Chem 467
Fall 2012
Review Sheet for Exam 2
Postulates of Quantum Mechanics •
The state of the system is described by a wavefunction that must satisfy certain conditions; continuous, single valued, continuous co ntinuous first derivative, square integrable, finite.
•
The product ∗ d gives the probability of finding the particle within a volume d .
•
All physical observables have a corresponding mathematical operator; specifically linear momentum is given by p x=
ℏ ∂ and position is given by x = x . The only only possible possible i ∂ x
values that can be observed are eigenvalues of the given operator. •
The expectation value of an observable J observable J is is given by
d . 〈 J 〉=∫ ∗ J
Solutions to Model Systems The Schrödinger equation can only be b e solved analytically for a handful of systems. Free Particle – A particle particle that experiences no interactions in all space in 1 dimension.
Potential: Hamiltonian:
Wavefunctions: Energy:
V = V = 0 for all x all x 2 2 −ℏ ∂ =T x = H 2 m ∂ x 2
x = c1 e
ikx
−i k x
c2 e
where
k =
1 /2
2 m E
ℏ2
All non-negative energies are allowed
The free particle can be anywhere with equal probability probability.. It has a well characterized momentum. If k If k is positive, it is traveling to the right; if k if k is is negative, it is traveling to the left. (Or, c2 is zero for rightmoving and c1 is zero for left-moving.) If the particle is not traveling traveling in a preferred direction, c1 = c2.
Walls of infinite potential restrict the particle to being in a box of length L length L.. Particle in a Box – Walls Potential:
V x = 0
for
V x =∞
for x 0 , x L
0 x L
∂2 2 m ∂ x 2
Hamiltonian (inside box):
=T x = −ℏ H
Wavfunctions (inside box):
x =
2
L
2
sin
n x L
n = 1, 2, 3 ...
(outside box):
x = 0 2
E n=
Energy:
h n
2
8m L
n = 1, 2, 3 ...
2
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:
Hamiltonian (outside box):
V x = 0
for
V x =V
for x 0 , x L
0 x L
2 2 −ℏ ∂ = V H 2 m ∂ x 2
Wavefunctions (outside box): =c e ± k x
k =
where
2 m V − E ℏ
The wavefunction decays exponentially. The + sign is for the left side, the – sign is for the right side. 2
∂2 2 m ∂ x 2
Hamiltonian (inside box):
= −ℏ H
Wavfunctions (inside box):
x ≈ N sin
n x n = 1, 2, 3 ... L
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 − 2 L
P t =e
probability of tunneling is given by
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 =∞ Hamiltonian:
for 0 x L x ,
V x , y , z = 0
0 y L y ,
0 z L z
for x 0 , y 0 , z 0 , x L x , y L y , z L z
= −ℏ H
2
[
2
2
2
∂ ∂ ∂ 2 2 2 2 m ∂ x ∂ y ∂ z
]
1 /2
8
Wavefunctions:
x , y , z =
Energy:
2 n x n y n z h 2 2 E n , n , n = 2 8 m L L y L z x x
y
L x L y L z
z
2
sin
2
2
If the box is cubic, meaning L x = L y = L z = L , then
n x x L x
sin
n y y L y
sin
n z z L z
E n , n , n = x
y
z
h
2
8m L
2
n n n 2
2
2
x
y
z
Now multiple states have the same energy; they are degenerate .
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:
2 2 −ℏ ∂ rot = H 2 I ∂ 2
Wavefunctions:
rot =
1
2 2
Energy:
E rot =
e
i ml
2
I = r
where
with ml = 0,± 1,± 2, . . .
2
ℏ ml 2 I
3-D Rigid Rotor, or Particle on a Sphere
Potential:
V =0 at r ,
Hamiltonian:
rot = −ℏ H
or:
2
V =∞ otherwise
[
= H
1 2 I
]
2 l
2 2 where l 2 is the angular momentum operator l =−ℏ
Wavefunctions:
2
1 ∂ 1 ∂ sin ∂ 2 2 I sin ∂ ∂ sin ∂ 2
[
ml
Spherical Harmonics Y l , 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
2
1 ∂ 1 ∂ sin ∂ 2 ∂ sin ∂ 2 sin ∂
ℏ2 2 I
In the absence of an external field, there are 2 l 1 degenerate states.
]
Harmonic Oscillator 1
2
Potential:
V = k x
Hamiltonian:
= −ℏ H
Wavefunctions:
n x =
2
2
∂2 1 2 kx 2 m ∂ x 2 2 1/ 4
1 /2
1
n
2 n!
where n = 0,1,2,...
=/ℏ , = k / , and y =1 / 2 x .
H n are polynomials of order n.
E n= n
Energy:
2
− y / 2
H n y e
1
ℏ = n
2
1 2
h
Solutions to Real Systems Hydrogenic Atom 2
Potential:
− Z e V r = 4 0 r
Hamiltonian:
= H
Wavefunctions:
r , , = R r Y ml ,
2
2 mr
l 2− ℏ
2
2
∂ r 2 ∂ − e 2 ∂ r 4 0 r 2 m r ∂ r l
[
R n , l r =−
with radial functions
where L
1
2 l 1 n 1
3
4 Z n −l −1 ! 4
3 0
3
n a [ n 1 ! ]
are polynomials and a0=
4 0 ℏ me e
] 1/ 2
l
2 Z r − Z r / n a 2l 1 2 Z r e L nl n a0 n a0 0
2
2
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 as atomic orbitals and are labeled by n and l , where l =
Energy:
0
1
2
3
s
p
d
f
− Z 2 e e 4
− Z 2 e 2 = E n= 2 2 2 2 8 0 h n 8 0 a 0 n
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 as spin. This has no classical m s=±1 / 2
analog. The two possible spin states are
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 (H 2+)
Hamiltonian:
ℏ2 ℏ2 ℏ2 2 2 H =− ∇ A− ∇ B − ∇ 2e − 2 m p
2 m p
2 me
e
2
4 0 r A
−
e
2
4 0 r B
e
2
4 0 R
If we consider the nuclei as being at fixed p ositions, we can treat R as a parameter and only worry about the behavior of the electron. This is known as the Born-Oppenheimer approximation . This is justified because nuclei are much more massive than electrons. Modified Hamiltonian:
ℏ2 ∇ 2e − H R=−
Wavefunctions:
± = N H 1 s ± H 1 s
2 me
A
e
2
4 0 r A B
−
e
2
4 0 r B
e
2
4 0 R
We use the atomic orbitals as our basis set, one on each proton. This is a linear 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 a bonding 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 an antibonding 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:
E = E j − E i = h
The energy of the light must equal the difference in energy of the states The transition dipole moment must not be zero
x ji =∫ ∗j x i dx = 0∫ j∗ x i dx
The second condition determines if a transition is allowed. It also defines the selection 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.)
ℏ2
Rotational Spectroscopy – We define the rotational constant as
ℏ2 = B= 2 I 2 r 20
The rotational states have energy
E rot = J J 1 B
Selection rules: molecule must have a permanent dipole, and
J =±1
Difference 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
2
I = r
Vibrational 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
Selection rules: motion must change the dipole moment, and
n=±1
Difference in energy levels:
E n n 1= h
1 h 2
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 infrared region 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.
2
1 1 and De is the dissociation energy. E n= n h − n x e h where x e = 4 D e 2 2
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 this fluorescence . If the molecule changes to a different excited state with a longer lifetime, the emission takes place over a longer time and is known as phosphorescence . 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 =
∑ e− / i
i
k T
. When we have different types of
molecular energy states, we can write the total internal energy as a sum;
T R V E i= i i i i .
Therefore, the partition function can be written as a product; q =q T q R qV q E .
Monatomic ideal gas : For this case, we only consider translational energy. The molecular partition
q=
function of a monatomic ideal gas is given by
2 m k T h
2
3/ 2
V
Polyatomic ideal gas : Use the same function for translations, degeneracy for electronic states.
q rot ,linear = where rot =
2 I k B
q vib=
1
σ Θrot
ℏ2
)
1/ 2
is the rotational temperature . σ is the symmetry number .
1 1−e
(
π T 3 q rot ,nonlinear = σ Θrot , x Θrot , y Θrot , z
T
−h / k B T
=
1 1−e
where
−vib / T
vib=
h is the vibrational temperature . k B
Canonical ensemble : A collection of microsystems that all have the same N, V and T . − E j / kT
The canonical distribution function is
j= e P
The canonical partition function for ideal gas is
q N Q= N !
Q
Thermodynamic Quantities of a Canonical Ensemble :
∑ M e−
E j / k T
We can determine the value of any thermodynamic variable M by M =
∑ M P = j
j
j 2
U = k T
∂ lnQ ∂ T
p = k T
N ,V
∂ lnQ ∂ V
S = k ln Q k T
N ,T
j
j
.
Q
∂ ln Q ∂ T
N ,V
Ergodic 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 =
1
− ⋅⋅⋅∫ e ∫ N !
E p / k T
d q1 d q2⋅⋅⋅d q N
If 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
2
f r dr where f =e
− E p k T
−1 .
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
6
r
Equations of State for Real Gases van der Waals EOS – Attempts to account for
Finite volume of molecules (b) Attractive interactions between molecules (a)
nRT n −a p= V − nb V
This EOS is written as
2
or p =
RT a − 2 V m −b V m
Virial EOS – Based on measured behavior of real gases. First we define a compression factor, Z
Z =
V m
p V m
V m
RT
= o
o
V m 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. 2
p V m= RT 1 B ' p C ' p ...
or
p V m= RT 1
B C 2 ... V m V m
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.
