Energy Profile and Rate Law of SN1 Reactions Steady State Approximation

1 Energy Profile and Rate Law of SN1 Reactions Steady State Approximation 

However, Equation 2.5 cannot be correlated with experimentally determined data. The reason for this is that the concentration of the carbenium ion intermediate appears in it. This concentration is extremely small during the entire reaction and consequently cannot be measured. However, it cannot be set equal to zero, either. In that case, Equation 2.5 would mean that the rate of product formation is also equal to zero and thus that the reaction does not take place at all. Accordingly, we must have a better approximation, one that is based on the following consideration  

The concentration of an intermediate in a multistep reaction is always very low when it reacts faster than it is produced. If this concentration is set equal to zero in the derivation of the rate law, unreasonable results may be obtained. In such a case, one resorts to a different approximation. The change of the concentration of this intermediate as afunction of time is set equal to zero. This is equivalent to saying that the concentration of the intermediate during the reaction takes a value slightly different from zero. This value can be considered to be invariant with time, i.e., steady. Consequently this approximation is called the steady state approximation. 

Equipped with this principle, let us now continue the derivation of the rate law for SN reactions. The approximation [carbenium ion] = 0 must be replaced by Equation 2.6. Let us now set the left-hand side of Equation 2.6, the change of the carbenium ion concentration with time, equal to the difference between the rate of formation of the carbenium ion and its consumption. Because the formation and consumption of the carbenium ion are elementary reactions, Equation 2.7 can be set up straightforwardly. Now we set the right-hand sides of Equations 2.6 and 2.7 equal and solve for the concentration of the carbenium ion to get Equation 2.8. With this equation, it is possible to rewrite the previously unusable Equation 2.5 as Equation 2.9. The only concentration term that appears in Equation 2.9 is the concentration of the alkylating agent. In contrast to the carbenium ion concentration, it can be readily measured. 

From Equation 2.9 we can also derive the following the SN1 product is produced with the rate constant of the first reaction step. Thus the rate of product formation does not depend on the rate constant kattlck of the second reaction step. In a multistep reaction, a particular step 

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