How Do You Calculate the Age of Plant Material Using Radioactive Decay?

[Keris]

Homework Statement

This problem involves calculating the age of plant material from which 3 radioactive disintergrations per minute per gram of carbon are detected. During its lifetime, the plant took in carbon from the atmosphere. Of this carbon, for every 10¹² atoms of normal ¹⁴C, there was 1 atom of radioactive ¹⁴C. When the plant died, the ¹⁴C decayed so the proportion of ¹²C to ¹⁴C increased.

The half life of ¹⁴C is 5570 years. The mass of a ¹²C is 1.993 x 10^-23kg.

How many carbon atoms (¹²C and ¹⁴C) arein 1 gram of carbon?

Homework Equations

t (time in seconds) = ln2/λ

( λ is decay constant)

n = n₀ exp (- λt)

n₀ is the number of nuclei present at time

The Attempt at a Solution

I'm sorry, I know the question seems well explained, but I'm still confused on how to solve it. I've worked out that the decay constant of ¹⁴C is 3.9 x 10^-12s.
After that I'm not sure what to do to find out how many atoms there are in 1g of carbon... I'msorry fornot attempting much, but I'm so confused. Please may I have some guidance on what to do? Thank you.

 

[daveb]

If there is 1 mole of atoms in 12 grams of carbon, how many atoms are in 1 gram?

 

[Keris]

If there is 1 mole of atoms in 12 grams of carbon, how many atoms are in 1 gram?

1/12..or 0.083mols?
Ah, thank you, I didn't know we had to use moles.

 

[tms]

You don't have to use moles. You are given the number of decays per second; in other words, you are given $dn/dt$. You can then calculate the number of C14 atoms, and then their mass. The rest of the 1 g are C12.

 

[asteriatic]

Dear student,

Thank you for your question. It seems like you have already made some progress in solving the problem by finding the decay constant of 14C. To find the number of atoms in 1g of carbon, we can use the following equation:

N = m/M

Where N is the number of atoms, m is the mass of the carbon sample (1g in this case), and M is the molar mass of carbon. The molar mass of carbon is 12.01 g/mol, so we can calculate the number of atoms in 1g of carbon as:

N = 1g / 12.01 g/mol = 8.33 x 10^22 atoms

Next, we need to find the number of 14C atoms in 1g of carbon. We can use the information given in the problem to calculate this:

For every 1.012 x 10^12 atoms of normal 14C, there is 1 atom of radioactive 14C. So, the ratio of 14C atoms to total atoms is:

1 / (1.012 x 10^12) = 9.88 x 10^-13

This means that out of the 8.33 x 10^22 atoms in 1g of carbon, only 9.88 x 10^-13 of them are 14C atoms. We can use this information to calculate the number of 14C atoms in 1g of carbon as:

N14C = 8.33 x 10^22 x 9.88 x 10^-13 = 8.23 x 10^10 14C atoms

Now, we can use the equation for radioactive decay to find the age of the plant material. We know that the plant material has 3 radioactive disintegrations per minute per gram of carbon. This means that 3 out of the 8.23 x 10^10 14C atoms decay every minute. We can use this information to find the decay constant (λ) as:

3 = 8.23 x 10^10 x exp(-λ x 60 seconds)

Solving for λ, we get:

λ = 5.79 x 10^-12 s^-1

Finally, we can use the equation t = ln2 / λ to find the age of the plant material:

t = ln2 / (5.79 x