C3303 Midterm 1 Flashcards
Classical thermodynamics
Relationship between mechanical and thermodynamic variables of a system
Mechanical Properties
Describe overall composition / position of a state; P, T, V
Thermodynamic Variables
Describe internal macroscopic state; U, H, A, G
Microcanonical Ensemble
Isolated system; no E and matter can exchange between the system and surroundings; V fixed
Canonical Ensemble
E can transfer across the boundary, but not matter; V is fixed
Isothermal-isobaric ensemble
Energy can transfer across boundary, but not matter; V of system can change such that the P is constant
Volume, units and in ensembles
m^3; in microcanonical and canonical, V is constant. A distribution of volumes is possible in isothermal-isobaric
Pressure of system, ensembles
in microcanonical and canonical, P depends on state of system; in isothermal-isobaric, the volume changes so the P of system is equal to P of surroundings
Internal energy
total E needed to create the system
What type of E describes intermolecular interactions?
U-pot
Enthalpy
Total E of system and E required to create a volume, V
Heat
thermal E transferred from surroundings to system
Work
E corresponding to expansion of system against surroundings
First Law of Thermo
dU=dq+dw
Entropy
Complexity; dS=dqrev/T
How is spontaneity determined
By change in E and entropy; Gibbs Energy and Hemholtz Energy
A equation
A=U-TS
G equation
G=H-TS
Pressure, T, heat capacity differential relations
p=-dU/dV
T=dU/dS
Cv=dU/dT
What are the limitations of classical thermodynamics?
No direct relationship between chemical structure and thermo; doesn’t allow us to predict dG of a reaction, but does explain why dG must be negative for spontaneity
Approximations to simplify models, ideal gases
Ideal gases have no intermolecular interactions and gas particles have no volume (point masses); ideal gases only have kinetic E
Equipartition Theorem
U=1/2*nDOFnRT
Degrees of Freedom
Number of independent ways the particle can move, resulting in a change to the original position. Depends on composition of gas particles
What are the 3 types of DOF?
- Translations: entire molecule in a direction
- Rotations: spinning along an axis
- Vibrations: stretching/bending of bonds
What are the DOF for a monoatomic gas?
Trans: 3 (x,y,z)
Rot: 0
Vib: 0
U=3/2*nRT
What are the DOF of diatomics?
Trans: 3
Rot: 2 (linear)
Vib: 1 (1 bond can vibrate)
Total: 6
Why do diatomics only have 2 rot DOF?
Rotation along the bond axis does not change appearance of molecule
What are the minimum number of atoms in a molecule needed to have 3 rot DOF?
3
Are there any molecules that are not diatomics that only have 2 rot DOF?
CO2
What is the general number of rot DOF for non-linear polyatomic molecule?
3
What is the relation between U and q for a change at constant volume?
dU=dq
What is the general expression of Cv?
Cv=dU/dT
What is the general expression for enthalpy? i.e. what is added ?
pV work term
H=U+pV
What is the general expression for Cp?
Cp=dH/dT
How can we relate Cv and Cp?
Cp=Cv+nR
Cp of monoatomic gases
Cp=5/2*nR
Cp of Diatomic gases (as predicted by equipartition)
Cp=7/2*nR
How does the Cp of diatomic gases present a failure of equipartition theory?
Equipartition performs less successfully for heat capacities of halogen gases, and gets worse as we get heavier. It approaches a value of R above predicted.
What prevents a vibrational mode from being populated at room temp?
Strong bond with light atoms
How much does a vibrational mode contribute to U?
Twice! U=nRT
Number of vib DOF, linear
3N-5
Number of vib DOF, non-linear
3N-6
What is U for linear molecules, including vibrations?
U=(3N-5/2)nRT
What is U for nonlinear molecules, including vibrations?
U=3(N-1)nRT
Review slide 30 calculation
Operators
mathematical functions that act on a wave function. Each corresponds to a physical property
Born Interpretation
Allows us to find probability density; that is, the probability density is the wavefunction multiplied by its complex conjugate and the probability of finding a particle in a certain range is the integral over this range.
Hamiltonian description
Eigenvalues of Hamiltonian operator give the E levels of the system.
Hamiltonian formula
see slide 33
What QM model describes translational motion?
Particle in a box
Describe the PIAB model
Assume a particle can move freely inside the box, thus potential E inside is 0, but cannot move outside of the box, thus potential E is infinite.
Quantum vs. Classical
Classical: particle can have any positive KE (continuous E levels) or particle can be motionless (0 E)
Quantum: only discrete/quantized E levels possible; lowest E level is non-zero (particle cannot be motionlesS)
Zero-E of PIAB
E=h^2/8ma^2
Why can PIAB not have 0 E?
Zero E contradicts uncertainty principle; no E means no momentum, i.e. the particle is stopped. Thus we would know both momentum and position exactly, which is not possible
What happens as we increase the E level (n) in translational motion?
Spacing becomes smaller and eventually will seem continuous.
Correspondence Principle
The behavior of a system described by QM should reproduce classical mechanics at large quantum numbers.
What states are degenerate with 2,1,1 for a 3D PIAB?
1,2,1 and 1,1,2
In what conditions does an energy level described by the 3D PIAB not have any degenerate E levels?
When all the quantum numbers are the same
True or False: the 3D translational states are very diffuse.
False, they are very dense
What happens to the density of translational E levels as the mass increases?
It becomes more dense, because as mass increases, E decreases and the gap decreases, so density increases
What is necessary to assume translational states can be summed (i.e. they do not affect other particles)?
The particles are ideal gas particles
Slide 51 calculation
Can translational transitions be observed spectroscopically?
No, the E spacings are too low
Is the correspondence principle applicable to translational E levels?
Yes, because they can be so low in E
Is rotational E kinetic or potential?
Kinetic
What assumption/model do we use to describe rotations?
Rigid rotor approximation
Describe the rigid rotor approximation
We assume the molecule is rigid, thus the change in bond length in a vibration is small relative to the length of the bond and there is no net change in PE
Moment of Inertia
sum of mass of each atom multiplied by distance from the axis of rotation
What is the equation for the moment of inertia of a triatomic linear rotor?
I=2mR^2 (only works for the same terminal atoms)
What are the 4 types of rigid rotors?
- Linear rotor, one moment equal to 0
- Spherical rotors: 3 equal moments of inertia
- Symmetric rotors: 2 equal moments
- Asymmetric rotors: 3 different moments
What is the reduced mass of a homonuclear diatomic?
m/2
General scale for bond lengths
angstroms; 10^-10m
What element can produce very short bonds?
H
Why are the bond lengths of H2, HD, and D2 so similar?
The difference between H and D is a neutron, which doesn’t change the bond length (would impact the mass)
Zero Point E of rigid rotor
Can have J=0; ground state E is 0.
Why can we have 0 E in the rigid rotor?
We would know its momentum but wouldn’t know anything about the positions/orientation
How many quantum numbers describe rotational E?
2! J and mJ
What does the secondary rot QN, mJ, describe?
Describes directionality, but not magnitude of angular momentum. Does not influence rotational E, but does describe degeneracy
Degeneracy of a rotational state:
g=2J+1
Density of Rotational States
spacing between rotational E increases quadratically as J is increased; degeneracy also increases. This means states are still dense, but not the same as translational levels
Will 15N2 or 15O2 have smaller rotational E spacings?
Mass is the same, so depends on bond length. Since E is inversely proportional to r^2, and O2 has a longer bond, O will have smaller spacings
Rotational Spectrum of Diatomics
Can see dJ=+-1 transitions. This falls within the microwave region and thus uses microwave spectroscopy (rotations only)
Rotational Spacings
Spacing of rotational transitions is constant!
dv=2B
What will microwave spectroscopy tell us if we know the masses of the diatomics?
Bond length!
Why don’t rotational spectra keep going to higher / lower frequencies?
- Low energy states are highly probable, but have low degeneracy
- High E transitions involve transitions between high E states; less likely to have that energy
What interaction is responsible for forming bonds?
Electrostatic interactions between protons and electrons
What type of electrostatic interactions are there? (3)
- Nuclear-electron attraction
- Electron-electron repulsion
- Inter-nuclear repulsion
For a diatomic, what does the PE depend on?
Distance between the two atoms
Where is the PE lowest?
Equilibrium bond length
What does the harmonic oscillator approximation involve?
We approximate the bond as a parabola centred around the equilibrium bond length and consider the bond as a spring.
What are 3 consequences of the harmonic oscillator approximation?
- The bond can never dissociate
- The repulsive wall isn’t repulsive enough
- The spacings of vib E levels are exactly equal, in reality they become slightly smaller as n increases
Why does the harmonic oscillator work?
The low PE structures (near equilibrium bond length) of molecules are the most important, and the harmonic oscillator approximation works well here.
If bonds really were harmonic, what would we not have?
Chemical reactions :(
What is the degeneracy of the states of a single QM harmonic oscillator?
All singly degenerate; degeneracy comes from multiple QNs, only 1 QN involved with vibrations
Vibrational Zero-Point Energy
1/2hv
Vibrational Absorption Spectra
Selection rule: dn=+-1
All vibrational transitions have the same E; dE=hv
What other type of transition is coupled with vibrations in IR?
Rotations
What is the total selection rule for rot-vib?
dn=+-1 and dJ=+-1 (need both)
What transitions occur in the P branch?
dJ=-1
What transitions occur in the R branch?
dJ=+1
Where is the pure vibration?
Q branch, dn=0, not visible in rot-vib spectrum
What can we expect of the force constant k for strong bonds?
Large k
What bonds have large vibrational E spacings?
Stiff bonds
Light atoms
Is the spacing of vib E for H2 or D2 larger?
H2, since H is lighter than D (smaller reduced mass = higher dE)
Will the density of vib states be larger or smaller for 15N2 or 15O2?
Smaller spacings = larger density.
The N-N bond is stronger, i.e. has a higher k. A higher k gives a higher dE, and thus is less dense. So, O2 has smaller spacings and a larger density.
How do we treat the vibrational states of polyatomics?
Treat each vibration as independent harmonic oscillators; have a QN for each vib mode
What are the 4 vibrational modes of CO2?
- Symmetric stretch
- Asymmetric stretch
3,4: bend (doubly degenerate)
What are the 3 vibrational frequencies of H2O?
- Bend
- Symmetric
- Asymmetric
With 4 or more atoms, what vibrational frequencies become possible?
Torsional rotation
What vibrations are stiffest?
Bond stretches; having a large dE; high frequency
Why are the bond stretch frequencies so high for CH, NH, and OH bonds?
freq (v) is proportional to sqrt(k), and inversely proportional to sqrt(1/u)
When H is involved, k is increased because it is small and forms short, strong bonds.
Additionally, H is light, so u is small and frequency increases.
Would you expect k for C-C stretch to be larger or smaller for ethane or ethene?
Ethene is shorter and stronger, so higher k
What are the 4 QNs for electronic E levels?
- Principal QN (n); n=1,2,3…
- Angular momentum QN (l); l=0,1,2,…,n-1
- Magnetic QN (ml); ml=-l,…,+l
- Spin QN (ms); ms=+-1/2
What QN(s) does the H atom depend on?
n
What do l and ml define in the H atom?
Degeneracy
All sets of ml and l with the same n are degenerate
What is the H model suitable for?
One electron systems
Besides ml and l, how else can electronic states be degenerate?
Electron spin
What is the spin degeneracy of a H atom in the ground state?
2; ms=+1/2, ms=-1/2
What is alpha and beta spin of an electron?
Alpha-spin = +1/2
Beta-spin = -1/2
For 2 degenerate unpaired electrons, what is the spin degeneracy?
3
2 Pure spin states (both up or both down)
1 Linear combination of spin states (a combination of 1 up, 1 down in both orders)
What is the spin degeneracy equation?
g=2S+1 where
S= sum(ms), unpaired electrons (take 1/2 for each unpaired electron)
What is the degeneracy of the ground state of helium?
S=0
g=1
What is the degeneracy of the ground state of N?
S=1/2+1/2+1/2=3/2
g=2(3/2)+1=4
How does the spacing of electronic states compare to trans, rot, and vib?
It is enormous;
only ground electronic state occupied at RT, but GS can be degenerate
What is an exception to electronic energy spacings?
It is possible to have a low lying electronic excited state if there is spin-orbit coupling (magnetic field created by electron)
Spin orbit coupling
Interaction between the magnetic moment of an unpaired electron with the angular momentum of its orbit can split the degeneracies of electronic states
Equation for J in term symbolds
J=|L-S| for shells less than half filled (L is sum of ml values)
J=L+S for half filled or greater
What is the main difference between term symbols for atoms and molecules?
For molecules, they are assigned a greek letter, and symmetry of MO is included
Spectroscopy
studies the absorption or emission of EMR by matter; deals with all interactions between light and matter, including scattering and rotation of plane of polarization
What are the 3 Einstein Classifications of Transitions?
- Stimulated absorption: transition from lower E state to higher E state due to absorption of photon
- Stimulated emission: transition from higher E to lower E due to absorption of photon
- Spontaneous emission: transition from higher E to lower E due to emission of photon
Vibrational Selection Rules
Gross selection rule: dipole moment of the molecule must change when atoms are displaced.
Specific selection rule: dn=+-1
Rotational Selection Rules
Gross selection rule for rotational spectroscopy: a molecule must possess a permanent dipole moment
Specific selection rule: dJ=+-1
Which of the following molecules will have a pure rotational spectrum? H2, HF, CH4, NO
HF, NO are polar - have pure rot spectrum
H2, CH4 are non-polar - no rot spectrum
Overtones
Anharmonicity of real bonds result in small, but non-zero, probability for dn=2,3,… transitions
Combination Bands
absorption of single photon can excite a combination of fundamental vibrational modes
Hot bands
At RT, majority of vib are in GS, which is why the first transition is dominant. At high T, transitions can occur between excited states
Scattering
when light passes through a gas, some photons change direction, even though freq is not absorbed; elastic (no change in E); equivalent to photon bouncing off molecule in diff direction
Why is the sky blue?
Short wavelength light is scattered at great intensity; human eye is not highly sensitive to wavelengths below blue
Raman Effect
When sample is exposed to intense, high E light source, most photons are scattered elastically, but 1 in 10^7 photons emerge at a higher or lower frequency
Rayleigh Line
Same frequency as source; most intense band
Stokes lines
lower frequency (excitation of molecule)
Anti-Stokes lines
Higher frequency (relaxation of molecule)
Raman Spectroscopy
gain or loss of E by incident light corresponds to transitions between the E levels of the molecules.
High E photon excites to “virtual state”, and the molecule returns to a different state on emission
Gross selection rule for raman spectroscopy
A molecule must be anisotropically polarizable (1 moment of inertia is not 0) to have a raman spectrum
Isotropic: same in every direction
Anisotropic: different in some directions
Works for diatomics
What molecules can’t be studies with Raman?
Spherical tops (tetrahedral, octahedral)
All others are Raman active
Specific Selection Rule of Rot Raman Spec
dJ=0,+ - 2
Double spacing of regular lines (dv=4B)
Selection Rules for Vib Raman
dn=+-1
Same as vib spec
O Branch
dJ=-2
Q Branch
dJ=0
S Branch
dJ=+2
Review Diagram of raman spectra (and equations for transitions)
What is the total energy of a general system
Sum of trans, rot, vib, and electronic energies
How do we define a microcanonical ensemble in stat mech?
Physical system where total number of particles (N), total volume (V), and total E are constant (abbrev NVE)
System cannot exchange E with surroundings
What configurations of NVE are possible?
Any configuration with a total energy E are possible
What is the distinguishability of configurations?
If you can assign a unique label to each particle, they are distinguishable
If particles are free to exchange (impossible to assign a unique label) they are indistinguishable
Are particles in gasses / liquids distinguishable?
No
Are particles in solids distinguishable?
Yes
What is the ground state E in stat mech?
Lowest accessible state is always E=0 (shifted)
Weight of Macrostates
The distinct sets of QNs (microstates) that give the same occupancy of states are part of the same macrostate
(i.e. (0,2) and (2,0) are 2 microstates of the same macrostate, whereas (1,1) would be a microstate of a different macrostate of the same E)
What is the number of ways N items can be arranged?
N!
When we are dealing with a large number of items, the number of ways of arranging them is ___?
Enormous
Weight of a Configuration
W=A!/(a1!a2!….)
where A is the total number of systems and ai represents the number of systems in a given level, i
Why are high E states less likely than lower E states?
There are fewer combinations of QNs that yield them; i.e. they have a low weight
What must we derive to consider systems that are not isolated?
The distribution of states when E is not constant
Mechanical Variables
Direct physical property of the system; V,N,E,P
Non-mechanical Variables
Require thermo to be defined/determined; T, S, A, G, U
Is the density of a system mechanical or non-m?
Mechanical
Is the heat capacity mech or non-mech?
Non-mech
Why is it a bad idea to take a time average of a single system?
Motion of particles and transitions is very complicated; timescale would be huge; number of particles is extremely large
What do we use instead of a time average?
Instantaneous average over many systems. This is a purely conceptual strategy to relate microscopic configurations to thermo properties for a single system.
First Postulate of Stat Mech
The time average of a mech variable M in the thermo system of interest is equal to the ensemble average of M in the limit A goes to infinity.
Gibbs Postulate
The internal Energy (U) is simply the average of the E of the system;
U=<E> and p=<p>
Can connect to other thermo parameters</E>
Second postulate of stat mech
For an ensemble rep of an isolated system, the systems of the ensemble are distributed uniformly.
All states will occur with equal probability.
