Reaction Mechanisms - derive rate laws

Skills to develop

Explain reaction mechanism.

Derive a rate law from a given mechanism.

Reaction Mechanisms - derive rate laws

A reaction mechanism is a collection of elementary processes or steps (also called elementary steps) that explains how the overall reaction proceeds.

A mechanism is a proposal from which you can work out a rate law that agrees with the observed rate laws. The fact that a mechanism explains the experimental results is not a proof that the mechanism is correct.

As a mechanism, no proof is required. Proposing a mechanism is an interesting academic exercise for a mature chemist. Students in general chemistry will not be required to propose a mechanism, but they are required to derive the rate law from a proposed mechanism.

Elementary Processes or Steps

A summary of the elementary process or steps is given in a table form here. This previous link provides details about these steps. In the table below, A, B, and C represent reactants, intermediates, or products in the elementary process.

Molecularity Elementary step Rate law
1 A -> products rate = k [A]
2 A + A -> products
A + B -> products rate = k [A]²
rate = k [A] [B]
3 A + A + A -> products
A + 2 B -> products
A + B + C -> products rate = k [A]³
rate = k [A] [B]²
rate = k [A] [B] [C]

Molecularity	Elementary step	Rate law
1	A -> products	rate = k [A]
2	A + A -> products A + B -> products	rate = k [A]² rate = k [A] [B]
3	A + A + A -> products A + 2 B -> products A + B + C -> products	rate = k [A]³ rate = k [A] [B]² rate = k [A] [B] [C]

Deriving Rate Laws from Mechanisms

You all have experienced that an accident or road construction on a free way slows all the traffic on the road, because limitations imposed by the accident and constructions apply to all cars on that road. The narrow passing stretch limits the speed of the traffic.

If several steps are involved in an overall chemical reaction, the slowest step limits the rate of the reaction. Thus, a slow step is called a rate determining step. The following examples illustrate the method of deriving rate laws from the proposed mechanism. Please learn the technique.

Example 1

If the reaction
2 NO₂ + F₂ = 2 NO₂F follows the mechanism,
i. NO₂ + F₂ = NO₂F + F (slow)
ii. NO₂ + F = NO₂F (fast) What is the rate law?

Solution
Since step i is the rate-determining step, the rate law is

    1  d[NO₂]
 - --- ------  =  k [NO₂] [F₂]
    2    dt

Since both NO₂ and F₂ are reactants, this is the rate law for the reaction.

Discussion:
Addition of i. and ii. gives the overall reaction, but the step ii does not affect the rate law. Note that the rate law is not derived from the overall equation either.

Example 2

i	Br₂	k₁ ® ¬ k_-1	2 Br	(both directions are fast)
ii	Br + H₂	k₂ ®	HBr + H	(slow)
iii	H + Br₂	k₃ ®	HBr + Br	(fast)

For the reaction
H₂ + Br₂ = 2 HBr, the following mechanism has been proposed

i Br₂ k₁
®
¬
k_-1 2 Br (both directions are fast)
ii Br + H₂ k₂
®
HBr + H (slow)
iii H + Br₂ k₃
®
HBr + Br (fast)
Derive the rate law that is consistent with this mechanism.

Solution
For a problem of this type, you should give the rate law according to the rate-determining (slow) elementary process. In this case, step ii is the rate-determining step, and the rate law is,

  1   d[HBr]
 --- -------- = k₂ [H₂] [Br]
  2    dt

The factor 1/2 results from the 2 HBr formed every time, one in step ii and one in step iii. Since [Br] is not one of the reactants, its relationship with the concentration of the reactants must be sought. The rapid reaction in both direction of step i implies the following relationship,

     k₁ [Br₂] = k_-1 [Br]²
or
     [Br] = ((k₁/k_-1) [Br₂])^(1/2)

Substituting this in the rate expression results in, rate = k₂ (k₁/k_-1)^(1/2) [H₂] [Br₂]^(1/2)

The overall reaction order is 3/2, 1 with respect to [H₂] and 1/2 with respect to [Br₂].

Discussion:
The important point in this example is that the rapid equilibrium in step i allows you to express the concentration of an intermediate, ([Br]) in terms of concentrations of reactants ([Br₂]) so that the rate law can be expressed by concentrations of the reactants.

The ratio k₁/k_-1 is often written as K, and it is called the equilibrium constant for the reversible elementary steps.

Example 3

Derive the rate law that is consistent with the proposed mechanism in the formation of phosgene from Cl₂ and CO. (K₁ = k₁/k_-1 and K₂ = k₂/k_-2 may be considered as equilibrium constants of the elementary processes, and M is any inert molecule.)
i. Cl₂ + M = 2 Cl + M (fast equilibrium, K₁) ii. Cl + CO + M = ClCO + M (fast equilibrium, K₂) iii. ClCO + Cl₂ = Cl₂CO + Cl (slow, k₃)
The overall reaction is Cl₂ + CO = Cl₂CO

Solution
From the rate-determining (slow) step,

  d[Cl₂CO]
 ---------  =  k₃ [ClCO] [Cl₂]  - - - (1)
    dt

You should express [ClCO] in terms of concentrations of Cl₂ and CO. This is done by considering step ii. [ClCO] = K₂ [Cl] [CO] - - - (2)

You should express [Cl] in terms of [Cl₂]. For this, you may use step i.

[Cl] = K₁^(1/2) [Cl₂]^(1/2) - - - (3) Substituting (3) in (2) and then in (1) gives the Rate, Rate = k₃ K₁^(1/2) K₂ [CO] [Cl₂]^(3/2)
= k [CO] [Cl₂]^(3/2)

where k = k₁ K₁^(1/2) K₂, the observed rate constant. The overall order of the reaction is 5/2, strange but that is the observed rate law.

Discussion:
This example shows how the concentrations of intermediates are related to those of the reactants in a two-step equilibrium.

If the third step is

iii. ClCO + Cl = Cl₂CO (slow, k₃) the rate law will be different from the result derived above. As an exercise, derive the rate law using this alternate step.

Example 4

In an acid solution, the mechanism for the reaction
NH₄⁺ + HNO₂ -> N₂ + 2 H₂O + H⁺ is:

i. HNO₂ + H⁺ = H₂O + NO⁺ (equilibrium, K₁) ii. NH₄⁺ = NH₃ + H⁺ (equilibrium, K₂) iii. NO⁺ + NH₃ -> NH₃NO⁺ (slow, k₃) iv. NH₃NO⁺ -> H₂O + H⁺ + N₂ (fast, k₄)
Derive the rate law.

Solution
From the rate-determining step, you have

      d[NH₃NO⁺]
     ------------- = k₃ [NO⁺] [NH₃]   - - - - (4)
        dt

Neither NO⁺ nor NH₃ is a reactant. You must express their concentrations in terms of [NH₄⁺] and [HNO₂] from elementary processes i and ii.

From i, [NO⁺] = K₁ [HNO₂] [H⁺] / [H₂O] - - (5)

From ii, [NH₃] = K₂ [NH₄⁺] / [H⁺] - - - - (6)

Substituting (6) and (5) in (4) gives,

                  [HNO₂] [NH₄⁺]
 Rate = k₃ K₁ K₂ -----------------
                    [H₂O]

      = k [HNO₂] [NH₄⁺]

where k = k₃ K₁ K₂ / [H₂O] is the overall rate constant.

Confidence Building Questions

If the reaction
2 A + B₂ = 2 AB follows the mechanism, i. A + B₂ = AB + B (slow)
ii. A + B = AB (fast) What is the order with respect to [B₂]?

Skill -
Figure out the order from a given mechanism.
What is the overall order?
Kinetic studies of the reaction A₂ + B₂ -> 2 AB suggested the mechanism as follows, i. A₂ = 2 A (fast equilibrium)
ii. A + B₂ -> AB + B (slow)
iii. B + A₂ -> AB + A (fast) What is the order of the reaction with respect to [A₂]?

Discussion -
If you got rate = k [A₂]^(1/2) [B₂], congratulations.
If the mechanism for the reaction 2 NO + O₂ -> 2 NO₂ is i. NO + NO = N₂O₂ (fast equilibrium)
ii. N₂O₂ + O₂ -> 2 NO₂ (slow) What is the power of [NO] in the differential rate law?

Skill -
Figure out the order from a given mechanism.
You should get: rate = k [NO]² [O₂]

If the reaction mechanism consists of these elementary processes,
i. A = 2 B (fast, equilibrium) ii. B + 2 C -> E (slow) iii. E -> F (fast)
Choose the correct differential rate law for the reaction
A + 4 C -> 2 F
(a) (1/2)(d[F]/dt) = k[A] [C]⁴ (b) rate = k[A] [C]² (c) -d[A]/dt = k[A]^(1/2) [C] (d) d[F]/dt = k[A] [C] (e) (d[F]/dt) = k[A]^(1/2)[C]²

The method -
The rate determining step is ii. Express [B] in terms of [A] from i.