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For a first-order reaction, how long will it take for the concentration of reactant to fall to one-eighth its original value? Express your answer in terms of the half-life $\left(t_{1}\right)$ and in terms of the rate constant $k$.

$$\begin{array}{l}t_{1 / 8}=\frac{\ln 8}{k} \\t_{1 / 8}=3 t_{1 / 2}\end{array}$$

Chemistry 102

Chapter 6

Chemical Kinetics

Kinetics

University of Central Florida

Drexel University

University of Kentucky

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In chemistry, kinetics is the study of the rates of chemical reactions. The rate of a reaction is the change in concentration of a reactant over time. The rate of reaction is dependent on the concentration of the reactants, temperature, and the activation energy of the reaction.

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Well, so today we're going to be talking about a situation where we have a first order reaction and we want to know when there will only be one days of the reactant left, while one thing that we should probably use is the half life. So if you are familiar with it, the half life is how long it takes for half of it to decay. So we'll go from one to a half, and that will be one half life. But now we're going to decay, and we're gonna have half of the half, so that would be 1/4. So the first half life we go from 1 to 1 half, so only 50% off the reactant will remain. Then we're gonna go toe one force that's another half life, Then half of 1/4 would be an ace, and we would have another half life. So as we can see the time it takes to get Teoh, 1/8 of the reactant left is three half life. But say we don't know the half life of this reaction. So how else could we find the time it takes to get to 1/8 of the reactant left. Well, hopefully you should remember that half life for first over order action is a natural log of two over the rate constant. So if time it takes to get to 1/8 of the act left is equal to three half life stun. It would just be three natural log of to over K. Or we could rewrite this as the natural log of eight because that's two to the third Power is eight natural log of eight over K. And so we see we have several different ways of finding out that time it takes for it to decay toe 1/8.

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