In general, using the integrated form of the first order rate law we find that:. 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. 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. Let's try a simple problem: A first order reaction has a rate constant of 1. What is the half life of the reaction? What is the rate constant?
What percentage of N 2 O 5 will remain after one day? The Activation Energy E a - is the energy level that the reactant molecules must overcome before a reaction can occur. In order to calculate the activation energy we need an equation that relates the rate constant of a reaction with the temperature energy of the system.
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Earlier in the chapter, reactions were discussed in terms of effective collision frequency and molecule energy levels. In , a Swedish scientist named Svante Arrhenius proposed an equation that relates these concepts with the rate constant:. The Arrhenius equation allows us to calculate activation energies if the rate constant is known, or vice versa.
As well, it mathematically expresses the relationships we established earlier: as activation energy term E a increases, the rate constant k decreases and therefore the rate of reaction decreases.
We can graphically determine the activation energy by manipulating the Arrhenius equation to put it into the form of a straight line. Taking the natural logarithm of both sides gives us:.
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