The half-life for a second order reaction

The half-life is proportional to the reciprocal of the concentration, i.e. con-centration, and so the larger the concentration the smaller is the half-life. The 

Question. Show that the following reaction follows second order kinetics, and find the rate constant and first half-life. 

The graph of 1/[A] versus t is linear reaction is second order. 

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Strategy (a) The relationship between the concentrations of a reactant at different times is given by the integrated rate law. Because this is a second-order reaction, we use Equation (13.7). (b) We are asked to calculate the half-life. The half-life for a second-order reaction is given by Equation (13.8). 

For a second-order reaction, the half-life formula is a little different because the rate equation is different. For a second-order reaction (A -l- B) where the initial concentration of A is the same as B (i.e., [A]q = [B]q), the time required for half the starting materials to react is inversely proportional to the initial concentration. The half-life for a second-order reaction is... 

The half-life of a reactant is the time required for half of that reactant to be converted into products. For a first order reaction, the half-life is independent of concentration so that the same time is required to consume half of any starting amount or concentration of the reactant. On the other hand, the half-life of a second-order reaction does depend on the starting amount of the reactant. 

What is the equation for the half-life of a second-order reaction ... 

We can obtain an expression for the half-life of a second-order reaction by substituting [A]f = [A]0/2 and t = ti/2 into the integrated rate law ... 

In contrast with a first-order reaction, the time required for the concentration of A to drop to one-half of its initial value in a second-order reaction depends on both the rate constant and the initial concentration. Thus, the value of h/2 increases as the reaction proceeds because the value of [A]0 at the beginning of each successive half-life is smaller by a factor of 2. Consequently, each half-life for a second-order reaction is twice as long as the preceding one (Figure 12.8). 

Whereas the half-life for a first-order reaction is independent of reactant concentration, that for a second-order reaction is not. If one inserts ajl for (ia - y) in Eq. (2.26), one obtains... 

The half-life of a second-order reaction is inversely proportional to [A]0. For type II second-order reactions, the rate can be expressed as ... 

We may observe that the half-life of a second-order reaction differs from that of a first-order reaction [Eq. (II.4.6)] in that the of the former depends on the initial concentration. As we shall see later, this provides a simple experimental test for distinguishing order. 

Unlike t for the first-order reactions, the half-life of the second-order reaction is dependent on the initial concentration of reactants. It is not possible to derive a simple expression for the half-life of a second-order reaction with unequal initial concentrations. 

The half-life for second-order and other reactions depends on reactant concentrations and therefore changes as the reaction progresses. We obtained Equation 14.15 for the half-life for a first-order reaction by substituting [ A]o for [A], and fi/2 for... 

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