Rates of Reaction Problem - Verification

[vertciel]

Hello everyone:

For the following problem, I arrived at the correct answer but am unsure about the method used. Could someone please check my work to see if it is legitimate or if it is a fluke?

Thank you.

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1. The age of wine can be determined by measuring the trace amount of radioactive tritium, ³H, present in a sample. Tritium gradually diminishes by a first-order radioactive decay with a half-life of 12.5 years. If a bottle of wine is found to have a tritium concentration that is 0.100 that of freshly bottled wine, what is the age of the wine?

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My Work:

Since it takes 12.5 years for the original sample to reach 50%, I would need to find the number of half-lives needed to reach 10%.

$$0.5^x = 0.1$$

$$x ln 0.5 = ln 0.1$$

$$x = \frac{ln 0.1}{ln 0.5}$$

Originally, I raised 12.5 to the power of
$$\frac{ln 0.1}{ln 0.5}$$, but this was incorrect since 4404 years would be unreasonable.

So instead:

$$12.5 x \frac{ln 0.1}{ln 0.5} = 41.5$$ years, which is the right answer.

Also, if I am indeed right, could someone please explain why I would multiply by the the number of half-lives?

 

[Ygggdrasil]

x represents the number of half lives elapsed. So, if three half lives have elapsed, 1/8 of the original sample will remain, and (1/2)³ = 1/8. Three half lives corresponds to 12.5 x 3 = 37.5 years, not 12.5³ years as the latter calculation makes no sense.