Statistical Mechanics

Question 1

What is the principle of equal priori probabilities?

Answer 1

All possible microstates of a system are equally probable

Conditions: isolated system, fixed E, N, and V

Question 2

What is n_i?

Answer 2

Distrbution number
Number of molecules in state i

Question 3

What is the weight, W(n)?

Answer 3

No of microstates associated with a given conformation n

Question 4

What is the total number of microstates?

Answer 4

Σ W(n)

Question 5

What is the probability that a microstate generated has a specific config?

Answer 5

p(n) = W(n) / Σ W(n)

No of microstates with config / overall microstates

Question 6

What is the probability that for a given config, the molecule is in state i?

Answer 6

P_i(n) = n_i/N

Fraction of molecules which are in state for given configuration

Question 7

What is the overall probability that any molecule is found in quantum state i?

Answer 7

P_i = Σ P_i(n) x p(n) = (Σ P_i(n)W(n)) / ΣW(n)

Question 8

What is a config?

Answer 8

Specific state of a system
i.e. how many molecules in each microstate

Question 9

How can you calculate a the weight of a config?

Answer 9

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

Question 10

What is the generalised formula for Weight?

Answer 10

W(n) = N! / Πn_i!
Where Π is the product

Question 11

What is n*?

Answer 11

Most probable config
Largest weight and overwhelms others in importance

Question 12

What does n* allow for probability calculation?

Answer 12

P_i = n_i* / N

As p(n*)=1, n=n*, p(n) = 0

So W is max

Question 13

What is Sterling’s approx?

Answer 13

lnx! = xlnx - x
only when x»1

Question 14

What is Sterling’s approx?

Answer 14

lnx! = xlnx - x
only when x»1

Question 15

What is the derivation for P_i when n* is present?

Answer 15

Find max W via dlnW = 0 (as easier than dW = 0)

N and E fixed, so dN = Σdn_i and same for E

Lagrange method used
αΣdn = 0, and βΣdn = 0
dlnW = Σ[ (δlnW/δn_i) + α - βε_i]dn_i = 0
From this can say dn_is are independent

From W = N!/ Πn_i!
lnW = lnN! - ln(Πn_i!) = lnN! - Σlnn_i!
As N is fixed, (δlnW/δn_i) = -Σ(δn_i!/ δn_i)

As n_i is large can use Sterling’s approx, cancels to lnx! = -lnn_i

-lnn_i + α - βε_i = 0

As n*= e^α e^{- βε_i}
N = e^{α Σe- βε_i}

So (finally)
P_i = n*/ N = e^{- βε_i} / Σe^{- βε_i} = e^{- βε_i} / q

Question 16

What are the implications of P_i being independent of time?

Answer 16

Confirms working with macroscopic system at eqm
So properties don’t change macroscropically over time

Question 17

What is the molecular partition function?

Answer 17

q = Σg_ie^{- βε_i}
Where the sum is over all levels (as g included)

Question 18

How do you derive E from q?

Answer 18

E = N x Σε_iP_i = (N/q) x Σε_ie^{- βε_i}

δlnq/δβ = (1/q)(δq/δβ) = (1/q)[(δ/δβ) Σε_ie^{- βε_i}] = (-1/q)Σε_ie^{- βε_i}]

E = - N(δlnq/δβ)
E = NkT²(δlnq/δβ)

Question 19

How does q relate to contributions?

Answer 19

q = q_tr x q_rot x q_vib x q_elec

Can do due to Born-Oppenheimer approx.

Question 20

Derive q_trans in 1-dimension

Answer 20

Particle in a box with length L_x

ε_nx = (n_x²h²)/(8mL_x²)

q_tr,x = Σ exp(-β (h²/8mL_x²)(n_x²h²)

As energy levels close to kT, can approx to an integral

q_tr,x = ∫ exp(- (h²/kT8mL_x²)(n_x²h²) dn
q_tr,x ≡ ∫ exp(-ax²) = (1/2)Sqrt(π/a)

so
q_tr,x = (Sqrt(2πmkT)/h) L_x

Question 21

Derive q_trans in 3-dimensions

Answer 21

ε_nx,ny,nz = (h²/8m) [ (n_x/L_x)² + (n_y/L_y)² + (n_z/L_z)²]

q_tr = Σ exp(-β[ε_nx + ε_ny + ε_nz) = q_tr,x * q_tr,y * q_tr,z

q_tr = ([2πmkT]^3/2/h³) L_xL_yL_z

L_xL_yL_z = Volume

q_tr = V / Λ³
where the de broglie thermal wavelength, Λ = h/Sqrt[2πmkT]

Question 22

What is the de Broglie thermal wavelength, and typical values?

Answer 22

Λ = h/Sqrt[2πmkT]

Λ ≈ 1-2 x10^-11 m = 10-2- pm
gives q_trs ≈ 10²⁸

Question 23

What does the parition function give an indication of?

Answer 23

Indication of average number of quantum states thermally accessible to a molecule @T

Question 24

What does Maxwell-Boltzmann stats depend on working for q_trans?

Answer 24

q_tr&raquo_space; N
i.e. (q_tr/N)&raquo_space; 1

Exceptions include light particles such as He

Question 25

Dervie the E_tr

Answer 25

q_tr = (V/h³)[2πmkT]^3/2

lnq_tr = (3/2)lnT + ln[(V/h³)(2πmkT)^3/2]

Then differentiate wrt T at constant V
E_tr = (3/2)NkT = (3/2)nRT

Question 26

Derive C_v just with trans energy only

Answer 26

Differentiate wrt T at constant V
E_tr = (3/2)nRT

C_v = (3/2)nR

Question 27

Derive q_elec

Answer 27

q_elec = exp[-βε₀] x Σg_iexp[-β[εi-ε₀]

Want this as specify states wrt ground state

q_elec = exp(-βε₀) x q⁰_elec

Usually q⁰_elec ≈ g₀
where g₀ is the degen of the ground state

Question 28

Derive E_elec

Answer 28

E_elec = NkT²(dlnq_elec/dT)
q_elec = g₀exp[-βε₀]
where β = 1/kT

E_elec = NkT² (1/kT²) = Nε₀

Question 29

What is the curie law?

Answer 29

Magnetic susceptibility inversely propertional to T
χ (chi) = c/T
c = N(gμ_B)²/4k

Question 30

Define a canonical ensemble

Answer 30

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

Question 31

What are microstates of an ensemble?

Answer 31

Complete spec of ensemble when each system is assigned to a quantum state of
Equally probable as isolated

Question 32

What is the P_i of a canonical distribution?

Answer 32

P_i = n_i^*/N
Analagous to standard definintion but not equal

Question 33

What is the canonical partition function?

Answer 33

Q = Σ exp[-βE_i]

Not equal to standard

Question 34

What is the energy of a canonical system?

Answer 34

E = - (δlnQ/δβ)_V = kT²(δlnQ/δT)_V

Question 35

How does the canonical partition function (Q) relate to molecular partition function (q)?

Answer 35

Ideal solid (independent distinguishable systems): Q = q^N

Ideal gas (independent indistuinguishable systems): Q = q^N/N!

Question 36

What is the entropy of an ensemble?

Answer 36

S = k lnW(n*)

Question 37

What is the entropy of an ensemble?

Answer 37

S = k lnW(n*)

Question 38

What is the entropy of a system?

Answer 38

S = S/N = (k/N)lnW(n*)
Where S (non-bold) is the entropy of an enseble

Question 39

Derive the entropy of a system from canonical ensemble

Answer 39

dE = dq_rev + dw_rev
(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*)

Question 40

What is the entropy of system including Q?

Answer 40

S = (E/T) + klnQ

Question 41

Derive the entropy of system including Q

Answer 41

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

Question 42

What is the Helmholtz free energy including Q?

Answer 42

A = E - TS = -kT lnQ

Question 43

What is pressure including Q?

Answer 43

P = -(δA/δV)T = kT(δlnQ/δV)T

Question 44

What is enthalpy including Q?

Answer 44

H = E + PV = kT²(δlnQ/δdT)V + VkT(δlnQ/δV)T

Question 45

What is the gibbs free energy including Q?

Answer 45

G = H - TS = A + PV
G = -kTlnQ + VkT(δlnQ/δV)T

Question 46

What is chemical potential including Q?

Answer 46

μ = (δA/δn)_V,T = N_A(δA/δN)_V,T = -RT(δlnQ/δN)_V,T

Question 47

What is S_tr?

Answer 47

S_tr = Nkln(e^5/2V/NΛ³)

From S = (E/T)+Nkln(qe/N), and definititons of E_tr & q_tr

Question 48

Derive S_elec

Answer 48

E_elec = Nε₀
q_elec = g₀ exp[-ε₀/kT]

From S = (E/T)+Nkln(qe/N)

S_elec = Nkln g₀

Question 49

What is the Sackur-Tetrode equation?

Answer 49

Overall entropy including elec and trs

S = Nk ln[e^5/2g₀V/NΛ³T)

Question 50

What is the rotational part of the parition function?

Answer 50

q_rot = Σg_J exp(-ε_J)

ε_J = BhcJ(J+1)
g_J = 2J+1

q_rot = Σ(2J+1)exp[-(θ_rot/T) J(J+1)]

Where θ_rot = Bhc/k

Question 51

What is θ_rot?

Answer 51

Rotational temperature
θ_rot = Bhc/k

T above which sufficient # of rot states become available
Can replace Σ with integral

Question 52

What is the high temp limit?

Answer 52

T&raquo_space;θ_rot
where B &laquo_space;kT/hc
Many rot states available so can integrate instead of sum

Question 53

What is q_rot under high temp limit?

Answer 53

q_rot = T/θ_rot

Question 54

Derive q_rot at high temperature limit

Answer 54

q_rot = ∫(2J+1)exp[(-θ_rot/T)J(J+11)] dJ
X = J(J+1)
q_rot= ∫ exp[(-θ_rot/T)X] dX

q_rot T/θ_rot

Question 55

What is E_rot and C_V,rot at the high temperature limit?

Answer 55

E_rot = NkT²(dlnq_rot/dT) = NkT

C_V,rot = dE_rot/dT = Nk

Question 56

What is S_rot at the high T limit?

Answer 56

S_rot = Nk ln(Te/θ_rot)

Question 57

What is E_rot at low temp?

Answer 57

Full sum of q_rot used
E_rot = (Nkθ_rot/q_rot)Σ(2J+1)J(J+1)exp[(-θ_rot/T)J(J+1)]

as T &laquo_space;θ_{rot</sub
Erot = Nkθrot[6exp(-2θrot/T) + …)
Decreasing exp with T}

Question 58

What is C_V,rot at low T?

Answer 58

E_rot = Nkθ_rot[6exp(-2θ_rot/T) + …)
So

C_V,rot = 12Nk(θ_rot/T)²
[exp(-2θ_rot/T) + …)

Therefore at low T, Cv -> 0

Question 59

How does C_V change with T?

Answer 59

Explained by low and high T limit

Question 60

What is q_rot for a homonuclear diatomic?

Answer 60

q_rot = T/σθ_rot

Where σ is symmetry number

Question 61

What is the symmetry number, σ?

Answer 61

# of indistinguishable forms in a molcule reached through rigid rotation

σ=1 -> heteronuclear diatomics
σ=2 -> homonuclear (as 180 degree)

Question 62

How is the Pauli principle applied to nucleii?

Answer 62

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)

Question 63

What is the magnitude and z-component of nuclear spin?

Answer 63

|I|(uppercase i) = Sqrt[I(I+1)ℏ]

I_z = m_Iℏ

m_{I is 2I+1 values}

Question 64

What are the nuclear spin states for homonuclear diatomics?

Answer 64

Can treat as 2e- problem: in triplet or singlet state

Question 65

What information can be found from a triplet nuclear spin state?
(when 2x I = 1/2 nuclei)

Answer 65

I_R = 1
m_I = 0, +/- 1

Wavefns are symmetric under exchange of nuclei 1<->2

α(1)α(2)>, M_I = +1
(1/sqrt2) α(1)β + β(1)α(2)>, M_I = 0
β(1)β(2)>, M_I = -1

Question 66

What information can be found from a singlet nuclear spin state?
(when 2x I = 1/2 nuclei)

Answer 66

I_R = 0
m_I = 0

Wavefns are antisymmetric under exchange of nuclei 1<->2

(1/sqrt2) α(1)β + β(1)α(2)>, M_I = 0

Question 67

What is the molecular wavefunction comprised of?

Answer 67

Ψ = Ψ_tr x Ψ_elec x Ψ_vib x Ψ_rot x Ψ_n.s

Ψ_n.s = Ψ_{nuclear spin}

Question 68

What is the effect of P^ on Ψ_tr?

Answer 68

P^(Ψ_tr) = +Ψ_tr

Depends only on centre of mass

Question 69

What is the effect of P^ on Ψ_vib?

Answer 69

P^Ψ_vib = +Ψ_vib
As fn of only nuclear separation

Question 70

What is the effect of P^ on Ψ_elec?

Answer 70

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

Question 71

What is the effect of P^ on Ψ_elec(¹Σ⁺_g)?

Answer 71

P^ Ψ_elec(¹Σ⁺_g) = +Ψ_elec

Question 72

What is the effect of P^ on Ψ_elec(³Σ^-_g)?

Answer 72

P^ Ψ_elec(³Σ^-_g) = -Ψ_elec

Question 73

What is the effect of P^ on Ψ_rot?

Answer 73

P^Ψ_elec = (-1)^JΨ

J = rot q.n., if even then symm

Question 74

What is the effect of P^ on Ψ_n.s?

Answer 74

P^ Ψ_n.s = -ve if (p+n) is odd, fermionic
+ve if (p+n) is even, bosonic

Question 75

What is the effect of P^Ψ when nuclei fermion has a closed shell?

Answer 75

Ψ_total is -ve
Ψ_tr, Ψ_vib, and Ψ_elec are +ve

Ψ_n.s. is +ve when triplet or -ve when singlet.
Ψ_rot is therefore -ve and J odd when triplet, and is +ve and J even when singlet

Question 76

What is the effect of P^Ψ when nuclei is boson when I>0 and closed shell?

Answer 76

Ψ_total is +ve
Ψ_tr, Ψ_vib, and Ψ_elec are +ve

Ψ_n.s. is +ve when triplet or -ve when singlet.
Ψ_rot is therefore +ve and J even when triplet, and is -ve and J odd when singlet

Question 77

What is the effect of P^Ψ when nuclei is boson when I=0?

Answer 77

Ψ_total is +ve
Ψ_tr & Ψ_vibare +ve
Ψ_elec is ve

Ψ_n.s. is +ve
Ψ_rot is therefore -ve and J odd

Question 78

Name examples of a diatomic nuclei fermions with a closed shell

Answer 78

H₂ or ¹⁹F₂

Question 78

Name examples of a diatomic nuclei fermions with a closed shell

Answer 78

H₂ or ¹⁹F₂

Question 79

Name examples of diatomic nuclei I>0 bosons with closed shells

Answer 79

D₂, ¹⁴N₂

Question 80

Name examples of diatomic I=0 bosons and the elec state

Answer 80

³Σ_g^-
16O₂, ¹⁸O₂

Question 81

What are the types of H₂?

Answer 81

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

Question 82

Why is there a peak in C_V with temperature?

Answer 82

Spacing increases with J so can do J=0->1, which increases degen and causes hump
Occurs at θ_rot

Question 83

What is the calc heat capacity of heteronuclear diatomics at different T?

Answer 83

@ 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

Question 84

What is the measured heat capacity of H₂?

Answer 84

Low T same as theorectical, (3/2)R
No hump but goes to (5/2)R
Then increases more linearly to populate vibration states

Question 85

How does the proportion of ortho and para H₂ change with T?

Answer 85

@ high T follows n.s. degen ratio
@ low T then all molcules occupt J=0 so para

Question 86

What are the energy levels for a simple harmonic oscillator?

Answer 86

ε_v = (v+1/2)ℏw = (v+1/2)hν
v (q.n. not wavelength as in 2nd) = 0,1,2..

Question 87

How can you evaluate q_vib for a harmonic vib?

Answer 87

q_vib = Σ_vg_vexp(-βε_v) = Σ_v=0^∞exp(-βε_v) = Σ_v=0^∞exp[-(v+1/2)ℏw/kT]

Evaluate instead of ∫
q_vib = exp[-ℏw/2kT] Σ_v=0^∞exp[-vℏw/kT]

Can use Maclurian series to give:
q_vib = exp(-θ_v/2T)/[1-exp(-θ_v/T)] = 1/[exp(θ_v/2T)-exp(-θ_v/2T)]

Question 88

What is the Macluirian series and how is it used in q_vib for harmonic oscillator?

Answer 88

1+x+x²+… = 1/(1-x)

Series in derivation has x = exp(-ℏw/RT)

Question 89

What is θ_vib?

Answer 89

θ_vib = ℏw/k = ℏν/k
Temperature at which ℏw = kT, so energy spacing comparable to thermal E

ℏw/kT = θ_vib/T

Question 90

What is low T limit of q_vib of harmonic oscillator?

Answer 90

q_vib = 1/[exp(θ_v/2T)-exp(-θ_v/2T)]

T«θ_vib so exp(-θ_v/2T) -> 0
so @ limit then q_vib = exp[-θ_v/2T]

Question 91

What is ε_vib @ low T limit (harmonic oscillator)?

Answer 91

E = Nk_BT²(dlnq/dT)_V
&
q_vib = exp[-θ_v/2T]

(dlnq/dT)_V =θ_v/2T²

Energy per molecule
E/N = (1/2)kθ_v = (1/2)ℏw = (1/2)hv
This is the ZPE of harmonic oscillator

Question 92

What is C_vib of harmonic oscillator?

Answer 92

C_vib = dε_vib/dT = 0
As is independent of T

Question 93

What is q_{vib</sup> @ high T limit?}

Answer 93

q_vib = 1/[exp(θ_v/2T)-exp(-θ_v/2T)]

@ T»θ_v, the exponents are small so can expand e^x ≈ 1+x+x²/2!+…

Results in
q_vib = T/θ_v

Question 94

What is ε_vib @ high T limit?

Answer 94

q_vib = T/θ_vib
E = NkT²(dlnq/dT)

E/N = kT² x 1/T = kT
Equipartition result

Question 95

What is C_v^vib @ high T limit?

Answer 95

E/N = kT
Cv = k per molecule or R per mol

Question 96

How does mass relate to popoulation of vib states?

Answer 96

Large mass means weaker bonds and smaller force constant
Results in high population of vib states
θvib = ℏw/k, lower when weaker bonds

Question 97

How many modes of vibration does a polyatomic molecule have?

Answer 97

linear: 3N-5 vib modes
non-linear: 3N-6 vib modes

Question 98

What is q_{vib, total} & ε_{vib, total} with a polyatomic molecule?

Answer 98

q_vib,total = q_vib¹ x q_vib² x …
ε_{vib, total} = ε_vib¹ + ε_vib² + …

Question 99

What occurs to energy levels at eqm?

Answer 99

E levels on reagent/product filled irrespective of if on either side
Just fills lower one

Question 100

Derive ΔG in terms of q

Answer 100

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(q^θ/N)
used as want standard molar of free energy
G = -NkT[ln(q^θ,0/N) + ln(exp(-ε⁰/kT)) = Nε⁰ - RTln(q^θ,0/N)

ΔG = ΔE₀ - RTΣ_Jv_Jln(q^θ,0_m,J/N)

ΔG = ΔE₀ - RTln[Π_J(q^θ,0/N)^Vj] = - RTln[exp(-ΔE₀/RT)
Π_J(q^θ,0/N)^Vj]

Question 101

What is the equilibrium constant with q?

Answer 101

ΔG = - RTln[exp(-ΔE₀/RT)
Π_J(q^θ,0/N)^Vj]

Generally ΔG = -RTlnK(T)
so
K(T) = Π_J(q^θ,0/N)^Vjexp(-ΔE₀/RT)

Question 102

What are the conditions of the equilibrium const with q?

Answer 102

q includes all dof for each J
For an ideal gas: θ applies to q_tr,m = V^θ_m/Λ³ = RT/P^θΛ³
q⁰ implies energies relative to ε₀ -> ground state @ T=0
ΔE₀ is from ground states between reactants and products
v_J is the stoichiometry coefficients

Question 103

How does translation Λ_J @ eqm relate to Λ₀?

Answer 103

Λ_J = h/Sqrt[2πm_JkT] = (h/Sqrt[2πmukT]) x M_J^-1/2 = Λ₀ x M_J^-1/2

Question 104

How do you go about finding the equilibrium constant theoretically?

Answer 104

Calc modes of motion individually before combining

trs: q^θ,0_J,m,tr = V^θ_m

elec: q⁰_J,m,elec = g₀ + g₁exp[-(ε₁-ε₀)/kT]+…

vib, rel to ZPE not bottom of potential well: q⁰_J,m,elec = 1/[1-exp(-hv/kT)]

rot, high T limit as usually : q_J,m,rot = T/σθ_rot

Question 105

What is molecular mass, M, used in calcs for isotope exchange?

Answer 105

Mass number of molecule - i.e. neutrons and protons (just integer values)

Question 106

What is the degen of an e- ground state?

Answer 106

g₀ = 2
Spin up or down -> must be retained in product/reactants

Question 107

What is the Saha equation?

Answer 107

Rate constant of thermal ionisation
K(T) = (kT/p^θ) (1/Λ³_e) exp[-I₁(atom)/RT]

Found as atoms/ions have no vibrations or rotations, same mass, ideal gas assumed

Question 108

How can you calculate the fraction of ionised atoms?

Answer 108

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) = [(P_M+/P^θ)(P_e-/P^θ)]/(P_M/P^θ)

Cancel to K(T) = (α²/1-α²) (p/p^θ)
If α small then can find value

Question 109

What is transition state theory?

Answer 109

R <-> ‡ -> P

for ‡->R, then K^‡
for ‡->P, then K^‡

Overall rate = d[P]/dt = K^‡[‡]

Question 110

How do you derive the reaction constants under TS theory?
AB + C <-> ‡ -> A + BC

Answer 110

K^‡ = (p^‡p^θ/p_ABp_C)

Assume ideal gas eqn:
K^‡ = (p^θ/RT) ([‡]/[AB][C])

[‡] = (RT/p^θ)K^‡[AB][C]

Therefore
d[P]/dt = k^‡[‡] = k_r[AB][C]
where k_r = k^‡(RT/p^θ) K^‡

And
K = [(N_Aq_m^‡)/(q_AB,m q_C,m)] exp(-ΔE₀^‡/RT)

Question 111

How do frequencies of intermediates relate to products?

Answer 111

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

Question 112

What is the q_vib for an intermediate?

Answer 112

q^‡ = ~q^‡ q_vib ‘
where ~q^‡ is the mpf for all other modes in ‡
q_vib’ is mpf for TS accounting only for E fo low-v vibration

q_vib’ = (1-exp[-hv/kT])^-1, relative to ZPE

When hv/kT &laquo_space;1 then can expand as series
q_vib ‘ = (1-1-(hv/kT) +…)^-1 = kT/hv
where v is freq of a loose vibration

Question 113

Derive the eyring equation for transition-state theory

Answer 113

k_r = k^‡ (RT/p^θ) K^‡

k_r = (κ-v)(RT/p^θ)(kT/hv) ~K^‡

Where ~K^‡ is eqm constant with q_vib’ removed

k_r = κ(RT/p^θ)(kT/hv)(N_A~q_‡/q_ABq_C) exp(-ΔE^‡/RT)

Question 114

What is the Eyring equation?

Answer 114

k_r = κ(RT/p^θ)(kT/hv)(N_A~q_‡/q_ABq_C) exp(-ΔE^‡/RT)

where all q’s are molar, 0, and θ

Question 115

What are the pros of the Eyring eqn?

Answer 115

v cancelled - validates can use this and other models

all terms can be calc or measuured so is calculable

simple cases -> good results

Question 116

What are the cons of the Eyring eqn?

Answer 116

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)