Activity of 1 g of carbon if 1 in 10^12 atoms are carbon-14

[WhiteWolf98]

Homework Statement

Carbon-14 decays by β emission and has a half-life of 5570 years.

1. What is the decay constant of carbon-14?
2. What is the activity of 1 g of carbon if 1 in 10¹² atoms are carbon-14?
3. After what time will the activity per gram have fallen to 3 Bq?

Homework Equations

λt_½ = ln(2)
A = A₀e^-λt
A = λN

The Attempt at a Solution

Question 1[/B]
λt_½ = ln(2)
λ= ln(2)/5570
= 1.24 * 10^-4 year^-1

Question 2
Mr of carbon-14 is 14 g mol^-1
1 mol = 6.022*10²³ atoms
Atoms in 1 g = (6.022*10²³)/14
= 4.30*10²² atoms

Don't know what to do from here. Any help would be appreciated.

Thanks

 

[gneill]

Hi WhiteWolf98, Welcome to Physics Forums!

Question 2
Mr of carbon-14 is 14 g mol^-1
1 mol = 6.022*10²³ atoms
Atoms in 1 g = (6.022*10²³)/14
= 4.30*10²² atoms

Remember that not all the atoms are C14. What's the main isotope and how big a contribution does C14 make to the given quantity of the carbon?

 

[WhiteWolf98]

Main isotope of carbon is carbon-12. Do I just divide the number of atoms by 1*10^-12?

 

[gneill]

Sure. But first calculate the number of atoms using the overwhelmingly more abundant C12 mass.

If you ever come across a case where the ratios of isotope abundance are closer to unity you'll have to use a more subtle mathematical approach, but in this case this approximation is all but perfect.

 

[WhiteWolf98]

Sure. But first calculate the number of atoms using the overwhelmingly more abundant C12 mass.

If you ever come across a case where the ratios of isotope abundance are closer to unity you'll have to use a more subtle mathematical approach, but in this case this approximation is all but perfect.

I don't understand what carbon-12 has to do with this question; the question is asking about carbon-14. Wouldn't introducing more isotopes make it more complicated than it already is?

 

[gneill]

The question asks about 1 gram of carbon, and says that only 1 in 10¹² atoms of it are C14. So what is the rest of it made up of? Will C14 make up the whole of the 1 gram mass?

 

[WhiteWolf98]

The question asks about 1 gram of carbon, and says that only 1 in 10¹² atoms of it are C14. So what is the rest of it made up of? Will C14 make up the whole of the 1 gram mass?

It seems it does. Well, I have no idea anymore; am I supposed to be working out the activity of carbon-12 or carbon-14? There lies also the existence of carbon-13, so if you consider that too, then this question is unanswerable. I will have to talk with the mastermind who created the question!

 

[gneill]

It seems it does. Well, I have no idea anymore; am I supposed to be working out the activity of carbon-12 or carbon-14? There lies also the existence of carbon-13, so if you consider that too, then this question is unanswerable. I will have to talk with the mastermind who created the question!

1 in 10¹² is an insignificant contribution to the overall mass. C13 makes a much larger contribution than C14 percentage wise, but it is still small compared to C12 which is the overwhelmingly largest contributor to the makeup of a given sample (unless it's specially prepared to enhance some particular isotope).

For this question you can assume that the only measurable contribution to the mass of the 1 gram is C12. Work out the number of atoms of C12 that would represent. You won't introduce any significant error by assuming that 1 x 10^-12 of them are actually the C14 that you're interested in.

 

[WhiteWolf98]

1 in 10¹² is an insignificant contribution to the overall mass. C13 makes a much larger contribution than C14 percentage wise, but it is still small compared to C12 which is the overwhelmingly largest contributor to the makeup of a given sample (unless it's specially prepared to enhance some particular isotope).

For this question you can assume that the only measurable contribution to the mass of the 1 gram is C12. Work out the number of atoms of C12 that would represent. You won't introduce any significant error by assuming that 1 x 10^-12 of them are actually the C14 that you're interested in.

In other words, the number of carbon-14 atoms is negligible? And all the information given about carbon-14 serves as a red herring?

 

[gneill]

In other words, the number of carbon-14 atoms is negligible? And all the information given about carbon-14 serves as a red herring?

It is negligible in that it makes up an insignificant portion of the mass of 1 gram of carbon. No scale could detect its contribution. So when measuring out the 1 gram sample, you can consider it to be pure C12. In order to determine the number of C14 atoms that came along for the ride, you look at the abundance ratio. One in 10¹² of the atoms in the sample will be C14 rather than C12.

 

[WhiteWolf98]

It is negligible in that it makes up an insignificant portion of the mass of 1 gram of carbon. No scale could detect its contribution. So when measuring out the 1 gram sample, you can consider it to be pure C12. In order to determine the number of C14 atoms that came along for the ride, you look at the abundance ratio. One in 10¹² of the atoms in the sample will be C14 rather than C12.

I appreciate the help, thanks.

 

[gneill]

You're very welcome.