Carbon-14 amount in an old tree

[dramadeur]

Carbon-14 is an unstable isotope of carbon, with a half-life of 5730 years. Suppose a tree sample is 8170 years old. What is the ratio of carbon-14 in the sample today, to the amount of carbon-14 when the tree was alive?
Suppose the amount of carbon-14 in a sample is observed to be 7 percent of the amount for a living plant. Then what is the age of the sample?




Alright, so if half-life is 5730, then full life is 11460 years

but the tree is just 8170 years old
so Carbon's atoms are still alive, right?

Does the problem ask me to find the ratio of atoms? Is it possible that some atoms got "diffused/fissioned"?
How do I know how many atoms ... got diffused... and into which elements
I'm reading blank with this problem, I don't get it at all...

 

[SteamKing]

In terms of radioisotopes, the term 'full life' is meaningless.

'Half life' is that amount of time it takes one-half of the original amount to decay. After a period of time equal to two half-lives, one-quarter of the original sample remains un-decayed.

This article explains in more detail:

http://en.wikipedia.org/wiki/Half-life

 

[dramadeur]

How do I know to which elements Carbon-14 will be diffused into?

 

[haruspex]

How do I know to which elements Carbon-14 will be diffused into?

Diffused into? Do you mean decay into? The ratio mentioned is between C14 and C12.
Carbon-14 decays into nitrogen-14 through beta decay (see http://en.wikipedia.org/wiki/Carbon-14), so the decay reduces the C14 without changing the C12.

 

[SteamKing]

How do I know to which elements Carbon-14 will be diffused into?

It doesn't matter. The problem is asking you to find out how much C14 is left after 8170 years in part 1.

 

[Basic_Physics]

An unstable isotope means that the specific carbon atoms will change into other elements by emitting radioactive rays. The nuclei of the carbon atoms are unstable and decay into more stable atoms. While the tree was still alive it absorbed carbon dioxide from the atmosphere and the amount of radioactivity emitted stayed constant, but once it died the carbon in it is not replaced anymore and the radioactivity will decrease - that is the amount of radioactive rays emitted decreases as the atoms decays the more stable forms. So the amount of carbon-14 in it decreases with time and after each half-life the amount of remaining carbon-14 in it is halved. This is an exponential decreasing quantity with time - the amount of remaining carbon 14

C = C_o e^-λt

where C_o is the intial amount of carbon-14 at time zero. λ is the decay constant and it is related to the half-life T by

λT = ln(2)

the natural logarithm of 2. So what the first question is asking you to find is the ratio of C/C_o

 

[dramadeur]

An unstable isotope means that the specific carbon atoms will change into other elements by emitting radioactive rays.

I'm confused, when does carbon-14 decay into nitrogen-14?
In this case, it decays into carbon-12? How do you know that?

The nuclei of the carbon atoms are unstable and decay into more stable atoms.

All of them are unstable? How does, its proton decays into a more stable atom? It stays the same, isn't it?
Maybe you mean number of neutrons? Because carbon-12 has 6.
So 2 neutrons from carbon-14 will ... disappear, right?

While the tree was still alive it absorbed carbon dioxide from the atmosphere and the amount of radioactivity emitted stayed constant

I still don't get this radioactivity emission? What does its value show? I take it, it has to do with decay...
so if there's no radioactivity ... then there's no decay, right?

but once it died the carbon in it is not replaced anymore and the radioactivity will decrease - that is the amount of radioactive rays emitted decreases as the atoms decays the more stable forms. So the amount of carbon-14 in it decreases with time and after each half-life the amount of remaining carbon-14 in it is halved. This is an exponential decreasing quantity with time - the amount of remaining carbon 14

C = C_o e^-λt

where C_o is the intial amount of carbon-14 at time zero. λ is the decay constant and it is related to the half-life T by

λT = ln(2)

the natural logarithm of 2. So what the first question is asking you to find is the ratio of C/C_o

So when radioactivity wasn't decreased, the atoms weren't decaying into the more stable forms?
And how do they decay into more stable forms?
>after each half-life
in this case is it 5700 years?

>the amount of remaining carbon-14
How do I calculate the amount of carbon-14? Is it the number of protons, neutrons, electrons summed up together... or?

 

[SteamKing]

You misunderstand C-14 dating.

There is a certain ratio of C-14 isotopes mixed in with C-12 isotopes in living things like trees. While the tree is alive, this ratio remains fixed because the tree is taking in CO2 during respiration to feed itself. When the tree dies, respiration ceases and no more CO2 is taken in by the tree. Over time, the C-14 isotopes which are contained in the tree's woody fibres start to decay, altering the ratio of C-14 to C-12. By measuring the ratio of C-14 to C-12 and knowing the half-life of C-14 decay, one can estimate how many years have elapsed since the tree died.

And yes, C-14 decays into N-14, which is a stable isotope.

 

[haruspex]

To add to SteamKing's explanation...
Atmospheric carbon has a higher proportion of C14 because it is created by radiation high in the atmosphere and diffuses down. It decays in the atmosphere just as fast, but the ratio remains higher because it is constantly replenished. Once trapped in organic matter, it is no longer replenished and only decays. (Note that if the tree lived a thousand years and died a thousand years ago, the ratio you get might depend on where your sample is taken from.)

 

[Basic_Physics]

One of the neutrons of a carbon-14 nucleus decays into a proton by emitting an electron (beta ray) and an anti-neutrino so that the nucleus now have one more proton - seven in total, but still 14 nucleons. This new nucleus is nitrogen-14, which is stable.

To say that an isotope is radioactive means that its nucleus is unstable and will eventually change (decay) into a more stable nucleus by emitting radioactive rays, alpha (particle), beta (particle) or gamma(photon). The instability is due to there being too many protons or too many neutrons (as apposed to the required amount to form a stable nucleus) or a nucleus in an excited state. It may go through several steps on its way to a stable nucleus. The time frame for these transitions vary. Some isotopes take longer to decay than others - they are less unstable. Not all of the unstable nuclei will decay at the same time, it is a random process, but over a certain period of time we can predict what percentage of all of the unstable nuclei will still be present in the sample.

The amount of decays of a sample of the material per second is called its activity. Another way of specifying the rate of decay of a sample is with its half-life. A shorter half-live indicates a sample with a higher activity.

How to calculate the amount of carbon-14 in a sample? From the radioactive decay law we can calculate the percentage that remains after a certain time period.