Further Properties of First-Order Reaction Kinetics
The integrated first-order rate law
is usually written in the form of the exponential decay equation
A different (but equivalent) way of considering first order kinetics is as follows: The exponential decay equation can be rewritten as:
where corresponds to a specific time period and is an integer corresponding to the number of time periods. At the end of each time period, the fraction of the reactant population remaining relative to the amount present at the start of the time period, will be:
Such that after time periods, the fraction of the original reactant population will be:
where: corresponds to the fraction of the reactant population that will break down in each time period. This equation indicates that the fraction of the total amount of reactant population that will break down in each time period is independent of the initial amount present. When the chosen time period corresponds to, the fraction of the population that will break down in each time period will be exactly ½ the amount present at the start of the time period (i.e. the time period corresponds to the half-life of the first-order reaction).
The average rate of the reaction for the nth time period is given by:
Therefore, the amount remaining at the end of each time period will be related to the average rate of that time period and the reactant population at the start of the time period by:
Since the fraction of the reactant population that will break down in each time period can be expressed as:
The amount of reactant that will break down in each time period can be related to the average rate over that time period by:
Such that the amount that remains at the end of each time period will be related to the amount present at the start of the time period according to:
This equation is a recursion allowing for the calculation of the amount present after any number of time periods, without need of the rate constant, provided that the average rate for each time period is known.
Read more about this topic: Rate Equation, First-order Reactions
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