. Require a 3- or 4-level system to achieve a population inversion and net amplification of photons.
Statistical Mechanics
Molecular partition function: Defined as q
=
∑
i
e−i/k T
. When we have different types of molecular energy states, we can write the total internal energy as a sum; i=i
T i R i V i E . Therefore, the partition function can be written as a product; q=qT q RqV q E .
Monatomic ideal gas: For this case, we only consider translational energy. The molecular partition function of a monatomic ideal gas is given by q
=
2
m k Th2
3/2
V
Polyatomic ideal gas: Use the same function for translations, degeneracy for electronic states. qrot ,linear
=
σ Θ
Trot qrot ,nonlinear
=
1
σ
(
π
T3
Θ
rot , xΘ
rot , yΘ
rot , z)
1/2
where
rot=
ℏ
2
2 I k B is therotational temperature. σ is thesymmetry number. qvib
=
11
−
e−h /k BT=
1
1
−
e−vib/T where
vib=
h
k B is the vibrational temperature.
Canonical ensemble: A collection of microsystems that all have the same N, V and T . The canonical distribution function is P
j=
e− E j/kT
Q The canonical partition function for ideal gas is Q
=
qN
N !
Thermodynamic Quantities of a Canonical Ensemble:
We can determine the value of any thermodynamic variable M by M
=
∑
j M j P
j=
∑
j M je− E j/k T Q . U=
k T 2
∂
lnQ∂
T
N ,V p=
k T
∂
lnQ∂
V
N ,T S=
k ln Q
k T
∂
ln Q∂
T
N ,VErgodic hypothesis: The statistical average of a number of microsystems in random configurations is equivalent to the time average of a single microsystem that is randomly fluctuating.
Configuration Integrals: Allows us to account for intermolecular interactions. Z N
=
1N !
∫
⋅⋅⋅
∫
e − E p/k Td q
1d q
2⋅⋅⋅
d q
NIf we assume that we only have pairwise interactions that do not depend on orientation, we can determine the second virial coefficient by evaluating B
=−
2
N A∫
0 f r 2dr where f=
e−k T E p−
Interaction Potentials
To account for behavior of real gases on a molecular level, we must model the interactions between molecules. These involve an interaction potential. One example is the Lennard-Jones 6-12 potential.
E p
r=
4
{
r o r
12−
r o r
6}
Equations of State for Real Gases
van der Waals EOS – Attempts to account for Finite volume of molecules (b)
Attractive interactions between molecules (a)
This EOS is written as p
=
nRTV
−
nb−
a
n V
2 or p=
RT V m−
b−
a V m2Virial EOS – Based on measured behavior of real gases. First we define a compression factor, Z Z
=
V mV mo
=
p V mRT V m
o
is the molar volume of ideal gas at a given T and p. We then write an expansion of Z in either pressure or reciprocal volume.
p V m= RT 1 B ' pC ' p2... or p V m
=
RT
1
B V m
C V 2m
...
Critical Points – Under particular conditions a gas will condense and ex ist as a liquid and a vapor. When this coexistence region reduces to a single point in T , V , p space, we refer to this as the critical point . Critical points are tabulated for most gases as T c, V c, pc. Below T c, a gas will eventually
condense. The critical points can be related to the parameters in certain equations of state.
Reduced Variables and Corresponding States – Reduced variables are obtained by dividing the
actual conditions by the critical values for the gas; reduced temperature is defined as T r = T
T c . Reduced variable have no units. Gases at the same reduced conditions will behave similarly.
