Although the rate law of the SN1 reaction is often regarded as being first order in alkyl halide and zero order in nucleophile, this is a simplification that holds true only under certain conditions. While it, too, is an approximation, the rate law derived from the steady state approximation (SSA) provides more insight into the kinetic behavior of the SN1 reaction. Consider the following reaction scheme for the mechanism shown above: Though a relatively stable tertiary
carbocation,
tert-butyl cation is a high-energy species that is present only at very low concentration and cannot be directly observed under normal conditions. Thus, the SSA can be applied to this species: (1) Steady state assumption: \frac{d[\text{tBu}^+]}{dt} = 0 = k_1[\text{tBuBr}] - k_{-1}[\text{tBu}^+][\text{Br}^-] - k_2[\text{tBu}^+][\text{H}_2\text{O}] (2) Concentration of t-butyl cation, based on steady state assumption: [\text{tBu}^+] = \frac{k_1[\text{tBuBr}]}{k_{-1}[\text{Br}^-] + k_2[\text{H}_2\text{O}]} (3) Overall reaction rate, assuming rapid final step: \frac{d[\text{tBuOH}]}{dt} = k_2[\text{tBu}^+][\text{H}_2\text{O}] (4) Steady state rate law, by plugging (2) into (3): \frac{d[\text{tBuOH}]}{dt} = \frac{k_1 k_2 [\text{tBuBr}][\text{H}_2\text{O}]}{k_{-1}[\text{Br}^-] + k_2[\text{H}_2\text{O}]} Under normal synthetic conditions, the entering nucleophile is more nucleophilic than the leaving group and is present in excess. Moreover, kinetic experiments are often conducted under initial rate conditions (5 to 10% conversion) and without the addition of bromide, so [\text{Br}^-] is negligible. For these reasons, k_{-1}[\text{Br}^-] \ll k_2[\text{H}_2\text{O}] often holds. Under these conditions, the SSA rate law reduces to: \text{rate} = \frac{d[\text{tBuOH}]}{dt} = \frac{k_1 k_2 [\text{tBuBr}][\text{H}_2\text{O}]}{k_2[\text{H}_2\text{O}]} = k_1[\text{tBuBr}] the simple first-order rate law described in introductory textbooks. Under these conditions, the concentration of the nucleophile does not affect the rate of the reaction, and changing the nucleophile (e.g. from H2O to MeOH) does not affect the reaction rate, though the product is, of course, different. In this regime, the first step (ionization of the alkyl bromide) is slow, rate-determining, and irreversible, while the second step (nucleophilic addition) is fast and kinetically invisible. However, under certain conditions, non-first-order reaction kinetics can be observed. In particular, when a large concentration of bromide is present while the concentration of water is limited, the reverse of the first step becomes important kinetically. As the SSA rate law indicates, under these conditions there is a fractional (between zeroth and first order) dependence on [H2O], while there is a negative fractional order dependence on [Br–]. Thus, SN1 reactions are often observed to slow down when an exogenous source of the leaving group (in this case, bromide) is added to the reaction mixture. This is known as the
common ion effect and the observation of this effect is evidence for an SN1 mechanism (although the absence of a common ion effect does not rule it out). == Scope ==