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A first-order reaction is one where the rate depends on the concentration of the species to the first power. For a general unimolecular reaction,

A -> products
the reaction rate expression for a first order reaction is
rate = k[A]0 = k[A]
The integrated rate law for a 1st order reaction can be easily shown to be
ln([A]0) - ln([A]) = kt

One interesting thing about first order reactions is that the time required for 1/2 of the beginning reagent to react is independant of the conentration of the reagent. This can easily be seen by combining the two logs in the above integrated rate expression

ln([A]0/[A]) = kt
If 1/2 of A reacts, the ratio inside the log is always just equal to ln(2) no matter what the initial concentration, and the time needed for 1/2 of the reagent to react is just t1/2 = ln(2)/k. This is known as the half life of a species.

Example: SO2Cl2 decomposes to sulfur dioxide and chlorine gas. The reaction is first order: if it takes 13.7 hours for a 0.250 M solution of SO2Cl2 to decompose into a 0.117M solution, what is the rate constant for the reaction and what is the half life of SO2Cl2 decomposition?

Solution: Use the integrated rate law above to determine the rate constant k.

ln([A]0) - ln([A]) = kt
ln(0.250) - ln(0.117) = k*13.7 hours
k = 0.0554 1/hour
Once k is known, use the relationship t1/2 = ln(2)/k to determine the half life
t1/2 = ln(2)/k
t1/2 = 0.693/0.0554/hour
t1/2 = 12.5 hours


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