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I've spent over an hour researching Carbon-14 decay for a Calculus problem, but I have one main problem when solving them: how do you solve for the k value (decay constant)?

Here is the problem

The method of carbon dating makes use of the fact that all living organisms contain two isotopes of carbon, carbon-12, denoted 12C (a stable isotope), and carbon-14, denoted 14C (a radioactive isotope). The ratio of the amount of 14C to the amount of 12C is essentially constant (approximately 1/10,000). When an organism dies, the amount of 12C present remains unchanged, but the 14C decays at a rate proportional to the amount present with a half-life of approximately 5700 years. This change in the amount of 14C relative to the amount of 12C makes it possible to estimate the time at which the organism lived.

A fossil found in an archaeological dig was found to contain 20% of the original amount of 14C. What is the approximate age of the fossil?

So, I'm not completely lost. I'm aware that the equation I need is:

$$\frac{[\ln\frac{N}{No}]}{k} * t_{1/2}$$

And I find many websites that insert -.693 for k when referencing Carbon-14 problems, but I have no idea why they use that value. I assume that the "approximately 1/10,000" part of the problem is significant, but I don't understand why.

Can someone please help me with understanding how to calculate this k value that some places have as -.693 and some sites have as .0001..., both referencing Carbon-14 problems?

Thanks!

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1 Answer 1

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I do not get the $-0.693$ value, but perhaps my answer will help anyway.

If we assume Carbon-14 decays continuously, then $$ C(t) = C_0e^{-kt}, $$ where $C_0$ is the initial size of the sample. We don't know this value, but we don't need it. Since it takes 5,700 years for a sample to decay to half its size, we know $$ \frac{1}{2} C_0 = C_0e^{-5700k}, $$ which means $$ \frac{1}{2} = e^{-5700k}, $$ so the value of $C_0$ is irrelevant.

Now, take the logarithm of both sides to get $$ -0.693 = -5700k, $$ from which we can derive $$ k \approx 1.22 \cdot 10^{-4}. $$

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Austin, thanks for the answer. Plugging that k value into my equation yields an incredibly large age (-9.17 * 10^7). Am I missing something here? Should I be using the first equation in your answer? –  David Dec 3 '12 at 2:38
    
So either the answer is that ridiculously big number (9.17e7) or 30,476 years, being calculated with the equation I provided and the first equation in your answer, respectively. I'm leaning toward the latter. –  David Dec 3 '12 at 2:41
    
Like you said, the equation I'm using was for a different type of problem. Thanks for your help! You made these kinds of problems incredibly simple! –  David Dec 3 '12 at 2:43
    
please check your Math. It seems you calculated the natural log of .5 to be -.301, which is incorrect. It seems to be -.693 –  David Dec 3 '12 at 2:47
    
@David You are correct. It appears I mistakenly hit the "log base 10" button on my calculator. Now the mystery is solved! –  Austin Mohr Dec 3 '12 at 2:58

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