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.

 

[Blueshift5]

Hello,

Your method for solving this problem appears to be correct. Half-life is a fundamental concept in radioactive decay, and your use of it in your calculations is valid. Multiplying by the number of half-lives is necessary because each half-life represents a 50% decrease in the original amount of tritium. Therefore, to reach a concentration of 0.1, which is 10% of the original amount, we need to find the number of half-lives that would result in a 90% decrease (since 100% - 90% = 10%). This is why you multiplied by the natural logarithm of 0.1 divided by the natural logarithm of 0.5, which is equivalent to the number of half-lives.

In general, it is always a good idea to double-check your work and make sure your calculations make sense. In this case, your initial answer of 4404 years seemed unreasonable because it would require an extremely large number of half-lives, which is not realistic. By adjusting your approach and multiplying by the number of half-lives instead of raising to the power of, you were able to arrive at a more reasonable and accurate answer.

I hope this explanation helps to clarify your understanding of the problem. Keep up the good work!