Final answer:
The rate law for the reaction of tert-butyl bromide with azide ion, under the steady-state concentration of (CH3)3C+ intermediate, is expressed as rate = k[(CH3)3CBr], where k is the effective rate constant.
Step-by-step explanation:
The student asked for the rate law of the reaction of tert-butyl bromide with azide ion in aqueous solution, which proceeds through a proposed mechanism involving a steady-state concentration of (CH3)3C+(aq). The rate law can be determined by analyzing the proposed mechanism and applying the steady-state approximation, where the concentration of the intermediate (CH3)3C+ remains constant over the course of the reaction.
Given the steps in the mechanism, the rate law for the formation of (CH3)3CN3 is:
rate = k2[(CH3)3CBr][N3-]
This expression is based on the idea that the formation of the tert-butyl cation is fast and reversible, so the rate-determining step is the reaction between the cation and the azide ion. Experimentally, it is found that the reaction is first order with respect to tert-butyl bromide. Therefore, simplifying the expression considering the steady-state approximation, the rate law is:
rate = k[(CH3)3CBr]
where k is the effective rate constant, which encompasses both the formation of the intermediate cation and its reaction with azide ion. This approach does not require assumptions about the relative magnitudes of the rate constants.