Chemical Kinetics and Transition States

Elementary Rate Lawsk(T)

Transition State TheoryCatalysis

I. Rate Equation for Elementary Rate Laws

• Rate =d[A]/dt = -k [A]n assuming nth order– [A] is reactant in A B (assume negligible

reverse rxn)– k is rate constant, units of 1/[concn-1-time]– [A] = [A(0)] exp (- kft)

• Now consider A ↔ B with kf rate and kr rate; i.e. there is a substantial back rxn– Then d[A]/dt = - kf[A] + kr[B] = - d[B]/dt

• Experiments rate laws, k and rxn order

Review of Elementary Rate LawsOrder Reaction Differential

Rate Law- d[A]/dt =

Integrated Rate Law[A] = f(t)

Units of k Plot Half-life

0 A P k [A] - [A]0 = -kt M/s [A] vs tSlope = - k

[A]0/2k

1 A P k[A] [A] = [A]0 e-kt 1/s ln[A] vs t

Slope = -k(ln 2)/k

2 A P k[A]2 1/[A]0 - 1/[A] = - kt (M-s)-1 1/[A] vs tSlope = k

1/k [A]0

2 A+ B P

k[A][B] kt = ([B]0 - [A]0)-1 *

ln {[A]0[B]/[B]0[A]}

(M-s)-1 Assume [A]0< [B]0

3 A P k[A]3 1/2 {[A]-2 - [A]0-2} =

kt

1/(M2-s) 3/{2k A]02}

n A P k[A]n α 1/[A]0 n-1

Equilibrium

• At equilibrium, d[A]/dt = 0 forward rate = kf[A] = kr[B] = reverse rate.

– This is the Principle of Detailed Balancing and leads to

– K =[B]eq/[A]eq = kf/kr (recall Eqn 13.23)

– Principle of Microscopic Reversibility

II. Arrhenius Eqn: k(T)

• In Ch 13, we combined the Gibbs-Helmholtz Eqn (G(T)) and the Gibbs Eqn (G = - RT ln K) – to get the van’t Hoff Eqn: d ln K/dT = ho/kT2

• Arrhenius combined the van’t Hoff eqn with K = kf/kb to get

– Differential eqn: d ln kf/dT = Ea/kT2 where Ea = forward activation energy; assume constant to integrate

– Integrated eqn: kf = A exp(-Ea/kT); as T ↑, kf ↑ if Ea > 0 (usual case shown in Fig 19.2 except see Prob 19.10)

– Therefore a plot of ln kf vs 1/T Eaand A (Fig 19.5)

Activation Energy Diagram (Fig 19.3)

• Ea= activation energy for forward rxn

• Ea‘= activation energy for reverse rxn

• ξ = rxn coordinate ho = Ea- Ea‘

• Note that Ea and Ea‘ > 0 (usual case) so ki ↑ with T for endo- and exothermic rxns

• Ex 19.1, Prob 8

III.Transition State Theory (TST)

• TS is at the top of the activation [‡] barrier between reactants and products.

• Energy landscape for chemical rxn– A + BC [A--B--C]‡ = TS AB + C

– Fig 19.7 for collinear rxn: D + H2 HD + H

• See handouts for – H + H2 H2 + H

– F + H2 HF + H

Saddle Point (Fig 19.7)

• Rxn starts in LHS valley (Morse potential) of H2 with D far away.

• D and H2 approach, potential energy ↑

• TS is at max energy along rxn coord; i.e. [H--H--D]‡ exists.

• Then H moves away and valley (another Morse potential) is HD.

Potential Energy Contour Diagram (Fig 19.8)

• The information in Fig 19.7 can be shown as a contour diagram (Fig 19.8).

• Follow rxn A + BC AB + C. Reactants are lower RH corner (energy min) and follow dotted line up to TS and then down to upper LH corner.

krsaddleshop.com

A 360 degree view

• http://my.voyager.net/~desotosaddle/saddle_pictures.htm

TST Rate Constant, k2

• A + B --k2 P overall rxn which proceeds via a TS: A + B K‡ (AB)‡ --k‡ P

• This 2-step mechanism involves an equilibrium between reactants and TS with eq. constant – K‡ = [(AB)‡]/[A][B] = [q‡/qAqB] exp (D‡/kT)

• and then the formation of products from the TS with rate constant k‡.

• d[P]/dt = k‡[(AB)‡] = k‡K‡[A][B] = k2[A][B]

TST: Reaction Coordinate

• d[P]/dt = k‡[(AB)‡] = k‡K‡[A][B] = k2[A][B]

• k2 = k‡K‡ is the connection between kinetics and stat. thermo (partition functions)

• In TST, we hypothesize a TS structure and assume that the reaction coord ξ is associated with the vibrational degree of freedom of the A—B bond that forms.

TST

• Define qξ as the partition function of this weak vibrational deg of freedom and separate it from other degs of freedom (q‡*)

• Then q‡ = q‡* qξ = product of TS q except rxn coord x q of rxn coord

• The reaction coord ξ is associated with a weak bond (small kξ and small ν ξ).

TST

• Then q‡ ≈ κkT/hνξ. • κ = transmission coefficient; 0 < κ ≤ 1.• K‡ = [q‡/qAqB] exp(D‡/kT)

= q‡*qξ/[qAqB] exp(D‡/kT)

= q‡* kT/hνξ /[qAqB] exp(D‡/kT) • k2 = k‡K‡=νξ(kT/hνξ){q‡*/[qAqB]} exp(D‡/kT)

= (kT/h) q‡*/[qAqB] exp(D‡/kT) = (kT/h) K‡*• Ex 19.2

Primary Kinetic Isotope Effect

• When an isotopic substitution is made for an atom at a reacting position (i.e. in the bond that breaks or forms in the TS), the reaction rate constant changes.

• These changes are largest for H/D/T substitutions.

• And can be calculated using eqn for k2 = (kT/h) q‡*/[qAqB] exp(D‡/kT)

Isotope Effect

• Example in text is for breaking the CH (kH) or CD (kD) bond.

• Assume that – q(CH‡)≈q(CD‡) and q(CH) ≈ q(CD)– C-H and C-D have the same force constants.

• Then C-H and C-D bond breakage depends on differences in vibration of reaction coord (ξ) or νCX or reduced mass.

Isotope Effect

• kH /kD = exp [(DCH‡ - DCD

‡ )/kT]

• = exp {-(h/2kT)[νCD - νCH ]}

• = exp {-(hνCH/2kT)[2-1/2 - 1]} since

• ν = (1/2π)√(ks/μ) ks= force const, μ = reduced mass

• Ex 19.3; see Fig 19.9

• Prob 3

Thermodynamic Properties of TS or Activated State (Arrhenius)

• K ‡* = equilibrium constant from reactants to TS without the rxn coordinate ξ.

• Define a set of thermody properties for the TS: G‡ = - kT ln K ‡* = H‡ - TS‡

– k2 = (kT/h) K‡* = (kT/h) exp(-G‡/kT)

= [(kT/h) exp(-S‡/k)] exp(-H‡/kT)

• [term] is related to Arrhenius A and H‡ is related to Ea Prob 6

• k vs T expts H‡, S‡, G‡

IV. Catalysis

• k0 = (kT/h) [AB‡*]/[A][B] = (kT/h) K0

‡* w/o catalyst

• kc = (kT/h) [ABC‡*]/[A][B][C] = (kT/h) Kc

‡* w/catalyst

• Rate enhancement = kc/k0 = [ABC‡*]/[AB‡][C] = measure of binding of C to TS = binding constant = KB*

• KB* increases as C-TS binding increases

• Fig 19.12, 19.13

Catalysis Mechanisms

• Catalysts stabilize ‡ relative to reactants; this lowers activation barrier.

• T 19.1: create favorable reactant orientation

• T 19.2 and Fig 19.14: reduce effect of polar solvents on dipolar transition state

Acid and Base Catalysis

• Consider a rxn R P catalyzed by H+ Then rate might be = ka [HA] [R]

• H+ is produced in AH ↔ H+ + A-

– Ka = [H+][ A-]/ [AH]

• These two rxns are coupled

Brønsted Law

• log ka = α log Ka + ca

– or log ka = - α pKa + ca

– α > 0 and ca = constant

– As Ka ↑ (stronger acid), rxn rate constant ka ↑

– Plot log ka vs pKa (Fig 19.15)

• This law proposes that presence of acid stabilizes the product.

• Omit pp 361-365