C3303 Final Flashcards
Classical thermodynamics
Relationship between mechanical and thermodynamic variables in a system
Mechanical properties
Describe composition/position (P, V)
Thermodynamic variables
Describe internal macroscopic state (U, H, A, G)
Microcanonical ensemble
System is totally isolated from surroundings; no E or matter can cross; V fixed (NVE); unrealistic
Canonical ensemble
E can transfer across boundary, but not matter; V fixed (NVT)
Isothermal-isobaric ensemble
Energy can transfer across boundary, but not matter; V can change to maintain P; (NPT)
Internal E of a system
Total E needed to create the system
Enthalpy
Total E of system and energy required to create a volume, V
Heat
Thermal E transferred from surr to sys
Work
E corresponding to expansion of system against surr
First Law of Thermo
dU=dq+dw
Entropy
change in degree of complexityE
Entropy relation
ds=dqrev/T
Hemholtz Energy
Changes in E at constant V
Gibbs Energy
Changes in E in the isothermal-isobaric ensemble at constant P
Limitations of Classical Thermo
No direct relationship between chemical structure and thermo
Ideal Gas Assumption
No intermolecular interactions
No volume
Point Masses
No potential E
Equipartition theorem
U=1/2 nDOF nRT
Degrees of Freedom (DOF)
Number of independent ways the particle can move resulting in a change to the original position
What are the 3 types of DOF?
- Translations
- Rotations
- Vibrations
Number of translations
3 (x,y,z)
Number of rotations
Linear: 2
Nonlinear: 3
Number of Vibrations
Linear: 3N-5
NonLinear: 3N-6
Why do diatomics only have 2 rot DOF?
Rotation along the bond axis does not change appearance of molecule
What are the min # atoms in a molecule needed to have 3 rot DOF?
3
Are there any molecules that are not diatomics that only have 2 DOF?
CO2
Heat capacity at volume eqn
Cv=(dU/dT)
Heat capacity at constant P eqn
Cp=(dH/dT)
Relation between Cp and Cv
Cp=Cv+nR
Why does equipartition theory perform less successfuly for heat capacities of halogen gases?
Equipartition ignores vibrations; these molecules are larger and heavier and their vibrations cannot be ignored
When can vibrations be ignored?
Strong bonds with light atoms
What do vibrations contribute to U?
nRT for each vib
What is the equipartition theory with vibrations included?
Linear: U = (3N - 5/2) nRT
NonLinear: U = 3(N-1)nRT
The Born Interpretation
The probability of finding a particle in a range [x1,x2] is the integral of the wavefunction multiplied by its complex conjugate over this range
Hamiltonial
Eigenvalues give E levels of the system
Particle in a Box
We consider the movement of an atom in a box (3D). The particle can move freely in the box, with PE=0, but cannot move outside the walls of the box, PE=infinity.
What is the E of particle in a 1D box?
E=h^2n^2 / 8ma^2
Zero-Point E for PIAB
E=h^2 / 8ma^2
Particle is always moving
Is the result of the Zero-Point E of PIAB consistent with Heisenberg uncertainty principle?
Yes, no E means no momentum; the particle is stopped. Since the Zero-point E is nonzero, this is allowed.
The Correspondence Principle
The behaviour of a system described by QM should reproduce classical mechanics at large QNs
Particle in a 3D Box
Extension of PIAB to 3 dimensions. Describes 3 QNs; nx, ny and nz. The E:
E = h^2 / 8ma^2 * (nx^2 + ny^2 + nz^2)
Degeneracy of PIA3DB
Any combination of QNs that give the same E; for example, (2,1,1) and (1,1,2) are degenerate.
What is the interpretation of degeneracy of PIA3DB?
Degenerate E levels have the same amount of trans E, just in different directions
How many degenerate states are possible for PIAC with QNs (1,2,3)?
6
(1,2,3)
(1,3,2)
(2,1,3)
(2,3,1)
(3,1,2)
(3,2,1)
Are the PIAC E states dense; why?
The spacing between E levels are small and the states can be degenerate. Thus, the 3D translation states are incredibly dense
Are the trans E levels of a particle more or less dense if the mass is larger?
More dense, because E is inversely proportional to mass, thus E is lower and the gaps decrease
Are the translational E levels of a particle more or less dense if the box is larger?
More dense because E is inversely proportional to the length of the box squared; lower E spacings = higher density
EX slide 51
Can we view translational transitions spectroscopically?
No, the E spacings are too low
Is the correspondance principle applicable to trans E levels?
Yes, can treat trans levels as continuous/classical
What is the first excited state for translational E (give QNs)
(2,1,1), triply degenerate
Rotational E is purely what?
Kinetic
What assumptions do we make in rotation of ideal gas molecules?
We assume the molecule is rigid, i.e. the change in bond length in a vibration is small relative to the total length of the bond.
Rigid Rotor Approximation
assuming rigidity means no net change in PE btwn atoms in a bond. Useful for description of rot E.
Moment of Inertia
I = ur^2
What is the eqn for the reduced mass of a homonuclear diatomic?
m/2
What is the general scale of bond lengths for diatomic molecules?
0.74 - 3 angstroms (1 angstrom = 10^-10 m)
Bond Length Trends
Increase down PT; larger atoms means longer distance between two nuclei in the bond
The bond lengths of H2, HD, and D2 are very similar. Why?
Diff btwn H and D is 1 neutron; changes the mass but not the bond length, as only p+ and e- impact the bond length.
Zero-Point E of Rigid Rotor
J=0; ground state E lvl is 0
Explain 0 E of Rigid Rotor with Uncertainty Principle
We know momentum with certainty since its zero, but we know nothing about its orientation/position, so no violation occurs
How many QN are in the Rigid Rotor Approx?
2; J and m.
The secondary QN, m, describes directionality. Does not influence magnitude of E but influences degeneracy.
Degeneracy of Rigid Rotor
For a given J, there are 2J+1 values of mJ (from - J to +J)
What is the relation of the spacing between rot E levels?
Quadratic
What is the density of rot E?
The degeneracy increases with J, so the states are still dense, but not to the same extent as trans lvls.
Which molecule will have smaller E spacings; 15N2 or 15O2?
Mass is the same. E is inversely proportional to r^2 (bond length). Since O2 has a longer bond length, there will be smaller spaces.
Can we see rotational transitions spectroscopically?
Yes, we can in the microwave or IR region
What transitions are allowed for rotational spectroscopy?
dJ=+ or - 1
Wavenumber equation
v~ = v / c
Note that variables with ~ have been divided by speed of light
Rotational spacings in spectroscopy
This is constant. The gap between two peaks is 2B~ where B~ = h / 8pi^2cI
Why do rotational spectra not keep going to higher frequencies?
There is no mathematical limit on how fast we can spin a molecule. We only see several lines because low E states are highly probable with low degeneracy. High E transitions involve transitions between high E states, which has less probability of occuring.
See ex page 73
Chemical bonds
Electrostatic interactions between protons and electrons
Types of PE interactions on surface of molecules (3)
- Nuclear-electron attraction (bond)
- e- - e- repulsion
- Inter nuclear repulsion
Equilibrium bond length
Where the PE is lowest; the bond is most commonly at this length in vibrations
Molecular vibration
Oscillation of bond length due to kinetic energy
Harmonic Oscillator Approximation
We approximate the bond as a parabola centred around the equilibrium bond length. Consider the bond a spring, such that when it is stretched or compressed, the PE increases.
What is k in the Harmonic Oscillator Approx?
Hooke’s constant, corresponding to stiffness of the spring, and i.e. strength of bond (higher k = stronger bond)
Justification of HO Approx
Low PE structure are most important. HO Approx works well here, fails at higher E states.
Anharmonic effects can be important
What is the degeneracy of HO?
Singly degenerate, since there is only 1 QN
Vibrational Zero-Point E
E0=1/2 hv
Describe Vib Zero-Point E with Heisenberg Principle
If a molecule had zero vib E, it would be at rest; we would know its position and momentum completely.
Vibrational Absorption Selection Rule
dn = + or - 1
What is the transition between vib E spacings?
dE = hv
Vib / Rot Spectra
Vibrational spectrum is coupled with rot transitions, since both are visible in IR region.
Total Vib / Rot Selection Rule:
dn= +-1, dJ = +-1; need both
P branch
dJ = -1
R branch
dJ = +1
Where do P/R branch fall on spectra?
If wavenumbers are increasing from L - R, P branch is on the left and R branch is on the right
What can we not see in Vib / Rot Spectra? Why?
Pure vibrational transition
This is dJ=0 which is not allowed, so it doesn’t appear.
What is the Q branch
Gap between P and R branches due to pure vibrational transition
How can we calculate the frequency of vibration from Vib / Rot spectra?
Take the average of the inner peaks of P and R branch and convert to appropriate units.
Ex Q Slide 87-90 Unit 1
What are the 3 consequences of the HO Approx?
- The bond can never dissociate (HO goes on forever)
- The repulsive wall isn’t repulsive enough (short distances are dominated by repulsive interactions)
- The spacings of vib E levels are exactly equal (in reality they become slightly smaller as n increases)
Do light atoms give large or small reduced masses?
Small
Is the spacing of vib E larger for H2 or D2?
dE is proportional to v, which is inversely proportional to u. Thus, since H2 is lighter, the dE is larger and the spacings are larger.
Will the density of vibrational states be larger or smaller for 15N2 or 15O2?
dE is proportional to k. Since N has a stronger bond, k is larger for N. Thus, dE is larger for N, and thus the density of states is smaller.
Types of vibrations
- Asymmetric Stretch
- Symmetric stretch
- Bends
- Others (for 4 or more atoms) such as torsions
What vibrations are stiffest? What does this mean?
Bond stretches; large k = large increase in E = high frequency
Why do C-H, N-H, and O-H bonds have large bond stretch vib frequencies.
The bond stretch vib frequency is inversely proportional to u and proportional to k. When H is involved, k is very high because H is small, forming very short / strong bonds. Reduced mass is also small since H is light. These both work toward increasing the dE.
Would you expect bond spring constant for C-C stretch to be larger or smaller for ethane or ethene?
Ethene, since a double bond is stronger and has higher k.
Electronic Energy Model
The Hydrogen Atom (1 proton and 1 electron)
What QN contribute to electronic E?
- Principle QN (n)
- ANgular momentum (l)
- Magnetic QN (ml)
- Spin QN (ms)
What QN(s) does electronic E depend on?
n
What is the Hydrogen model suitable for?
One-electron systems; He+, Li2+, Be3+, etc.
How do we adjust the electronic E formula for one-electron systems other than H?
Add a Z^2 term corresponding to the atomic number in the numerator
Why are electronic transitions not very common at RT?
E spacings are so large that the probability of this transition is so low at RT. However, electronic ground state E levels can be degenerate.
Electron spin
The electron behaves as though it is a spinning sphere; intrinsic angular momentum is due to spinning
Which angular momenta for spin can be defined?
Lz; Lx and Ly cannot be specified
What letter do we represent spin up with?
alpha
What letter do we represent spin down with?
beta
What is the most important thing to know about spin degeneracy?
Electronic states can be degenerate due to energetically equivalent combos of unpaired electrons.
How do we find electron degeneracy?
g = 2S+1
Where S is the total spin angular momentum (sum of all unpaired e-)
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= 3/2 ; g = 4
About how much more E will it require to excite a molecule of H2 from GS to 1st ES for vib vs. electronic energy?
140 times more
Spin-orbit coupling
an electron orbiting a nucleus effectively has a magnetic field due to its motion in the electric field of the nucleus. This can split the spin states slightly.
Term symbols
2S+1LJ
What is L in the term symbol
sum of ml values of each e- in the subshell; use the letter
What is J in term symbols
J = |L-S| for shells less than half-filled
J = L+S for shells half filled or greater
What should be ignored when considering term symbols?
Paired electrons (inner shells)
Review term symbol examples
Slide 118 ish
What is the degeneracy in a term symbol?
The superscript
Spectroscopy
studies the absorption or emission of EMR by matter
Raman Spectroscopy
scattering of light
Einstein Classification of Transitions (3)
- Stimulated absorption: transition from lower E to higher E due to absorption of photon
- Stimulated emission: transition from higher E to lower E due to absorption of photon (one photon absorbed, 2 emitted)
- Spontaneous emission: transition from higher E to lower E due to emission of photon
Vibrational Selection rules
Gross selection: dipole moment of molecule must change
Specific: dn = +-1
Can we see homonuclear diatomics in IR?
No
Rotational Selection Rules (Microwave)
Gross selection rule: a molecule must possess a permanent dipole moment
Specific: dJ = +-1
What molecules have no pure rotational spectrum?
Non-polar
Which of the following molecules will have a pure rot spectrum: H2, HF, CH4, NO
HF, NO are polar
IR Overtones
Occur at multiples for fundamental frequency. Less intense than the band
IR Combination Bands
Excitation of a combination of fundamental vibrational modes. Usually at lower absorbances.
IR Hot Bands
At very high T, transitions can occur btwn excited states; in an ideal harmonic oscillator, these would be the same as the fundamental frequency. Anharmonicity results in differences in these higher transitions
Light Scattering
when light passess through a gas, some photons change direction even though frequency of radiation is not absorbed by the molecule.
What does EMR create around a molecule?
AN electric field
What effect does EMR have on electron density of molecules?
They are polarized by the electric field; emit radiation in diff direction than incoming radiation. Scattering is elastic; no change in E or freq of radiation. Equivalent to photon bouncing off the molecule
Why is the sky blue?
Short wavelength light is scattered at greater intensity, toward us. Human eye is not highly sensitive to wavelengths below blue, so the most intense color we see is blue
Raman Effect
When sample is exposed to intense, high E light sources (lasers), most photons are scattered elastically, but a small portion emerge at higher or lower freq.
Rayleigh Line
Elastically scattered photons. Most intense line in Raman spectra. Frequency of source; no change in frequency of light added.
About how many photons emerge at higher or lower frequencies in Raman?
1 in 10^7 photons
Stokes lines
lower frequency (Excitation)
Anti-stokes lines
higher frequency (Relaxation)
Raman spectroscopy
Gain or loss of E by incident light corresponds to transitions btwn E levels of molecules.
How does Raman spec work?
High E photon excites molecule to ‘virtual state’; molecule returns to a different state on emission. The difference is related to the difference in the E levels.
Gross selection rule for Raman spectroscopy
a molecule must be anisotropically polarizable to have a Raman spectrum. This means 1 moment of inertia
What molecules cannot be studied using Raman?
Molecules that are spherical tops (all 3 moments of inertia are equal). These are true tetrahedral or octahedral point group molecules.
What can be seen in Raman?
All non-spherical molecules, including homonuclear diatomics (which cannot be seen in vib/rot).
Specific Selection Rule for Raman Spec
dJ=0, +-2
What is the separation between Raman peaks?
4B~
O Branch
dJ=-2
Q branch
dJ=0
S branch
dJ=+2
What do rot-vib Raman usually show?
3 branches with fine structure.
Review the Raman diagrams and examples (study sheet)
Slides 143 - 145 unit 1
Configurations
At every instant, there is a set of QNs that define the system.
What does NVE mean in the microcanonical ensemble?
Number of particles, volume, and total E are constant
What configurations are acceptable?
Configurations where the total energies are equal
Distinguishability
Distinguishable: if you can assign a unique label to each particle
Indistinguishable: if particles are free to exchange so it is impossible to assign a unique label
What states of matter are distinguishable/indistinguishable?
Solids are distinguishable
Liquids and gases are indistinguishable
What do we define the GS E in Stat Mech?
0
Macrostate
Gives the same occupancy of states; weighted average of all possible configurations
Microstate
each possible arrangement of E level occupations
What is the number of ways N items can be arranged?
N!
What is the weight of a configuration (eqn)?
W = A! / (a1! * a2! * …)
A is total number of systems
ai is the number of systems in level i
If we have a system with 5 molecules, 3 in state 1 and 2 in state 2, what is the weight?
10
There are 10 ways of exchanging the molecules that would yield the same occupation of states
Do higher E states have higher or lower weights?
Lower
Is the density of a system mechanical or non-mechanical?
Mechanical
Is the heat capacity of a system a mechanical or non-mechanical variable?
Non-mechanical
What is the time average of a single system?
Consider a single system and calculate to simulate its take on different states over time
Why are time averages of single systems a bad idea?
Motion of particles is very complicated; timescale would need to stretch from atomic scale to macroscopic scale; number of particles for a real system is very large
Instantaneous average
invent a mental ensemble of a large number of systems of same composition, frozen in time. Note this is a conceptual strategy to relate properties in a single system
First Postulate of Stat Mech
The time average of a mechanical variable M in the thermodynamic system of interest is equal to the ensemble average of M in the limit A goes to infinity
Ensemble avg ex
slide 29 unit 2
Is energy mechanical or non-mechanical?
Mechanical
Gibbs Postulate
The internal E is the average E; U = <E></E>
Second Postulate of Stat Mech
for an ensemble representative of an isolated system, the systems of the ensemble are distributed uniformly. I.e. all states with specified N,V,E will occur with equal probability/frequency.
Heat baths
we describe system as being immersed in a heat bath at T. Heat bath is much larger than system; no heat by system will significantly change T of surroundings.
How does the canonical ensemble differ from microcanonical in the development of stat mech?
Systems of canonical ensemble have a range of energies; micro is fixed
What happens after the canonical ensemble is placed in a heat bath?
The ensemble is removed from the bath so that it is isolated again; the systems can exchange E, and thus the other systems serve as a heat bath for each system
What is the weight of a configuration as a function of N?
W = A! / N! (A-N)!
Most Probable Distribution
for large A, configuration with largest weight will be much more heavily weighted than all other configurations. Thus, we only need to determine the most probable set to describe the canonical ensemble; max of W
What are the 2 main constraints on configurations of canonical ensemble?
- sigma aiEi = E
SUm over all E levels has to equal total E - sigma ai = A
Sum of all occupancies must equal A
Boltzmann Distribution derivation*
Review lagrange multipliers and the ideas in the derivation (don’t need to know the full derivation)
For a positive function f, the location of max of lnf is in the same location as ____
f
What are 3 examples of equations in chemistry derived from Boltzmann distribution?
- Equilibrium thermo
- NMR spectroscopy
- Kinetic theory of gases
Saturation in spectroscopy
rate coefficients for stimulated absorption and emission equal; no net change in signal
Partition Function
denominator of the Boltzmann distribution is a sum over all the states of the system. Called Q; PF.
Probability
weight of a state divided by sum of possible weights (Q)
Two Variations of PF
- Sum Over states; each E state has its own index; degenerate states have their own index
- Energy levels; only E levels with distinct E have their own index; degeneracy is included
What does a large partition function value mean?
There are a large number of accessible states at given T.
What is the minimum value of a PF?
Lowest E is E=0; this gives Q=1.
Molecular PF
total E of a specific system can be decomposed into sum of energies of different degrees of freedom
Distinguishable particle PF eqns
Distinguishable: Q = q^N
Indistinguishable: Q = q^N / N!
What is the general eqn for the molecular partition function, q?
q=qtrans qrot qvib qelec
Statistical Interpretation of Heat
Heat absorbed by the system corresponds to a change in the population of E levels; the E levels stay the same but higher E levels are populated more
Statistical Interpretation of Work
Work done by the system corresponds to a change in the E levels, but the populations of the E levels stay the same; i.e. the box expands to a greater volume so gas particles can translate in a larger box
Review derivations
E unit 2 slide 79
Blackbody radiation model
Heated materials emit electromagnetic radiation; we define black bodies as a model system for thermal radiation.
What is a black body?
A material that absorbs all frequencies of light completely
Describe the black body radiation model
EM waves can form inside cavity inside body, known as photon gas. Imagine small hole in sphere to let small amt of radiation escape; the spectrum of radiation can be measured from this light. Energy in EM waves inside cavity absorbed and reemitted by material of walls; thermal equilibrium btwn materials and radiation
Planck Black Body Radiation
Intensity of emitted light varies with frequency of light and temperature of body. Max occurs at longer wavelength, distribution becomes broader at high T.
What does black body accurately describe?
Emission of radiation from stars. Different colours due to different surface T; light from hot stars is more intense at lower wavelengths
Blackbody Radiation of Earth
Earth thermally radiates like a blackbody; dissipates E out to space
Greenhouse Effect
High amt of heat dissipated from Earth as IR, since most of atm does not absorb IR (O2 and N2). Small components do absorb IR (H2O, CO2, CH4, O3). Increased conc of greenhouse gases from anthropogenic sources result in increased absorption of IR by atmosphere
Fluctuations
The total E of the system fluctuates and holds different values at diff points in time
What do variance and fluctuations depend on?
Number of particles and Cv (more DOF = more fluctuations )
Which system would have a higher entropy, a system with a large or small number of accessible states?
A lot of states = more complexity = higher Q, thus higher S
Keeping all other properties constant, would the entropy of a gas in a sealed container be higher or lower if the number of particles was increased?
A lot of molecules has more complexity; q^N increases faster than N!, so Q is higher and S is higher.
Hemholtz E eqn
A = -KbT lnQ
Chemical Potential
Change in Hemholtz E when a particle is added to the system; dA/dN
Open form PF
An infinite sum; very difficult to evalutate
How do we solve open form problems?
Approx function as a closed form without a sum
Review derivations of PF
starts unit 2 slide 106
How do we approx a closed form PF for trans and rot?
Approximate the infinite sum as an integral. This is okay because the states are dense and can be considered continuous
DeBroglie Wavelength
Specific for a molecule at a given temp. Used to simplify Trans PF
When we evaluate the integral for trans PF, we ignore zero-point E and set lower bound to be 0. Why?
Corrects for the ground state being shifted to have 0 E.
What factor is introduced in rot PF to correct for symmetry?
Sigma, the symmetry factor; 1 for heteronuclear and 2 for homonuclear (or molecules with an inversion centre)
Why are translational PF much larger than rotational PF?
PF is sum of states. Potential weights for trans is so large (Can populate many states), but rot cannot populate nearly as many
How do we solve the open form vibrational PF?
Use a geometric sum to arrive at an exact solution
Which PF is challenging to derive a closed form?
Electronic
What do we approximate as the electronic PF?
Degeneracy of GS, g0
Why can we approximate the electronic PF as g0?
Gaps between E lvls are so large that upper levels are much harder to populate in normal temperature conditions
What is the electronic PF of O2?
3
Characteristic Temperatures
Combine constants in rot and vib PF expressions; units K.
Chemical Equilibrium Description
A mixture of reactive gases inside a closed vessel in thermal contact w surr; through chemical rxns, the gases will reach their eq conc
What is the system PF for a mixture of gases?
Product of the molecular PF of each molecule in the mixture
Why is chemical potential useful in equilibrium discussions?
The chemical potential describes the flow of a rxn; i.e. how some molecules and others decrease during a process
How do we define the extent of reaction?
dNj=njdlambda
A change in lambda corresponds to a change in conc of reactants and products. At equilibrium,
dA/dlambda = 0, so a shift toward r/p is nonspontaneous
What is the eqn for the equilibrium constant based on PF?
See notes
How does the electronic PF change in chemical rxns?
We must consider relative probability of a bond being broken or formed. We introduce a weighted term with the bond E of the diatomic, D0.
qelec = g1 exp (D0 / kBT)
How can molecular PF be simplified for monoatomics?
No rots or vibs; these terms are not included
What is the electronic degeneracy of Na?
S=1/2; g=2S+1 = 2
What is the electronic degeneracy of Na2?
S=0; g=1
What is the symmetry factor of Na2?
sigma=2
If bond E is in J/mol, how do we adjust the electronic PF weight term?
exp (D0 / RT) instead of kBT
Why does the equilibrium of 2Na = Na2 highly favour the reactants?
Two separated Na atoms have more entropy than one Na2 molecule
Why is there error between the experimental Kc values and the calculated Kc values?
Calculated values predicated on models, which involve approximations. Closed form PF was approximated. Experimental error
Why is the change in bond E so small for Isotopic Exchange (H and D)?
Electronically, H and D are the same
Which term or factor favours the equilibrium of isotopic exchange toward HD?
The symmetry factor; entropy is higher for 2 heteronuclear atoms compared to 2 homonuclear atoms, driving it forward
Adsorption of Gases
some materials can adsorb molecules on their surface; the simplest example is adsorption of gas molecules on the surface of a solid
Chemisorption
Formation of chemical bond btwn surface and sorbent
Physisorption
Sorbent interacts with surface through non-bonded intermolecular forces
Adsorption isotherms
amt of gas adsorbed on surface measured for a range of P at constant T
What is the trend for gas adsorption isotherms?
Non-linear; partial coverage at moderate P, full coverage requires very high P.
Why is it harder to completely cover surface than to partially cover it?
Adsorption is an equilibrium; think of LCP, where higher P drives equilibrium to the right
Why do higher T adsorption isotherm curves have lower rates of adsorption?
Gases have too much E to stick (adsorb) at higher E/T
Langmuir Model of Surface Adsorption
Imagine surface has many sites where molecules adsorb; only one molecule adsorbs in each site; molecules adsorb and desorb dynamically
Assumptions of the Langmuir Model
-only one gas species interacts with surface
-adsorption occurs non-dissociatively
-no interaction btwn molecules in adjacent sites
How is the Langmuir equation derived?
Considering chemical potential of gas and solid phase, separately, and noting that solid is distinguishable
What materials made good sorbents?
Equilibrium favours strong adsorption when there are strong interactions between gas and surface. The number of sites (higher SA) also correlates with higher adsorption
Are uniform materials good sorbents?
No, low SA, limited adsorption sites
What are a couple of materials that are good sorbents?
Charcoal
Zeolites
MOFs
Transport Properties (4)
- Diffusion
- Effusion
- Thermal Conductivity
- Viscosity
Why does diamond exist even though graphite is more thermodynamically stable at STP?
The conversion of diamond to graphite has incredibly slow kinetics; kinetically disfavoured
Reaction extend equation
e = NJ - NJ,0 / nJ
What is the difference btwn rate laws and integrated rate laws?
Rate laws express the change in concentration of a species with respect to time. Integrated rate laws allow us to predict concentrations at a given time
Arrhenius Theory
Empirical relationship btwn rate constant and T
Arrhenius Eqn
k = A exp (-Ea / RT)
Limitations of Arrhenius Theory
-Empirical model; no rigorous physical interpretation, i.e. some rxns can show more complex dependencies
-A and Ea must be determined experimentally, cannot be determined theoretically
Simplest Chemical Rxn (Model for deriving rate laws)
H + H-H = H-H + H
How is the TS Geometry Determined
Make a simplifying assumption that we only need to consider the minimum E path
Potential E Surface
2D plot (PES); there is a minimum E path btwn reactants and products
Conventional TST
Treat the TS structure as a distinct chemical species in equilibrium with the reactants. TS species goes through a unimolecular rxn to form products, defined by k double dagger. This allows us to use stat mech to calculate the rate
Assumptions of TST
-reactants and TS are present in equilibrium distributions
-molecules that reach the TS proceed to products without recrossing
Why is the asymmetric stretch of the TS imaginary?
The TS only completes half the vibration (never goes back)
What weighting term is present in the rate constant expression?
exp ( -dE double dagger / kBT)
where
-dE double dagger = D0,TS - nAD0,B - nBD0,B
The greater the number of reactant species leads to a larger product of PF in the denominator of the rate eqn. Does this result in a larger or smaller rate constant?
Smaller
Which rates are faster; bimolecular or trimolecular?
Bimolecular; would need a proper collision, E and orientation, with 3 atoms instead of 2 for trimolecular; much less likely
Transmission coefficient
Generally ignored; this coefficient corrects for the fraction of reaction events that recross back to the reactants
What is an issue with TST in practice?
We need to know the molecular PFs of the TS, and this generally cannot be observed spectroscopically.
How are molecular PFs for TS determined?
Computational chemistry
Principle of Microscopic Reversibility
Corresponding to every individual process there is a reverse process, and in a state of equilibrium the average rate of every process is equal to the average rate of its reverse process
Why is the rotational PF of TS generally larger than the reactant (specifically in the H2 simple rxn)?
Difference in inertia; H2 is small and short (lower I) whereas H - H - H is larger and has a greater value of I.
What are the possible sources of error that would cause the calculated rate constant to be different than experiment?
-Derivations based on models (assume 3D box, HO, rigid rotor)
-integration approx to get closed form
-difficult to isolate TS; values are computational, not experimental
Kinetic Isotope Effect
H + H2 = H2 + H
vs.
H + D2 = HD + D
Kinetic isotopic effect describes the difference in rate between these two rxns
What is the same / different for the parameters when considering the isotopic effect?
-I is different
-Different barrier height dE
-diff vib frequencies
What most contributes to the kinetic isotopic effect for H and D?
Difference in barrier heights
Explain the origin of KIE for H and D
The zero energy for H is higher than D, because ZPE is inversely proportional to reduced mass. Reduced mass is larger for D so ZPE is smaller. Since ZPE is smaller, D0 is larger in magnitude
How can the KIE be used to Interpret mechanisms?
A large KIE indicates that the RDS involves breaking the isotopic species
Enthalpy of activation
Bond breaking needed to form TS; generally positive and large
Entropy of activation
amount of complexity in TS compared to reactants (number of accessible states); generally negative. Bimolecular have large entropies of activation
Why do bimolecular rxns have large entropies of activation?
There is a loss of trans entropy to form a single TS from two species
Associative mechanism
dS double dagger is negative
Entropy at TS is smaller than in reactants; TS more ordered
Dissociative mechanism
dS double dagger is positive
ENtropy at TS is larger than in reactant; TS is less ordered
Eyring Analysis equations
See Notes (Unit 3 slide 95)
What does a large positive dH double dagger indicate?
TS has weaker bonding than reactants
What does a small negative dS double dagger indicate?
Loss of translational E
Kinetic Theory of Gases
Gas molecules move in random directions with a distribution of molecular speeds; collisions occur when molecules overlap
Maxwell Distribution
The distribution of translational E of gas particles determines a velocity probability density
What is the Maxwell Distribution Function
See Notes
What factor is employed when we convert to spherical polar coordinates?
4 pi v^2
What does the Maxwell distribution describe?
The fraction of molecules at a speed, v
Average of a distribution function
Integrate the property over the full range of the distribution (0 to infinity)
Which would have a greater average speed, N2 or Ne?
Ne has less mass and is faster
How does the collision flux change with T, gas density, and mass?
Increases with gas density
Decreases with mass and T
Effucion
Consider a container at a pressure, containing a small hole in the container. Molecules of gas that pass through the hole to irreversibly escape the container is the rate of effusion
Which gas would effuse from a container faster, N2 or O2?
O2 is heavier; lower rate of effusion. N2 effuses faster
Why does the rate of effusion decline over time?
Less particles, less probability to effuse
What do we assume of particles in description of transport properties?
They are assumed to behave like hard spheres, having elastic collisions within the radius of another molecule and no interaction outside this radius.
Collision diameter
Any time another molecule is within this diameter, there is a collision
Collision cross section
Area on which collisions can occur (pi * sigma^2 where sigma is the collision diameter)
Mean Free Path
average distance that a molecule can travel before it will undergo a collision
Which gas would have a larger mean free path, He or Ne?
Ne is bigger than He, so mean free path is smaller for Ne
Viscosity
Imagine two parallel planes separated by a gap, with a bottom stationary plate and top plate moving forward in x direction at constant velocity. The drag exerted on the motion of the upper plate by the gas in between the plates is the viscosity
Shear Stress
Drag force per unit area
How does momentum exchange in viscosity model?
Molecules immediately next to plates undergo rapid collisions and exchange momentum; these then move through the gas. The viscosity coefficient determines how quickly momentum will be transferred along the z-axis
Properties of the Viscosity coefficient
-proportional to drag on the moving plate due to the gas
-viscosity is larger for gases with larger masses
-viscosity is smaller for gases with large molecular diameters
-viscosity is larger at high T
-viscosity is independent of P
CO2 has a higher molecular weight than N2 but a smaller viscosity, why?
CO2 is much larger than N2, which has a much more profound impact on the viscosity than mass.
Thermal Conductivity
Consider when there are two systems at different T, separated by a material barrier. E will be transported through the atoms of the material from the warmer system to the cooler system.
Is thermal conductivity higher for solids or gases?
Gases is much lower than most solids
Interpretation of Thermal Conductivity
-increases with T
-larger for gases with a high heat capacity
-smaller for gases with larger masses
-smaller for gases with big collision diameters
Rates of transport are generally slower for ____ molecules.
Large
What is a flux?
Rate of transfer per unit area
Fick’s Law
Relates the concentration gradient to the rate of flux
Diffusion flux
Amount of substance to cross a unit area during a small time interval
Diffusion
movement of a chemical from a region of higher concentration to lower concentration due to random movement of molecules
What effects aren’t true diffusion that seem like they may be?
Smoke or perfume moving through a room, drop of ink into a glass of water; due to fluid dynamics, such as convection and buoyancy
Mean Free Path of Mixed Gases
more complex; more than one type of collision can occur. Collision cross-section of each species is involved.
Self Diffusion
Where there is only one component; a gas molecule moves around in its container. Only observable through isotopic labelling
Interpreting the diffusion coefficient
-proportional to the net rate of flow of gas
-slower at high P
-faster at high T
-slower for heavier particles
-lower if collision cross section is large
Review units, equations for transport properties, coefficients, fluxes, rates, etc.
Slide 75 unit 4
Rigorous expression for transport equations
More rigorous derivations are made by assuming hard sphere collisions; only differences are the prefactors
Would the rate of diffusion be higher or lower for 15N2 or 15O2?
The collision cross section is slightly higher for N2, so the diffusion for N2 will be slightly slower
Would the rate of self diffusion be higher for D2 or H2?
Higher for H2. Collision cross section are similar, so big difference is the mass of D2 is higher than H2.
