Question

Define the energy of activation and how to determine it by graphical method.

Answer 1

going on to the Activation Energy, let's look some more at Integrated Rate Laws. Specifically, the use of first order reactions to calculate Half Lives.

Let's review before going on...

Integrated forms of rate laws:

In order to understand how the concentrations of the species in a chemical reaction change with time it is necessary to integrate the rate law (which is given as the time-derivative of one of the concentrations) to find out how the concentrations change over time.

1. First Order Reactions

Suppose we have a first order reaction of the form, B + . . . . → products. We can write the rate expression as rate = -d[B]/dt and the rate law as rate = k[B]b . Set the two equal to each other and integrate it as follows:



The first order rate law is a very important rate law, radioactive decay and many chemical reactions follow this rate law and some of the language of kinetics comes from this law. The final Equation in the series above iis called an "exponential decay." This form appears in many places in nature. One of its consequences is that it gives rise to a concept called "half-life."

Half-life

The half-life, usually symbolized by t1/2, is the time required for [B] to drop from its initial value [B]0 to [B]0/2. For Example, if the initial concentration of a reactant A is 0.100 mole L-1, the half-life is the time at which [A] = 0.0500 mole L-1. In general, using the integrated form of the first order rate law we find that:



Taking the logarithm of both sides gives:



The half-life of a reaction depends on the reaction order.

First order reaction: For a first order reaction the half-life depends only on the rate constant: 

Thus, the half-life of a first order reaction remains constant throughout the reaction, even though the concentration of the reactant is decreasing.

Second order reaction: For a second order reaction (of the form: rate=k[A]2) the half-life depends on the inverse of the initial concentration of reactant A:

Since the concentration of A is decreasing throughout the reaction, the half-life increases as the reaction progresses. That is, it takes less time for the concentration to drop from 1M to 0.5M than it does for the drop from 0.5 M to 0.25 M.