Rate Processes Flashcards
What is the Arrhenius equation?
kr = A exp(-Ea/RT)
Determined by gradient of plot of lnkr against 1/T
What is the assumption of Arrhenius equation?
A is independent of T
What is simple collision theory (SCT) based on?
Kinetics of bimolecular gas-phase reactions
What are the assumptions of SCT?
Frequent collisions between reactants
Fraction of collisions have E >= Ea
Steric factor
What is the pre-exponential factor in SCT?
A = N2 σ crel
where
σ = πd2, collision cross section
crel = Sqrt[8kT/πμ], mean rel speed of reactants
What is d in SCT calculations?
Mean molec diameter
d = 1/2(da + db)
What is the collision density in SCT?
p(r) = NA2 σ
crel [A][B]
How does A in SCT compare to experimental values?
Experimental A is sig larger than calculated by SCT
What is the steric factor in SCT?
Correction for exp values
P = Aexp / Acalc
Fudge factor
What is the dividing surface in a PE map?
Group of tipping points from one compound to another
TS on the surface
What is the thermal rate coefficient?
kr -> rate at thermal eqm, specified T
Includes rot, vib, and trans
What is the transition-state theory (TST) overview?
A + B <-> C‡ -> P
Assumes pre-equilibrium, fixed E, all reactive states accessible, motion classical and separable from orthoganol reaction coord
What is the Eyring equation?
kr = NA κ(kT/h) (qC‡~,0/V) / [ (qA0/V)(qB0/V)] exp(-ΔE0‡)
What is q and how does ~,0 effect them?
q is molecular partition function
~ means vib modes for motion through TS removed
0 means using energies from ground state (T=0)
What is κ, kappa, in Eyring eqn?
Transmission coefficient
Prob an activated complex will lead to product
What is the energy term in the Eyring equation?
ΔE0‡ = Ea + ZPE(C‡) - ZPE(Reactants)
How would you derive the Eyring equation?
d[P]/dt = κv[C‡]
where kappa is trans coefficient and v is freq of low vib mode for motion over saddle point
use K‡ = (pθ/RT) [C‡]/[A][B]
as assume perfect gas, then use result from stat mech and q of C‡ relation to ZPE to give values
What is qC‡0?
qC‡0 = qC‡~,0 qC^,‡0
As a harmonic oscillator, qC^,‡0 = 1 - exp(-hv/kT) ~ kT/hv
This is as hv «_space;kT
What does q of a species consist of?
qtotal = qtrans x qrot x qvib x qelec
What is the equation for qtrans in terms of T?
qtrans/V = (2πmjkT/h2)3/2
For relationship, q/V α T3/2
What is the equation for qrot in terms of T and B?
qrot ~ kT/B = 8π2kTμd2/h2
this is because B = hbar2/2I and I = μd2
How can you estimate A in TST?
Find approx values of q as average mps per degree of freedom
Then put into kr
What is the approx value for qtrans?
qtrans ~ qT3
Due to 3x degrees of freedom when moving in 3D
What is the approx value for qrot?
qrot ~ qRn
Where n=2 for linear & n=3 for non-linear
What is the approx value for qvib?
qvib = qvn
n = No of vib modes
3N-5 for linear, 3N-6 is for non-linear
What occurs to qvib if remove vib mode at TS?
qvib = qvn
qvib~ = qvn-1
How can you estimate A in SCT?
SCT is equal to linear TS in TST
qA0 = qB0 = qT3
qAB‡0 = qT3 x qR2 x qV
Simplifies to qR2 / qT3
Then insert into eyring
Estimate A for a non-linear TS in TST
AB + CD -> ABCD‡ -> Prod
qAB0 = qCD0 = qT3 x qR2 x qV
qABCD‡0,~ = qT3 x qR3 x qV5
Simplifies to qV3 / [qT3 x qR]
What is the steric factor in terms of q for a diatomic TS?
ATST α qR2 / qT3
ASCT α qV3 / [qT3 qR]
P = (qV/qR)3
What are some typical values of qR and qV?
qR = 10
qV = 1
No of accessible states at T
What is the temp dependence of qV, qT, qR?
Common assumption
qV = 1
qT α Sqrt(T)
qR α Sqrt(T)
What is the T dependence of kr in TST?
kr α T-1 exp[-ΔE0‡/RT]
How does Ea relate to ΔE0‡ in TST?
Ea = ΔE0‡ + nRT
Mostly ΔE0‡»_space;> RT so Ea has little T-dependence
When do isotopic efffects dominate?
Low masses
Such as H/D as largest % difference in mass
How do rates compare when H/D?
(in terms of ln)
ln[krH/krH] = ln[AH/AD] - [ΔE0,H‡ - ΔE0,D‡]/RT
2nd term dominates
ln() α 1/T
How does activation energy relate comparing H/D?
ΔE0,H‡ - ΔE0,D‡ = ZPE(D) - ZPE(H)
Assumes TS has same E, assume mH/D much smaller than other atom so μHB/D = mH/D
ΔE0,H‡ - ΔE0,D‡ = 0.5NAhbar x Sqrt[kf/mH] x (1/Sqrt2 - 1)
What is the comparison in rates between H and D?
(absolute values)
krH/krD = 8
For typical diatomics @ 298K
H transfer is faster
How does A change when different isotope?
Mass change effects:
qtrans α m3/2
qrot α I
Means A larger when lighter
AH/AD = 2.8
Why do curved Arrhenius plots occur?
3 ways which causes this:
* T-dependence of A
* kr depends on reactant quantum states
* Quantum mech tunneling at low T
How does kr depend on the quantum state of a reactant?
If the reactant is in a higher energy state then the reaction is faster
e.g. v=1 instead of v=0, then the rate is 100x faster
What is the problem of TST?
Structure & vib frequency of TS may not be known so need to use a simpler equation
How can you simplify TST?
Use ideal gas, Vmθ = RT/Pθ
so kr = κ(kT/h)(RT/Pθ)K~‡
Where K~‡ is partial pressure eqm const
Then relate to ΔG‡
kr = κ(kT/h)(RT/Pθ) exp(ΔS‡/R) exp(-ΔH‡/RT)
And as Ea = RT2 x dlnkr/dT, Ea = ΔH‡ + 2RT
kr = κ(kT/h)(RT/Pθ)e2 exp(ΔS‡/R) exp(-Ea/RT)
What does the simplification of TST change dependence of A?
A is therefore entropy-based difference between reactants & TS
Most important is difference in no of vib modes
When greatest loss of rot energy (Δn most -ve) then smaller A
What is the difference in vib modes (Δn) for different reactions?
A = atom, D = diatomic, P = Polyatomic
A + A -> TS, Δn=2
A + D -> non-linear TS, Δn=1
A + D -> linear TS, Δn=0
D + D -> non-linear TS, Δn=-1
D + D -> linear TS, Δn = -2
non-linear P x2 -> non-linear TS, Δn=-3
How do reactions in both solution and gas occur?
Similar rates in both phases
Little solvent dependence
How does separation of molecules, mean free path and more, change in gas and solution?
Gas - rare collisions with rel free motion, low density
Solution - frequent solvent-solute collisions with rare solute-solute encounters
What is the cage effect for gas and solution?
Gas - collisions between molecules occur singly
Solutions - collisions occur in bunches with rel long intervals between them
What are scavenging reactions evidence for?
R-N=N-R -> {R* R} -> RR* -> R-R
Without scavenger then recombines to form R-R
Low conc competition between scavenging and bulk recomb of free radicals
What is diffusion or activation control?
Diffusion -> RDS is formation of encounter pair
Activation -> RDS is reaction pair, overall will depend on KAB (eqm const for encounter formation)
What is flux?
#of moles passing through a surface area of 1m2 per second
What is the rate in a diffusion controlled reaction dependent on?
Rate = total flux of B molecules diffusing towards & colliding with A molecules
[B]r = (1-R*/R) [B]
where encounter distance, R* = RA + RB
and bulk conc is [B]
What is Fick’s 1st law for diffusion controlled reactions?
Jr = DB(d[B]r/dr) = DB(R*/r2)[B]
where Jr = flux of B down a conc gradient
DB = diffusion coefficient
What is the reaction rate under Fick’s 1st law for 1x static A?
Rate = 4πR*2 Jr=R* = (4πR*2)(DB[B]/R*) = 4πR*[B]
What is the reaction rate under Fick’s 1st law for A and B?
Rate = 4πDR* [A][B]= kd [A][B]
where kd = 4πDR* NA
What is the Stokes-Einstein equation?
DA = kT/6πηRA
Where viscocity, η = η0 exp(Ea/RT)
What is the Stokes-Einstein equation?
DA = kT/6πηRA
Where viscocity, η = η0 exp(Ea/RT)
so kd α η-1 α exp(-Ea/RT)
What is the Stokes-Einstein equation when they are ions?
Coulomb interaction adds drift term
kd = 4πDReff NA
where Reff= Rc/[exp(Rc/R*)-1]
Rc = zazbe2 / 4πε0εkT
What is the Onsanger distance?
In stokes-einstein equation and diffusion control
Rc, separation @ which Coulomb interaction = kT
What is the rates of steps in activation controlled reactions?
A+B <-> {AB} -> P
ka is {AB} ->, and kd is forwards of eqm and kd’ is backwards reaction
ka «_space;kd’ so kr=ka(kd/kd’) KAB
What is influence of solvent perimittivity on ionic reactions in activation-controlled reactions?
Add coulomb interaction to ΔG
lnkr = lnkr0 - (1/RT)(zazbe2NA/4πεε0R*)
where lnkr0 is rate without ionic interaction
zazb > 0 then kr<kr,0
Gives lnkr vs 1/ε is linear
How does rate change when ε changes under activation controlled reactions?
As ε decreases, shielding of ions by solvents decreases
Means less stable encounter complex and a slower reaction
How does ionic effects change rate in activation controlled reactions?
Ionic can stabilise or destabilise the TS which effects the kr
What is the influence of pressure on activation controlled reactions?
Use dG = Vdp - SdT
(dlnkr/dp)T = (-1/RT)(dΔG‡/dp)T = -ΔV‡/RT
Where ΔV‡ is molar vol of reactants and TS
What occurs when ΔV‡ is independent of T in activation controlled reactions?
(dlnkr/dp)T = -ΔV‡/RT
lnkr = lnkrθ - (ΔV‡/RT)(p-pθ)
where krθ is kr at p=pθ
For bimol reactions -> ΔV‡ dominated by ΔVm of reactants & encounter pair
For ionic reactions -> dominated by ΔV occupied solvent, called electrostriction
If TS more charged than reactants, solvent around TS tighter packed & ΔV‡ -ve
How can you use Ea for kr definition?
Ea = ΔH‡ + RT
kr = κNA(kT/h) exp[ΔS‡/R] exp[-Ea/RT]
How does ΔS‡ depend on zazb?
zazb > 0 then ΔS‡ < 0, higher TS charge so solvent more ordered
zazb < 0, then ΔS‡ > 0
What is ionic strength effects in an activation controlled reaction?
K = aAB/aAaB = [{AB}]/[A][B] * (γAB/γAγB) = KAB (γAB/γAγB)
where
aJ = activity of species J
γJ = activity of coefficient of J
What is the Debye Huckle limiting law for activation controlled reactions?
log γJ = -Az2J Sqrt[I]
Where I = 0.5 ΣJ (bJ/bθ)z2J
Where bJ = molarity
How do you derive the Debye-Huckle limiting law?
kr = kaKAB = ka K (γAγB/γAB) = kr0 (γAγB/γAB)
logkr = logkr0+ logγA + logγB - γAB = logkr0 - ASqrt[I](z2A + z2B - z2AB
simplifies to
logkr = logkr0 + 2AzAzBSqrt[I]
What is an unforseen factor in electron transfer (ET) reactions?
No bond breaking in condensed phases but still has an Ea
What causes Ea in electron transfer (ET) reactions?
Effect similar to Franck-Condon factor
e- transfer faster than change in bond length or orientation change of solvent
What is the Franck-Condon factor?
Absorption of light nearly instantaneous
No time for change in position of nuclei
What is an assumption in Marcus theory?
All occurs in aqueous
G is in 1D, and treats it as harmonic
How does ET occur in Marcus theory?
Where () is in one solvent sphere, [] is another and {} is a third
(2+) + {3+}<-> [(2+) + {3+}] <-> [2+ 3+] <-> [3+ 2+] <->{3+} + (2+)
What is the process of ET in Marcus theory?
D + A -> D+ + A-
Before: solvent reorganisation & bond reorgansiation so reactants and products have same energy
ET: Tunelling in rigid nuclear framework then relaxation
What is the rate of tunnelling dependent on in Marcus theory?
Depends on overlap of donor & acceptor elec wavefn decreases wrt r
ket α exp(-βr)
r = edge-to-edge separation of D & A
What is the Δ‡G in ET according to Marcus theory?
When identical PE curves and ΔGF(D+A) = ΔGF(D+ + A-)
Then
Δ‡G = ( ΔrGθ +λ)2 / 4λ
where λ = reorganisation energy
What is λ in Marcus theory?
Reorganisation energy
Gibbs energy to distort eqm nuclear framework of prods to that of reactants without electron transfer
What is the reaction rate in Marcus theory?
kr α exp[-βr]exp[-Δ‡G/RT]
plot of lnkr vs ΔrG gives normal on left of λ, and inverted on RHS
What is the normal Marcus reigon?
-ΔrGθ < λ
More -ve ΔrGθ is the driving force, so rate increases
What occurs when -ΔrGθ = λ in Marcus theory?
Activation-less ET
ΔrGθ = 0 so kr is at max
Little change in bond length or molec geometry
What is the inverted Marcus reigon?
ΔrGθ > λ
More -ve ΔrGθ so larger driving force
However, k decreases
What is potential, φ?
φ = potential @ point is work done in moving unit +ve charge from infinity to a point
Units of V
What is the potential difference?
E = φ2 - φ1
Made by charge separation
Measure by estabilishing eqm when charged species exchanged across a surface
What is current?
Rate of flow of charge
j = dQ/dT
What is G at an electrode?
When charge ze & subject to φ
G = G(0) + zFφ
Where F is faraday const
What is the faraday constant?
Charge on mole of e-
How does rate of an electrode reaction change when the potential, φ, changes?
As φ increases, the reactant is destabilised
But destab TS by smaller amount so reaction occurs faster
What is the Butler-Volmer eqn for current density (j)?
j = j0( [Red]0/[Red]∞) exp[βFη/RT] - j0( [Ox]0/[Ox]∞) exp[-αFη/RT]
Where:
[]0 is conc @ electrode surface
[]∞ is conc in bulk
η is activation overpotential
What is η in the Butler-Volmer eqn?
Activation overpotential
η = E - E0
What are α&β in the Butler-Volmer eqn?
β is transfer coefficient
α = 1-β
if β~1 then TS resembles reactants
if β~0 then TS resembels products
Derive the Butler-Volmer eqn
for reaction in aq: Oxn+ + e- <-> OxOx(n-1)+
Forward reaction is kred & backwards is kox
φm & φs for in metal and solutions respectively
Gr = Gr(0) + (n-1)Fφs - F(φm-φs)
Gp = Gp + (n-1)Fφs
G‡ = G‡(0) + (n-1)Fφs - βF(φM-φS)
gives
ΔG‡red = G‡ - Gr
ΔG‡oxn = G‡ - Gp
j = F d[e-]/dt = F(kox[Red]0 - kred[Ox]0]
Include overpotential to give Butler-Volmer
j = j0( [Red]0/[Red]∞) exp[βFη/RT] - j0( [Ox]0/[Ox]∞) exp[-αFη/RT]
What is the exchange current density, j0?
j0 = Fk0 [Red]α∞ [Ox]β∞
where k0 is rate const for forward and backwards
How does rate of red and oxn depend on overpotential, η?
When η large & +ve then v slow redn but fast oxn
exp(βFη/RT)~0 j~j0exp(βFη/RT) so
lnj = lnj0 + βFη/RT
When η large & -ve then v slow oxn but fast redn
exp(-αFη/RT)~0
j~ -j0exp(-αFη/RT)
ln-j = lnj0 - αFη/RT)
How can you find j0 & E0?
Tafel plot of η vs ln|j|
Get α&β from gradients
j0 from extrapolated intercept of two lines
E0 is from
How can you find j0 & E0?
Tafel plot of η vs ln|j|
Get α&β from gradients
j0 from extrapolated intercept of two lines
E0 is from η-axis intercept
What is a cyclic voltammetry plot?
Time vs Measured current (function of potential)
X <-> Y + e-
More +ve potential gives oxn
More -ve potential gives redn
How does potential change rate of oxn?
When low, kox small so rate of reaction governed by rate of e- transfer
When high, kox higher but electrolysis laready carried out depleted surface [X]
So k determined by rate of diffusion of X to electrode
