Kinetics 5: Complex reactions: composite Arrhenius parameters; chain reactions
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Composite Arhenius parameters Study the image. Assuming each elementary rate constant obeys an Arhenius law, with individual values of A and Ea, derive a composite expression for kobs. Use a reaction profile to illustrate what the apparent Ea represents.
What is a chain reaction?
A reaction comprised of non-linear elementary steps, which form chains in which the output of one step may be the input of an earlier step.
What is a chain carrier?
An intermediate formed during a chain reaction, which is an input into earlier steps, and thus propogates the reaction.
What are the general stages of a chain reaction?
• Initiation generates chain carriers • Propogation maintains chain carriers • Inhibition reduces reaction rate by destroying product (but not chain carriers) • Termination destroys chain carriers • Chain branching (sometimes): one chain carrier reacts to give 2+ carriers (results in explosion, eg combustion)
The thermal chain reaction H2 + Br2 –> 2HBr is well-characterised. Illustrate the stages of this reaction.
Initiation Br2 + M –> 2Br + M where M is a molecule Or photodissociation: Br2 + hv –> 2Br Propogation Br + H2 –> HBr + H H + Br2 –> HBr + Br Inhibition HBr + H –> H2 + Br Termination 2Br + M –> Br2 + M (M absorbs excess energy from new Br2 molecule, such that the Br2 doesn’t immediately fall apart)
The thermal chain reaction H2 + Br2 –> 2HBr is well-characterised. Which species would you expect to follow the steady state approximation?
Br and H, i.e. the chain carriers, since they are intermediates. Propogation • Br + H2 –> HBr + H* • H + Br2 –> HBr + Br*
The rate law for the thermal chain reaction H2 + Br2 –> 2HBr was found to be: d[HBr]/dt = (ka[H2][Br2]3/2)/([Br2] + kb[HBr]) What does this indicate about the mechanism?
• Complex • Fractional power indicates radicals are involved • [HBr] on denominator indicates HBr inhibits the reaction, ie it goes slower as [HBr] increases — but when [HBr] is small in the initial phase, it can be omitted from the denominator and inhibition doesn’t occur
Assuming the thermal chain reaction H2 + Br2 –> 2HBr has the following steps, use the ss approximation to derive an expression for the rate of formation of HBr. Initiation Br2 + M –> 2Br + M (k1) [1] Propogation Br + H2 –> HBr + H (k2) [2] H + Br2 –> HBr + Br (k3) [3] Termination 2Br + M –> Br2 + M (k4) [4] note other reactions could be involved, but the question is whether they compete effectively enough to alter the rate law
Assume chain carriers Br and H are in the steady state Expression for [H] d[H]/dt = k2[Br][H2] - k3[H][Br2] = 0 [H] = (k2[Br][H2]) / (k3[Br2]) Expression for [Br] d[Br]/dt = 2k1[M][Br2] - (k2[Br][H2] - k3[H][Br2]) - 2k4[M][Br]2 = 0 but the quantity in brackets is just d[H]/dt = 0 = 2k1[M][Br2] - (d[H/dt) - 2k4[M][Br]2 = 0 = 2k1[M][Br2] - 2k4[M][Br]2 = 0 [Br] = (k1[Br2]/k4)1/2 Note the 2-coefficients are from the fact that there are 2Br in the reagents/products, so the rate of change of Br is doubled. Also the squared term for k4 is due to the stoichoimetric coefficient = 2 Plug [Br] into [H] expression [H] = (k2[Br][H2]) / (k3[Br2]) = (k2[H2]/k3[Br2]) x (k1[Br2]/k4)1/2 = (k2/k3) x (k1/k4)1/2 x ([H2]/[Br2]1/2) Expression for HBr, then plug in [H] and [Br] d[HBr]initial/dt = k2[Br][H2] + k3[H][Br2] = k2[H2] x (k1[Br2]/k4)1/2 + k3[Br2] x (k2/k3) x (k1/k4)1/2 x ([H2]/[Br2]1/2) = 2k2 x (k1/k4)1/2 x [H2] x [Br2]1/2
Define chain length. Give its expression.
The average number of times that the closed cycle of steps producing products is repeated per chain carrier. chain length, l = (overall reaction rate)/(rate of initiation)
Expansion of the H2 + Br2 –> 2HBr reaction scheme to include [5], an inhibition step, gives the overall rate equation shown. Derive an expression for the chain length in the limit of low [HBr].
Overall reaction rate, A [HBr] low so can be removed from denominator, so: d[HBr]/dt = 2k2 x (k1/k4)1/2 x [H2] x [Br2]1/2 Rate of initiation, B refer back to scheme = 2k1[M][Br2] Chain length = A/B l = (2k2 x (k1/k4)1/2 x [H2] x [Br2]1/2) / (2k1[M][Br2]) = (k2 / (k4k1)1/2) x ([H2] / ([Br2]1/2[M])) under typical conditions of pressure, chain length is order 10^13, indicating one Br produces many HBr before being terminated