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3rd Year Winter > C3303 Midterm 1 > Flashcards

C3303 Midterm 1 Flashcards

1
Q

Classical thermodynamics

A

Relationship between mechanical and thermodynamic variables of a system

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2
Q

Mechanical Properties

A

Describe overall composition / position of a state; P, T, V

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3
Q

Thermodynamic Variables

A

Describe internal macroscopic state; U, H, A, G

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4
Q

Microcanonical Ensemble

A

Isolated system; no E and matter can exchange between the system and surroundings; V fixed

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5
Q

Canonical Ensemble

A

E can transfer across the boundary, but not matter; V is fixed

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6
Q

Isothermal-isobaric ensemble

A

Energy can transfer across boundary, but not matter; V of system can change such that the P is constant

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7
Q

Volume, units and in ensembles

A

m^3; in microcanonical and canonical, V is constant. A distribution of volumes is possible in isothermal-isobaric

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7
Q

Pressure of system, ensembles

A

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

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8
Q

Internal energy

A

total E needed to create the system

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9
Q

What type of E describes intermolecular interactions?

A

U-pot

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10
Q

Enthalpy

A

Total E of system and E required to create a volume, V

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11
Q

Heat

A

thermal E transferred from surroundings to system

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12
Q

Work

A

E corresponding to expansion of system against surroundings

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13
Q

First Law of Thermo

A

dU=dq+dw

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14
Q

Entropy

A

Complexity; dS=dqrev/T

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15
Q

How is spontaneity determined

A

By change in E and entropy; Gibbs Energy and Hemholtz Energy

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16
Q

A equation

A

A=U-TS

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17
Q

G equation

A

G=H-TS

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18
Q

Pressure, T, heat capacity differential relations

A

p=-dU/dV
T=dU/dS
Cv=dU/dT

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19
Q

What are the limitations of classical thermodynamics?

A

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

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20
Q

Approximations to simplify models, ideal gases

A

Ideal gases have no intermolecular interactions and gas particles have no volume (point masses); ideal gases only have kinetic E

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21
Q

Equipartition Theorem

A

U=1/2*nDOFnRT

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22
Q

Degrees of Freedom

A

Number of independent ways the particle can move, resulting in a change to the original position. Depends on composition of gas particles

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23
Q

What are the 3 types of DOF?

A
  1. Translations: entire molecule in a direction
  2. Rotations: spinning along an axis
  3. Vibrations: stretching/bending of bonds
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24
Q

What are the DOF for a monoatomic gas?

A

Trans: 3 (x,y,z)
Rot: 0
Vib: 0
U=3/2*nRT

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25
Q

What are the DOF of diatomics?

A

Trans: 3
Rot: 2 (linear)
Vib: 1 (1 bond can vibrate)
Total: 6

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26
Q

Why do diatomics only have 2 rot DOF?

A

Rotation along the bond axis does not change appearance of molecule

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27
Q

What are the minimum number of atoms in a molecule needed to have 3 rot DOF?

A

3

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28
Q

Are there any molecules that are not diatomics that only have 2 rot DOF?

A

CO2

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29
Q

What is the general number of rot DOF for non-linear polyatomic molecule?

A

3

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30
Q

What is the relation between U and q for a change at constant volume?

A

dU=dq

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31
Q

What is the general expression of Cv?

A

Cv=dU/dT

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32
Q

What is the general expression for enthalpy? i.e. what is added ?

A

pV work term
H=U+pV

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33
Q

What is the general expression for Cp?

A

Cp=dH/dT

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34
Q

How can we relate Cv and Cp?

A

Cp=Cv+nR

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35
Q

Cp of monoatomic gases

A

Cp=5/2*nR

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36
Q

Cp of Diatomic gases (as predicted by equipartition)

A

Cp=7/2*nR

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37
Q

How does the Cp of diatomic gases present a failure of equipartition theory?

A

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.

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38
Q

What prevents a vibrational mode from being populated at room temp?

A

Strong bond with light atoms

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39
Q

How much does a vibrational mode contribute to U?

A

Twice! U=nRT

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40
Q

Number of vib DOF, linear

A

3N-5

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41
Q

Number of vib DOF, non-linear

A

3N-6

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42
Q

What is U for linear molecules, including vibrations?

A

U=(3N-5/2)nRT

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43
Q

What is U for nonlinear molecules, including vibrations?

A

U=3(N-1)nRT

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44
Q

Review slide 30 calculation

A
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45
Q

Operators

A

mathematical functions that act on a wave function. Each corresponds to a physical property

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46
Q

Born Interpretation

A

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.

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47
Q

Hamiltonian description

A

Eigenvalues of Hamiltonian operator give the E levels of the system.

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48
Q

Hamiltonian formula

A

see slide 33

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49
Q

What QM model describes translational motion?

A

Particle in a box

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50
Q

Describe the PIAB model

A

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.

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51
Q

Quantum vs. Classical

A

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)

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52
Q

Zero-E of PIAB

A

E=h^2/8ma^2

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53
Q

Why can PIAB not have 0 E?

A

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

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54
Q

What happens as we increase the E level (n) in translational motion?

A

Spacing becomes smaller and eventually will seem continuous.

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55
Q

Correspondence Principle

A

The behavior of a system described by QM should reproduce classical mechanics at large quantum numbers.

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56
Q

What states are degenerate with 2,1,1 for a 3D PIAB?

A

1,2,1 and 1,1,2

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57
Q

In what conditions does an energy level described by the 3D PIAB not have any degenerate E levels?

A

When all the quantum numbers are the same

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58
Q

True or False: the 3D translational states are very diffuse.

A

False, they are very dense

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59
Q

What happens to the density of translational E levels as the mass increases?

A

It becomes more dense, because as mass increases, E decreases and the gap decreases, so density increases

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60
Q

What is necessary to assume translational states can be summed (i.e. they do not affect other particles)?

A

The particles are ideal gas particles

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61
Q

Slide 51 calculation

A
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62
Q

Can translational transitions be observed spectroscopically?

A

No, the E spacings are too low

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63
Q

Is the correspondence principle applicable to translational E levels?

A

Yes, because they can be so low in E

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64
Q

Is rotational E kinetic or potential?

A

Kinetic

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65
Q

What assumption/model do we use to describe rotations?

A

Rigid rotor approximation

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66
Q

Describe the rigid rotor approximation

A

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

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67
Q

Moment of Inertia

A

sum of mass of each atom multiplied by distance from the axis of rotation

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68
Q

What is the equation for the moment of inertia of a triatomic linear rotor?

A

I=2mR^2 (only works for the same terminal atoms)

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69
Q

What are the 4 types of rigid rotors?

A
  1. Linear rotor, one moment equal to 0
  2. Spherical rotors: 3 equal moments of inertia
  3. Symmetric rotors: 2 equal moments
  4. Asymmetric rotors: 3 different moments
70
Q

What is the reduced mass of a homonuclear diatomic?

A

m/2

71
Q

General scale for bond lengths

A

angstroms; 10^-10m

72
Q

What element can produce very short bonds?

A

H

73
Q

Why are the bond lengths of H2, HD, and D2 so similar?

A

The difference between H and D is a neutron, which doesn’t change the bond length (would impact the mass)

74
Q

Zero Point E of rigid rotor

A

Can have J=0; ground state E is 0.

75
Q

Why can we have 0 E in the rigid rotor?

A

We would know its momentum but wouldn’t know anything about the positions/orientation

76
Q

How many quantum numbers describe rotational E?

A

2! J and mJ

77
Q

What does the secondary rot QN, mJ, describe?

A

Describes directionality, but not magnitude of angular momentum. Does not influence rotational E, but does describe degeneracy

78
Q

Degeneracy of a rotational state:

A

g=2J+1

79
Q

Density of Rotational States

A

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

80
Q

Will 15N2 or 15O2 have smaller rotational E spacings?

A

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

81
Q

Rotational Spectrum of Diatomics

A

Can see dJ=+-1 transitions. This falls within the microwave region and thus uses microwave spectroscopy (rotations only)

82
Q

Rotational Spacings

A

Spacing of rotational transitions is constant!
dv=2B

83
Q

What will microwave spectroscopy tell us if we know the masses of the diatomics?

A

Bond length!

84
Q

Why don’t rotational spectra keep going to higher / lower frequencies?

A
  • 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
85
Q

What interaction is responsible for forming bonds?

A

Electrostatic interactions between protons and electrons

86
Q

What type of electrostatic interactions are there? (3)

A
  1. Nuclear-electron attraction
  2. Electron-electron repulsion
  3. Inter-nuclear repulsion
87
Q

For a diatomic, what does the PE depend on?

A

Distance between the two atoms

88
Q

Where is the PE lowest?

A

Equilibrium bond length

89
Q

What does the harmonic oscillator approximation involve?

A

We approximate the bond as a parabola centred around the equilibrium bond length and consider the bond as a spring.

90
Q

What are 3 consequences of the harmonic oscillator approximation?

A
  1. The bond can never dissociate
  2. The repulsive wall isn’t repulsive enough
  3. The spacings of vib E levels are exactly equal, in reality they become slightly smaller as n increases
91
Q

Why does the harmonic oscillator work?

A

The low PE structures (near equilibrium bond length) of molecules are the most important, and the harmonic oscillator approximation works well here.

92
Q

If bonds really were harmonic, what would we not have?

A

Chemical reactions :(

93
Q

What is the degeneracy of the states of a single QM harmonic oscillator?

A

All singly degenerate; degeneracy comes from multiple QNs, only 1 QN involved with vibrations

94
Q

Vibrational Zero-Point Energy

A

1/2hv

95
Q

Vibrational Absorption Spectra

A

Selection rule: dn=+-1
All vibrational transitions have the same E; dE=hv

96
Q

What other type of transition is coupled with vibrations in IR?

A

Rotations

97
Q

What is the total selection rule for rot-vib?

A

dn=+-1 and dJ=+-1 (need both)

98
Q

What transitions occur in the P branch?

A

dJ=-1

99
Q

What transitions occur in the R branch?

A

dJ=+1

100
Q

Where is the pure vibration?

A

Q branch, dn=0, not visible in rot-vib spectrum

101
Q

What can we expect of the force constant k for strong bonds?

A

Large k

102
Q

What bonds have large vibrational E spacings?

A

Stiff bonds
Light atoms

103
Q

Is the spacing of vib E for H2 or D2 larger?

A

H2, since H is lighter than D (smaller reduced mass = higher dE)

104
Q

Will the density of vib states be larger or smaller for 15N2 or 15O2?

A

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.

105
Q

How do we treat the vibrational states of polyatomics?

A

Treat each vibration as independent harmonic oscillators; have a QN for each vib mode

106
Q

What are the 4 vibrational modes of CO2?

A
  1. Symmetric stretch
  2. Asymmetric stretch
    3,4: bend (doubly degenerate)
107
Q

What are the 3 vibrational frequencies of H2O?

A
  1. Bend
  2. Symmetric
  3. Asymmetric
108
Q

With 4 or more atoms, what vibrational frequencies become possible?

A

Torsional rotation

109
Q

What vibrations are stiffest?

A

Bond stretches; having a large dE; high frequency

110
Q

Why are the bond stretch frequencies so high for CH, NH, and OH bonds?

A

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.

111
Q

Would you expect k for C-C stretch to be larger or smaller for ethane or ethene?

A

Ethene is shorter and stronger, so higher k

112
Q

What are the 4 QNs for electronic E levels?

A
  1. Principal QN (n); n=1,2,3…
  2. Angular momentum QN (l); l=0,1,2,…,n-1
  3. Magnetic QN (ml); ml=-l,…,+l
  4. Spin QN (ms); ms=+-1/2
113
Q

What QN(s) does the H atom depend on?

A

n

114
Q

What do l and ml define in the H atom?

A

Degeneracy
All sets of ml and l with the same n are degenerate

115
Q

What is the H model suitable for?

A

One electron systems

116
Q

Besides ml and l, how else can electronic states be degenerate?

A

Electron spin

117
Q

What is the spin degeneracy of a H atom in the ground state?

A

2; ms=+1/2, ms=-1/2

118
Q

What is alpha and beta spin of an electron?

A

Alpha-spin = +1/2
Beta-spin = -1/2

119
Q

For 2 degenerate unpaired electrons, what is the spin degeneracy?

A

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)

120
Q

What is the spin degeneracy equation?

A

g=2S+1 where
S= sum(ms), unpaired electrons (take 1/2 for each unpaired electron)

121
Q

What is the degeneracy of the ground state of helium?

A

S=0
g=1

122
Q

What is the degeneracy of the ground state of N?

A

S=1/2+1/2+1/2=3/2
g=2(3/2)+1=4

123
Q

How does the spacing of electronic states compare to trans, rot, and vib?

A

It is enormous;
only ground electronic state occupied at RT, but GS can be degenerate

124
Q

What is an exception to electronic energy spacings?

A

It is possible to have a low lying electronic excited state if there is spin-orbit coupling (magnetic field created by electron)

125
Q

Spin orbit coupling

A

Interaction between the magnetic moment of an unpaired electron with the angular momentum of its orbit can split the degeneracies of electronic states

126
Q

Equation for J in term symbolds

A

J=|L-S| for shells less than half filled (L is sum of ml values)
J=L+S for half filled or greater

127
Q

What is the main difference between term symbols for atoms and molecules?

A

For molecules, they are assigned a greek letter, and symmetry of MO is included

128
Q

Spectroscopy

A

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

129
Q

What are the 3 Einstein Classifications of Transitions?

A
  1. Stimulated absorption: transition from lower E state to higher E state due to absorption of photon
  2. Stimulated emission: transition from higher E to lower E due to absorption of photon
  3. Spontaneous emission: transition from higher E to lower E due to emission of photon
130
Q

Vibrational Selection Rules

A

Gross selection rule: dipole moment of the molecule must change when atoms are displaced.
Specific selection rule: dn=+-1

131
Q

Rotational Selection Rules

A

Gross selection rule for rotational spectroscopy: a molecule must possess a permanent dipole moment
Specific selection rule: dJ=+-1

132
Q

Which of the following molecules will have a pure rotational spectrum? H2, HF, CH4, NO

A

HF, NO are polar - have pure rot spectrum
H2, CH4 are non-polar - no rot spectrum

133
Q

Overtones

A

Anharmonicity of real bonds result in small, but non-zero, probability for dn=2,3,… transitions

134
Q

Combination Bands

A

absorption of single photon can excite a combination of fundamental vibrational modes

135
Q

Hot bands

A

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

136
Q

Scattering

A

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

137
Q

Why is the sky blue?

A

Short wavelength light is scattered at great intensity; human eye is not highly sensitive to wavelengths below blue

138
Q

Raman Effect

A

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

139
Q

Rayleigh Line

A

Same frequency as source; most intense band

140
Q

Stokes lines

A

lower frequency (excitation of molecule)

141
Q

Anti-Stokes lines

A

Higher frequency (relaxation of molecule)

142
Q

Raman Spectroscopy

A

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

143
Q

Gross selection rule for raman spectroscopy

A

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

144
Q

What molecules can’t be studies with Raman?

A

Spherical tops (tetrahedral, octahedral)
All others are Raman active

145
Q

Specific Selection Rule of Rot Raman Spec

A

dJ=0,+ - 2
Double spacing of regular lines (dv=4B)

146
Q

Selection Rules for Vib Raman

A

dn=+-1
Same as vib spec

147
Q

O Branch

A

dJ=-2

148
Q

Q Branch

A

dJ=0

149
Q

S Branch

A

dJ=+2

150
Q

Review Diagram of raman spectra (and equations for transitions)

A
151
Q

What is the total energy of a general system

A

Sum of trans, rot, vib, and electronic energies

152
Q

How do we define a microcanonical ensemble in stat mech?

A

Physical system where total number of particles (N), total volume (V), and total E are constant (abbrev NVE)
System cannot exchange E with surroundings

153
Q

What configurations of NVE are possible?

A

Any configuration with a total energy E are possible

154
Q

What is the distinguishability of configurations?

A

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

155
Q

Are particles in gasses / liquids distinguishable?

A

No

156
Q

Are particles in solids distinguishable?

A

Yes

157
Q

What is the ground state E in stat mech?

A

Lowest accessible state is always E=0 (shifted)

158
Q

Weight of Macrostates

A

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)

159
Q

What is the number of ways N items can be arranged?

A

N!

160
Q

When we are dealing with a large number of items, the number of ways of arranging them is ___?

A

Enormous

161
Q

Weight of a Configuration

A

W=A!/(a1!a2!….)
where A is the total number of systems and ai represents the number of systems in a given level, i

162
Q

Why are high E states less likely than lower E states?

A

There are fewer combinations of QNs that yield them; i.e. they have a low weight

163
Q

What must we derive to consider systems that are not isolated?

A

The distribution of states when E is not constant

164
Q

Mechanical Variables

A

Direct physical property of the system; V,N,E,P

165
Q

Non-mechanical Variables

A

Require thermo to be defined/determined; T, S, A, G, U

166
Q

Is the density of a system mechanical or non-m?

A

Mechanical

167
Q

Is the heat capacity mech or non-mech?

A

Non-mech

168
Q

Why is it a bad idea to take a time average of a single system?

A

Motion of particles and transitions is very complicated; timescale would be huge; number of particles is extremely large

169
Q

What do we use instead of a time average?

A

Instantaneous average over many systems. This is a purely conceptual strategy to relate microscopic configurations to thermo properties for a single system.

170
Q

First Postulate of Stat Mech

A

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.

171
Q

Gibbs Postulate

A

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>

172
Q

Second postulate of stat mech

A

For an ensemble rep of an isolated system, the systems of the ensemble are distributed uniformly.
All states will occur with equal probability.

3rd Year Winter (8 decks)

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