Steady-State Approximation (College Board AP® Chemistry)

Steady-State Approximation

Mechanisms with an Initial Fast Step

• When the first step of a reaction mechanism is the rate-limiting step, the rate law can be easily determined

• However, it is less straightforward to derive rate laws for a mechanism where an intermediate is a reactant in the rate-limiting step

  • This happens when the rate-limiting step follows an initial first step

  • For example, consider the gas-phase reaction between nitric oxide, NO and bromine gas, Br₂

2NO (g) + Br₂ (g) → 2NOBr (g)

• The experimental rate law for this reaction is given as:

Rate = k[NO]²[Br₂]

• A possible reaction mechanism consistent with this rate law could be an elementary reaction, involving three reactant molecules— termolecular reaction

NO (g) + NO (g) + Br₂ (g)  → 2NOBr (g)                        Rate = k[NO]²[Br₂]

• This however does not seem likely because such termolecular reactions are rare

• So let’s consider an alternative mechanism which does not involve termolecular reactions:

NO (g) + Br₂ (g) ⇋ NOBr₂ (g)                    (fast step)

NOBr₂ (g) + NO (g) → 2NOBr (g)                        (slow step)  

• From the proposed elementary reactions above, we see that

  • The first step is fast and reversible

  • The rate-determining step involves an intermediate, NOBr₂

• Based on the equation of the rate-limiting step, the rate law for the reaction will be:

Rate = k[NOBr₂][NO]

• This is not consistent with the experimental rate law and involves an intermediate

• To eliminate the intermediate and convert it into one of the reactants, we use the steady-state assumption or approximation

  • This assumption states that if the second step is the rate-limiting step, then the first step must be relatively fast and reversible

• This means that the rate at which the intermediate is formed— forward reaction— is equal to the rate at which it is consumed — backward reaction

  • Using the fast step elementary equation proposed for the reaction between nitric oxide and bromine:

NO (g) + Br₂ (g) ⇋ NOBr₂ (g)                    (fast step)

• The rate laws for the forward and backward reverse reactions are:

Rate_forward = k_f[NO][Br₂]

Rate_backward = k_b[NOBr]

• Since the Given that rate of forward and backward reactions are equal / at steady-state, we can write:

k_f[NO][Br₂] = k_b[NOBr₂]

• By rearranging the above expression, we can express the concentration of the NOBr₂ intermediate, NOBr₂ in terms of the concentration of the reactants:

[NOBr₂] = k_f/k_b[NO][Br₂]

• We then rewrite the rate law equation for the rate-limiting step by substituting the above expression for the intermediate:

Rate = k[NOBr₂][NO]

Rate = k(k_f/k_b)[NO][Br₂][[NO]

• All the k terms are constants, which means that they can be combined

This gives a rate law expression consistent with the experimental rate law:

Rate = k[NO]²[Br₂]

• In general, whenever a fast step precedes a slow one, we can solve for the concentration of an intermediate by assuming that an equilibrium is established in the fast step