The integrated rate law allows chemists to predict the reactant concentration af

Question

The integrated rate law allows chemists to predict the reactant concentration after a certain amount of time, or the time it would take for a certain concentration to be reached.

The integrated rate law for a first-order reaction is:
[A]=[A]0ekt

Now say we are particularly interested in the time it would take for the concentration to become one-half of its initial value. Then we could substitute [A]02 for [A] and rearrange the equation to:
t1/2=0.693k  

This equation calculates the time required for the reactant concentration to drop to half its initial value. In other words, it calculates the half-life.

Half-life equation for first-order reactions:
t1/2=0.693k  
where t1/2 is the half-life in seconds (s), and k is the rate constant in inverse seconds (s1).

Question: A certain first-order reaction has a rate constant of 3.90×103 s1. How long will it take for the reactant concentration to drop to 18 of its initial value?

Explanation / Answer

Ao = initial

At = Ao /18

k = 1 / t * ln [ Ao /At ]

k = 1/t * ln (Ao / Ao /18)

3.90×103 = 1 / t x 2.89

t = 741 sec

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