Conceptual half-life Question

Question

So I have this word problem and I’m a bit confused about it. I have the answer and explanation but I still don’t understand:

The half-life of carbon-14 is approximately 5730 years, while the half-life of carbon-12 is essentially infinite. If the ratio of carbon-14 to carbon-12 in a certain sample is 25% less than the normal ratio in nature, how old is the sample?

    A. Less than 5730 years

    B. Approximately 5730 years

    C. Significantly greater than 5730 years, but less than 11460 years

    D. Approximately 11460 years

Correct Answer: A

Explanation:

Because the half-life of carbon-12 is essentially infinite, a 25 percent decrease in the ratio of carbon-14 to carbon-12 means the same as a 25 percent decrease in the amount of carbon-14. If less than half of the carbon-14 has deteriorated, then less than one half-life has elapsed. Therefore, the sample is less than 5730 years old. Be careful with the wording here—the question states that the ratio is 25% less than the ratio in nature, not 25% of the ratio in nature, which would correspond to choice (D).

How is the ratio of the carbon isotopes relevant to half-lives? What is the purpose of saying the half-life of an isotope is infinite? What is meant by “the normal ratio in nature”? Just based on the answer, it seems like the question said “25% of a carbon sample decayed, how old is this sample?” Obviously it’s younger than 5,730 years (the time for its first half life) because if only 25% decayed that means half a half-life has passed. I don’t see how isotopes and ratios would change this problem at all.

Edit: Every answer here was very informative and helpful. It was tough to pick a best answer. I picked Cosma’s because it clicked with me.

If you’re reading this and want to understand radiocarbon dating, look at every answer because there is some useful information in every answer

Answer 1

How is the ratio of the carbon isotopes relevant to half-lives? What is the purpose of saying the half-life of an isotope is infinite? What is meant by “the normal ratio in nature”?

You might read up on radiocarbon dating. Cosmic rays constantly create C-14 in the atmosphere, with then decays, and so there is an equilibrium value of C-14/C-12 in nature, the "normal ratio". When living things capture carbon, it stops equilibrating with atmospheric CO2, so the C-14 decays, while the C-12 stays put ("infinite half-life"), crucially providing a normalization for the amount of carbon involved: you can only measure ratios in a sample; you can't monitor initial amounts and wait for millennia to monitor their decay.

So, indeed, a decrease in the ratio from the natural value is tantamount to a decrease in the initial amount of C-14 in the sample when it stopped photosynthesizing and breathing. So you read it right that

“25% of a carbon sample decayed, how old is this sample?” Obviously it’s younger than 5,730 years (the time for its first half life) because if only 25% decayed that means half a half-life has passed.

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Answer 2

It does indeed look like the words are dressing up the fact '25% of the C-14 decayed', for which the answer is obvious. But there is a twist.

In a real case you don't know how much C-14 was there originally. You've got a sample of something organic: bone or wood or suchlike dug out of the ground somewhere. You can measure the amount of C-14 it contains now, after years/centuries of decay. But to determine how long that's been, you would need to know how much C-14 was there to begin with.

So you count the C-12 as well and use that to normalise. A fresh sample has 1.25 parts in $10^{12}$ of C-14 (that's the 'normal ratio' bit). As time goes by that fraction will fall: after 5730 years it will be 0.625 in $10^{12}$ - the $10^{12}$ stays constant, that's the 'infinite half life' bit. In your question it will have fallen to 0.94 in $10^{12}$. That's the value of the actual measurement that was (supposedly) made.

Answer 3

Defining half-time
I think a part of the question is answered by rigorously defining half-life: if we have $N$ atoms of a certain isotope, which decays itno a different isotope (e.g., C-14 decaying into C-12, then the quantity of these atomes changes with time as $$ N = N_0\left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}}, $$ where $t_{1/2}$ is the half-time. In other words, after $t=t_{1/2}$ years the quantity of the isotope is halved (reduced by a factor of 2).

Infinite half time
If an isotope doesn't decay (i.e., if it is stable=non-radioactive), we say that its half-time is infinite.

Carbon dating
Carbon dating is mainly used for fossils. The basic fact behind it is that fraction of C-14 in living organisms is higher than in non-live objects. After the organism dies, this ratio is not maintained anymore and the radioactive deacy leads to its decrease towards the ration in non-living organisms. Thus, the normal ratio in nature in the question cited in the OP is supposed to mean the normal ratio in living organisms.

Answer 4

The ratio of carbon isotopes is used to determine age because we have a pretty good idea what the initial ratio was. Without using the initial ratio how could we know how much carbon-14 was initially present? C-14 decays to nitrogen-14 which, being a gas I expect doesn't stick around particularly in a porous organic sample, so comparing the ratio of C-14 to N-14 isn't feasible.

Rocks can sometimes be dated by looking at the ratio of potassium-40 and argon-40 to which it decays. In that case, once the rock has recrystallized the gaseous argon can not escape and begins accumulating in the rock. This makes it possible in this case to calculate the age by comparing the ratio of the parent and the daughter nuclei.