Statistical Mechanics Flashcards
What is the principle of equal priori probabilities?
All possible microstates of a system are equally probable
Conditions: isolated system, fixed E, N, and V
What is ni?
Distrbution number
Number of molecules in state i
What is the weight, W(n)?
No of microstates associated with a given conformation n
What is the total number of microstates?
Σ W(n)
What is the probability that a microstate generated has a specific config?
p(n) = W(n) / Σ W(n)
No of microstates with config / overall microstates
What is the probability that for a given config, the molecule is in state i?
Pi(n) = ni/N
Fraction of molecules which are in state for given configuration
What is the overall probability that any molecule is found in quantum state i?
Pi = Σ Pi(n) x p(n) = (Σ Pi(n)W(n)) / ΣW(n)
What is a config?
Specific state of a system
i.e. how many molecules in each microstate
How can you calculate a the weight of a config?
e.g.
4 molecules (N=4) with 3 localised energy levels (ε0 = ε, ε1 = 2ε, ε2 = 3ε)
If E = 8ε, a possible config is n(1,2,1)
This gives 3 microstates when one is equal to 0, so overall
W(1,2,1) = 12
What is the generalised formula for Weight?
W(n) = N! / Πni!
Where Π is the product
What is n*?
Most probable config
Largest weight and overwhelms others in importance
What does n* allow for probability calculation?
Pi = ni* / N
As p(n*)=1, n=n*, p(n) = 0
So W is max
What is Sterling’s approx?
lnx! = xlnx - x
only when x»1
What is Sterling’s approx?
lnx! = xlnx - x
only when x»1
What is the derivation for Pi when n* is present?
Find max W via dlnW = 0 (as easier than dW = 0)
N and E fixed, so dN = Σdni and same for E
Lagrange method used
αΣdn = 0, and βΣdn = 0
dlnW = Σ[ (δlnW/δni) + α - βεi]dni = 0
From this can say dnis are independent
From W = N!/ Πni!
lnW = lnN! - ln(Πni!) = lnN! - Σlnni!
As N is fixed, (δlnW/δni) = -Σ(δni!/ δni)
As ni is large can use Sterling’s approx, cancels to lnx! = -lnni
-lnni + α - βεi = 0
As n*= eα e- βεi
N = eα Σe- βεi
So (finally)
Pi = n*/ N = e- βεi / Σe- βεi = e- βεi / q
What are the implications of Pi being independent of time?
Confirms working with macroscopic system at eqm
So properties don’t change macroscropically over time
What is the molecular partition function?
q = Σgie- βεi
Where the sum is over all levels (as g included)
How do you derive E from q?
E = N x ΣεiPi = (N/q) x Σεie- βεi
δlnq/δβ = (1/q)(δq/δβ) = (1/q)[(δ/δβ) Σεie- βεi] = (-1/q)Σεie- βεi]
E = - N(δlnq/δβ)
E = NkT2(δlnq/δβ)
How does q relate to contributions?
q = qtr x qrot x qvib x qelec
Can do due to Born-Oppenheimer approx.
Derive qtrans in 1-dimension
Particle in a box with length Lx
εnx = (nx2h2)/(8mLx2)
qtr,x = Σ exp(-β (h2/8mLx2)(nx2h2)
As energy levels close to kT, can approx to an integral
qtr,x = ∫ exp(- (h2/kT8mLx2)(nx2h2) dn
qtr,x ≡ ∫ exp(-ax2) = (1/2)Sqrt(π/a)
so
qtr,x = (Sqrt(2πmkT)/h) Lx
Derive qtrans in 3-dimensions
εnx,ny,nz = (h2/8m) [ (nx/Lx)2 + (ny/Ly)2 + (nz/Lz)2]
qtr = Σ exp(-β[εnx + εny + εnz) = qtr,x * qtr,y * qtr,z
qtr = ([2πmkT]3/2/h3) LxLyLz
LxLyLz = Volume
qtr = V / Λ3
where the de broglie thermal wavelength, Λ = h/Sqrt[2πmkT]
What is the de Broglie thermal wavelength, and typical values?
Λ = h/Sqrt[2πmkT]
Λ ≈ 1-2 x10-11 m = 10-2- pm
gives qtrs ≈ 1028
What does the parition function give an indication of?
Indication of average number of quantum states thermally accessible to a molecule @T
What does Maxwell-Boltzmann stats depend on working for qtrans?
qtr»_space; N
i.e. (qtr/N)»_space; 1
Exceptions include light particles such as He
Dervie the Etr
qtr = (V/h3)[2πmkT]3/2
lnqtr = (3/2)lnT + ln[(V/h3)(2πmkT)3/2]
Then differentiate wrt T at constant V
Etr = (3/2)NkT = (3/2)nRT
Derive Cv just with trans energy only
Differentiate wrt T at constant V
Etr = (3/2)nRT
Cv = (3/2)nR
Derive qelec
qelec = exp[-βε0] x Σgiexp[-β[εi-ε0]
Want this as specify states wrt ground state
qelec = exp(-βε0) x q0elec
Usually q0elec ≈ g0
where g0 is the degen of the ground state
Derive Eelec
Eelec = NkT2(dlnqelec/dT)
qelec = g0exp[-βε0]
where β = 1/kT
Eelec = NkT2 (1/kT2) = Nε0
What is the curie law?
Magnetic susceptibility inversely propertional to T
χ (chi) = c/T
c = N(gμB)2/4k
Define a canonical ensemble
Collection of v.many systems, each replicating
@ thermo level disclosed in isothermal system of interest
Fixed E, large fixed N, fixed volume, assumed knowledge of each ensemble units
What are microstates of an ensemble?
Complete spec of ensemble when each system is assigned to a quantum state of
Equally probable as isolated
What is the Pi of a canonical distribution?
Pi = ni*/N
Analagous to standard definintion but not equal
What is the canonical partition function?
Q = Σ exp[-βEi]
Not equal to standard
What is the energy of a canonical system?
E = - (δlnQ/δβ)V = kT2(δlnQ/δT)V
How does the canonical partition function (Q) relate to molecular partition function (q)?
Ideal solid (independent distinguishable systems): Q = qN
Ideal gas (independent indistuinguishable systems): Q = qN/N!
What is the entropy of an ensemble?
S = k lnW(n*)
What is the entropy of an ensemble?
S = k lnW(n*)
What is the entropy of a system?
S = S/N = (k/N)lnW(n*)
Where S (non-bold) is the entropy of an enseble
Derive the entropy of a system from canonical ensemble
dE = dqrev + dwrev
(dE)V = TdS
As: E = ΣEiPi = (1/N) ΣEini*
if heat @ constant V then Ei’s fixed but population alters
ni* -> ni* + dni and E-> E + (dE)V
(dE)V = (1/N) ΣEi dni
dS = (1/NT)ΣEi dni
S = (k/N) lnW(n*)
What is the entropy of system including Q?
S = (E/T) + klnQ
Derive the entropy of system including Q
S = (k/N) lnW(ni*) = (k/N) (lnN! - Σln ni*!)
Sterling’s approx can be used as N and ni* large
S = (k/N) (NlnN - N - Σni*lnni* + Σni*) = (k/N) (NlnN - Σni*lnni*)
S = -k Σ(ni*/N) ln(ni*/N)
as ni*/N = [exp(-βEi)/Q]
S = -(k/Q)Σ (exp(-βEi))(-βEi - lnQ) = (βk/Q)(Σ Ei exp(-βEi)) + klnQ
where βk = 1/T and (1/Q)Σ Ei exp(-βEi)) is E
so
S = (E/T) + klnQ
What is the Helmholtz free energy including Q?
A = E - TS = -kT lnQ
What is pressure including Q?
P = -(δA/δV)T = kT(δlnQ/δV)T
What is enthalpy including Q?
H = E + PV = kT2(δlnQ/δdT)V + VkT(δlnQ/δV)T
What is the gibbs free energy including Q?
G = H - TS = A + PV
G = -kTlnQ + VkT(δlnQ/δV)T
What is chemical potential including Q?
μ = (δA/δn)V,T = NA(δA/δN)V,T = -RT(δlnQ/δN)V,T
What is Str?
Str = Nkln(e5/2V/NΛ3)
From S = (E/T)+Nkln(qe/N), and definititons of Etr & qtr
Derive Selec
Eelec = Nε0
qelec = g0 exp[-ε0/kT]
From S = (E/T)+Nkln(qe/N)
Selec = Nkln g0
What is the Sackur-Tetrode equation?
Overall entropy including elec and trs
S = Nk ln[e5/2g0V/NΛ3T)
What is the rotational part of the parition function?
qrot = ΣgJ exp(-εJ)
εJ = BhcJ(J+1)
gJ = 2J+1
qrot = Σ(2J+1)exp[-(θrot/T) J(J+1)]
Where θrot = Bhc/k
What is θrot?
Rotational temperature
θrot = Bhc/k
T above which sufficient # of rot states become available
Can replace Σ with integral
What is the high temp limit?
T»_space;θrot
where B «_space;kT/hc
Many rot states available so can integrate instead of sum
What is qrot under high temp limit?
qrot = T/θrot
Derive qrot at high temperature limit
qrot = ∫(2J+1)exp[(-θrot/T)J(J+11)] dJ
X = J(J+1)
qrot= ∫ exp[(-θrot/T)X] dX
qrot T/θrot
What is Erot and CV,rot at the high temperature limit?
Erot = NkT2(dlnqrot/dT) = NkT
CV,rot = dErot/dT = Nk
What is Srot at the high T limit?
Srot = Nk ln(Te/θrot)
What is Erot at low temp?
Full sum of qrot used
Erot = (Nkθrot/qrot)Σ(2J+1)J(J+1)exp[(-θrot/T)J(J+1)]
as T «_space;θrot</sub
Erot = Nkθrot[6exp(-2θrot/T) + …)
Decreasing exp with T
What is CV,rot at low T?
Erot = Nkθrot[6exp(-2θrot/T) + …)
So
CV,rot = 12Nk(θrot/T)2
[exp(-2θrot/T) + …)
Therefore at low T, Cv -> 0
How does CV change with T?
Explained by low and high T limit
What is qrot for a homonuclear diatomic?
qrot = T/σθrot
Where σ is symmetry number
What is the symmetry number, σ?
# of indistinguishable forms in a molcule reached through rigid rotation
σ=1 -> heteronuclear diatomics
σ=2 -> homonuclear (as 180 degree)
How is the Pauli principle applied to nucleii?
neutron (n) and protons (p) are both fermions
P^Ψ = -Ψ if p+n odd, fermionic nucleus
but = Ψ if p+n even, bosonic nucleus
as -1(p+n)
What is the magnitude and z-component of nuclear spin?
|I|(uppercase i) = Sqrt[I(I+1)ℏ]
Iz = mIℏ
mI is 2I+1 values
What are the nuclear spin states for homonuclear diatomics?
Can treat as 2e- problem: in triplet or singlet state
What information can be found from a triplet nuclear spin state?
(when 2x I = 1/2 nuclei)
IR = 1
mI = 0, +/- 1
Wavefns are symmetric under exchange of nuclei 1<->2
α(1)α(2)>, MI = +1
(1/sqrt2) α(1)β + β(1)α(2)>, MI = 0
β(1)β(2)>, MI = -1
What information can be found from a singlet nuclear spin state?
(when 2x I = 1/2 nuclei)
IR = 0
mI = 0
Wavefns are antisymmetric under exchange of nuclei 1<->2
(1/sqrt2) α(1)β + β(1)α(2)>, MI = 0
What is the molecular wavefunction comprised of?
Ψ = Ψtr x Ψelec x Ψvib x Ψrot x Ψn.s
Ψn.s = Ψnuclear spin
What is the effect of P^ on Ψtr?
P^(Ψtr) = +Ψtr
Depends only on centre of mass
What is the effect of P^ on Ψvib?
P^Ψvib = +Ψvib
As fn of only nuclear separation
What is the effect of P^ on Ψelec?
Summarised by electronic state
2S+1Λ+/-g/u
Product of the inversion and reflection symmetries
g x +, u x -, give symmetric
g x -, u x +, gives anti-symmetric
What is the effect of P^ on Ψelec(1Σ+g)?
P^ Ψelec(1Σ+g) = +Ψelec
What is the effect of P^ on Ψelec(3Σ-g)?
P^ Ψelec(3Σ-g) = -Ψelec
What is the effect of P^ on Ψrot?
P^Ψelec = (-1)JΨ
J = rot q.n., if even then symm
What is the effect of P^ on Ψn.s?
P^ Ψn.s = -ve if (p+n) is odd, fermionic
+ve if (p+n) is even, bosonic
What is the effect of P^Ψ when nuclei fermion has a closed shell?
Ψtotal is -ve
Ψtr, Ψvib, and Ψelec are +ve
Ψn.s. is +ve when triplet or -ve when singlet.
Ψrot is therefore -ve and J odd when triplet, and is +ve and J even when singlet
What is the effect of P^Ψ when nuclei is boson when I>0 and closed shell?
Ψtotal is +ve
Ψtr, Ψvib, and Ψelec are +ve
Ψn.s. is +ve when triplet or -ve when singlet.
Ψrot is therefore +ve and J even when triplet, and is -ve and J odd when singlet
What is the effect of P^Ψ when nuclei is boson when I=0?
Ψtotal is +ve
Ψtr & Ψvibare +ve
Ψelec is ve
Ψn.s. is +ve
Ψrot is therefore -ve and J odd
Name examples of a diatomic nuclei fermions with a closed shell
H2 or 19F2
Name examples of a diatomic nuclei fermions with a closed shell
H2 or 19F2
Name examples of diatomic nuclei I>0 bosons with closed shells
D2, 14N2
Name examples of diatomic I=0 bosons and the elec state
3Σg-
16O2, 18O2
What are the types of H2?
ortho and para H2
Can only access only odd or even J-state, but odd is 3x more likely as triplet
ortho - odd J
para - even J
Why is there a peak in CV with temperature?
Spacing increases with J so can do J=0->1, which increases degen and causes hump
Occurs at θrot
What is the calc heat capacity of heteronuclear diatomics at different T?
@ v.low T then (3/2)R from translations
then (5/2)R when rot and trans, and hump due to degen
@ high T then (7/2)R when all previous and vib
What is the measured heat capacity of H2?
Low T same as theorectical, (3/2)R
No hump but goes to (5/2)R
Then increases more linearly to populate vibration states
How does the proportion of ortho and para H2 change with T?
@ high T follows n.s. degen ratio
@ low T then all molcules occupt J=0 so para
What are the energy levels for a simple harmonic oscillator?
εv = (v+1/2)ℏw = (v+1/2)hν
v (q.n. not wavelength as in 2nd) = 0,1,2..
How can you evaluate qvib for a harmonic vib?
qvib = Σvgvexp(-βεv) = Σv=0∞exp(-βεv) = Σv=0∞exp[-(v+1/2)ℏw/kT]
Evaluate instead of ∫
qvib = exp[-ℏw/2kT] Σv=0∞exp[-vℏw/kT]
Can use Maclurian series to give:
qvib = exp(-θv/2T)/[1-exp(-θv/T)] = 1/[exp(θv/2T)-exp(-θv/2T)]
What is the Macluirian series and how is it used in qvib for harmonic oscillator?
1+x+x2+… = 1/(1-x)
Series in derivation has x = exp(-ℏw/RT)
What is θvib?
θvib = ℏw/k = ℏν/k
Temperature at which ℏw = kT, so energy spacing comparable to thermal E
ℏw/kT = θvib/T
What is low T limit of qvib of harmonic oscillator?
qvib = 1/[exp(θv/2T)-exp(-θv/2T)]
T«θvib so exp(-θv/2T) -> 0
so @ limit then qvib = exp[-θv/2T]
What is εvib @ low T limit (harmonic oscillator)?
E = NkBT2(dlnq/dT)V
&
qvib = exp[-θv/2T]
(dlnq/dT)V =θv/2T2
Energy per molecule
E/N = (1/2)kθv = (1/2)ℏw = (1/2)hv
This is the ZPE of harmonic oscillator
What is Cvib of harmonic oscillator?
Cvib = dεvib/dT = 0
As is independent of T
What is qvib</sup> @ high T limit?
qvib = 1/[exp(θv/2T)-exp(-θv/2T)]
@ T»θv, the exponents are small so can expand ex ≈ 1+x+x2/2!+…
Results in
qvib = T/θv
What is εvib @ high T limit?
qvib = T/θvib
E = NkT2(dlnq/dT)
E/N = kT2 x 1/T = kT
Equipartition result
What is Cvvib @ high T limit?
E/N = kT
Cv = k per molecule or R per mol
How does mass relate to popoulation of vib states?
Large mass means weaker bonds and smaller force constant
Results in high population of vib states
θvib = ℏw/k, lower when weaker bonds
How many modes of vibration does a polyatomic molecule have?
linear: 3N-5 vib modes
non-linear: 3N-6 vib modes
What is qvib, total & εvib, total with a polyatomic molecule?
qvib,total = qvib1 x qvib2 x …
εvib, total = εvib1 + εvib2 + …
What occurs to energy levels at eqm?
E levels on reagent/product filled irrespective of if on either side
Just fills lower one
Derive ΔG in terms of q
ΔG = -NkTln(K)=-NkTln(q/N) for ideal gas
q must be written in terms of ZPE @ T=0
q = q0 x exp(-ε0/kT)
G = -NkTln(qθ/N)
used as want standard molar of free energy
G = -NkT[ln(qθ,0/N) + ln(exp(-ε0/kT)) = Nε0 - RTln(qθ,0/N)
ΔG = ΔE0 - RTΣJvJln(qθ,0m,J/N)
ΔG = ΔE0 - RTln[ΠJ(qθ,0/N)Vj] = - RTln[exp(-ΔE0/RT)
ΠJ(qθ,0/N)Vj]
What is the equilibrium constant with q?
ΔG = - RTln[exp(-ΔE0/RT)
ΠJ(qθ,0/N)Vj]
Generally ΔG = -RTlnK(T)
so
K(T) = ΠJ(qθ,0/N)Vjexp(-ΔE0/RT)
What are the conditions of the equilibrium const with q?
q includes all dof for each J
For an ideal gas: θ applies to qtr,m = Vθm/Λ3 = RT/PθΛ3
q0 implies energies relative to ε0 -> ground state @ T=0
ΔE0 is from ground states between reactants and products
vJ is the stoichiometry coefficients
How does translation ΛJ @ eqm relate to Λ0?
ΛJ = h/Sqrt[2πmJkT] = (h/Sqrt[2πmukT]) x MJ-1/2 = Λ0 x MJ-1/2
How do you go about finding the equilibrium constant theoretically?
Calc modes of motion individually before combining
trs: qθ,0J,m,tr = Vθm
elec: q0J,m,elec = g0 + g1exp[-(ε1-ε0)/kT]+…
vib, rel to ZPE not bottom of potential well: q0J,m,elec = 1/[1-exp(-hv/kT)]
rot, high T limit as usually : qJ,m,rot = T/σθrot
What is molecular mass, M, used in calcs for isotope exchange?
Mass number of molecule - i.e. neutrons and protons (just integer values)
What is the degen of an e- ground state?
g0 = 2
Spin up or down -> must be retained in product/reactants
What is the Saha equation?
Rate constant of thermal ionisation
K(T) = (kT/pθ) (1/Λ3e) exp[-I1(atom)/RT]
Found as atoms/ions have no vibrations or rotations, same mass, ideal gas assumed
How can you calculate the fraction of ionised atoms?
M <-> M+ + e-
α is degree of dissociation
M: 1-α, M+: α, e-: α
partial pressures (molefraction x pressure)
M: (1-α/1+α)P, M+: (α/1+α)P = e-
K(T) = [(PM+/Pθ)(Pe-/Pθ)]/(PM/Pθ)
Cancel to K(T) = (α2/1-α2) (p/pθ)
If α small then can find value
What is transition state theory?
R <-> ‡ -> P
for ‡->R, then K‡
for ‡->P, then K‡
Overall rate = d[P]/dt = K‡[‡]
How do you derive the reaction constants under TS theory?
AB + C <-> ‡ -> A + BC
K‡ = (p‡pθ/pABpC)
Assume ideal gas eqn:
K‡ = (pθ/RT) ([‡]/[AB][C])
[‡] = (RT/pθ)K‡[AB][C]
Therefore
d[P]/dt = k‡[‡] = kr[AB][C]
where kr = k‡(RT/pθ) K‡
And
K = [(NAqm‡)/(qAB,m qC,m)] exp(-ΔE0‡/RT)
How do frequencies of intermediates relate to products?
v of forming products proportional to natural v of loose vib mode (assumed to be ‡)
Each time oscillation occurs it gives a chance for moleucle to escape and form products
k‡ = κv
where κ= transiction coefficient, prob of forming products when oscillation occuring
What is the qvib for an intermediate?
q‡ = ~q‡ qvib ‘
where ~q‡ is the mpf for all other modes in ‡
qvib’ is mpf for TS accounting only for E fo low-v vibration
qvib’ = (1-exp[-hv/kT])-1, relative to ZPE
When hv/kT «_space;1 then can expand as series
qvib ‘ = (1-1-(hv/kT) +…)-1 = kT/hv
where v is freq of a loose vibration
Derive the eyring equation for transition-state theory
kr = k‡ (RT/pθ) K‡
kr = (κ-v)(RT/pθ)(kT/hv) ~K‡
Where ~K‡ is eqm constant with qvib’ removed
kr = κ(RT/pθ)(kT/hv)(NA~q‡/qABqC) exp(-ΔE‡/RT)
What is the Eyring equation?
kr = κ(RT/pθ)(kT/hv)(NA~q‡/qABqC) exp(-ΔE‡/RT)
where all q’s are molar, 0, and θ
What are the pros of the Eyring eqn?
v cancelled - validates can use this and other models
all terms can be calc or measuured so is calculable
simple cases -> good results
What are the cons of the Eyring eqn?
q‡ not readily accessible (shapes&modes) so must assume and estimate
Strong assumption: pre-equilibrium between R&‡, and all ‡ -> P
Must have a fitting parameter κ (kappa)
