Foundations of Chemical Kinetics Lecture 16: Unimolecular reactions - - PowerPoint PPT Presentation

Foundations of Chemical Kinetics Lecture 16: Unimolecular reactions in the gas phase: The Lindemann mechanism

Marc R. Roussel Department of Chemistry and Biochemistry

Examples of unimolecular reactions in the gas phase

Isomerizations:

• N

C C C N H3 CH3

• Decompositions:

CH2 CH2 CH3 CH2 CH3 CH2 C CH H3 CH2 CH3 CH3 CH2 C C O O +

What makes these reactions happen?

General behavior of unimolecular reactions

◮ These reactions are generally carried out with a “bath gas”

(M) for which [M] ≫ [A] ([A] = reactant).

◮ At low pressures, the rate law has partial orders with respect

to A and M of 1 each, so an overall order of 2.

◮ At high pressures, first-order kinetics in [A] is observed.

These observations are both

• 1. a clue as to the mechanism, and
• 2. a puzzle to be solved.

The Lindemann mechanism

A + M

k1

− − ⇀ ↽ − −

k−1

A∗ + M A∗ k2 − → products Implied potential energy profile:

x A A* P E

The Lindemann mechanism

Rate equations A + M

k1

− − ⇀ ↽ − −

k−1

A∗ + M A∗ k2 − → products Rate equations: d[A] dt = −k1[A][M] + k−1[A∗][M] d[A∗] dt = k1[A][M] − k−1[A∗][M] − k2[A∗]

◮ There is no way to solve these equations exactly.

The Lindemann mechanism

Steady-state approximation

◮ There is a low barrier for reaction of A∗, so we expect

k−1 ≫ k1 and k2 ≫ k1[M].

◮ Expected time course for [A∗]:

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 100 200 300 400 500 [A*] (arbitrary units) t (arbitrary units)

The Lindemann mechanism

Steady-state approximation

◮ During the long, slow decline of [A∗],

d[A∗] dt ≈ 0

◮ This is called the steady-state approximation.

d[A∗] dt = k1[A][M] − k−1[A∗][M] − k2[A∗] ≈ 0 ∴ [A∗] ≈ k1[A][M] k−1[M] + k2 ∴ v = k2[A∗] ≈ k1k2[A][M] k−1[M] + k2

The Lindemann mechanism

v = k1k2[A][M] k−1[M] + k2

◮ Low-pressure limit: ◮ High-pressure limit:

The Lindemann mechanism

Interpretation of the high-pressure limit

◮ Suppose that the reaction A + M

k1

− − ⇀ ↽ − −

k−1

A∗ + M was the only

• ne occurring.

◮ When this reaction reached equilibrium, we would have

da dt = da∗ dt = 0 ∴ k1[A][M] = k−1[A∗][M] ∴ [A∗] [A] = k1 k−1 = K1 where K1 is the equilibrium constant for this reaction.

◮ The high-pressure limit of the Lindemann rate equation is

therefore v ≈ K1k2[A]

The Lindemann mechanism

A slight rewrite v = k1k2[A][M] k−1[M] + k2 = kL[A] with kL = k1k2[M] k−1[M] + k2 . Define k∞ = k1k2/k−1 (high-pressure limit of kL) Then, kL = k∞[M] [M] + k∞/k1 .

Experimental determination of the Lindemann parameters

kL = k∞[M] [M] + k∞/k1 . ∴ 1 kL = 1 k∞ + 1 k1 1 [M] A plot of k−1

L

vs [M]−1 therefore allows us to recover k∞ and k1.

Example: cis-trans isomerization of 2-butene at 740 K

C H H C C H C H H 3 CH 3 CH 3 CH 3 C

[2-butene]/10−5mol L−1 0.25 0.3 0.6 1.2 5.9 k/10−5 s−1 1.05 1.14 1.43 1.65 1.82 Note: The theory is the same even if the “bath gas” is the reactant itself.

Example: cis-trans isomerization of 2-butene at 740 K

(continued)

0.5 0.6 0.7 0.8 0.9 1 0.5 1 1.5 2 2.5 3 3.5 4 4.5 k-1/105s [2-butene]-1/105L mol-1

Example: cis-trans isomerization of 2-butene at 740 K

(continued) slope = 0.106 mol s L−1 ∴ k1 = 1 slope = 9.4 L mol−1s−1 intercept = 0.53 × 105 s ∴ k∞ = 1.9 × 10−5 s−1

The value of k1

◮ If we vary the temperature, we can get the preexponential

factor and activation energy corresponding to k1.

◮ We might guess that k1 (rate constant for A + M → A∗ + M)

is collision limited. We should therefore be able to predict the pre-exponential factor from collision theory.

Collision-theory preexponential factor for the collisional activation of cyclopropane at 760 K

◮ The hard-sphere radius of cyclopropane is 2.2 ˚

A.

◮ σ = π(2r)2 = 6.1 × 10−19 m2 ◮ The mean relative speed of cyclopropane molecules is

¯ vr =

• 8RT

πµm =

• 8(8.314 472 J K−1mol−1)(760 K)

π(70.134 × 10−3 kg mol−1) = 479 m s−1.

◮ Act = σ¯

vrL = 1.75×108 m3mol−1s−1 ≡ 1.75×1011 L mol−1s−1

◮ Experimental value: 9 × 1018 L mol−1s−1

Curved Lindemann plots

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 50 100 150 200 250 300 350 400 k-1/105s p-1/torr-1

Lindemann plot for the decomposition of cyclobutane to ethene at 449 ◦C Source: Butler and Ogawa, JACS 85, 3346 (1963).

Summary: the two problems with Lindemann theory

• 1. The rate constant k1 exceeds the collision theory value, which

should be an upper limit according to the theory studied so far.

• 2. Plots of k−1 vs p−1, which should be straight, deviate from

linearity at low pressures.