Proportion of atmospheric Carbon 14

[poiuy]

The half-life of Carbon 14 is 5730 years and the activity of atmospheric carbon due to Carbon 14 is 255Bq/Kg. What proportion of atmospheric carbon atoms are Carbon 14.

Attempted solution

Decay constant for C14 = ln(2)/t(1/2) = ln(2)/5730 = 1.21x10^-4

Current activity A = A(0)exp(-1.21*10^-4 *t )
A = 255
t = age of universe = 10*10^9 years


After this I start getting silly numbers and it just doesn't work, presumably A/A(0) would give the proportion of the current atmosphere which is C14, but then again this doesn't seem to work and I am all out of ideas.

All ideas appreciated (I don't really want the answer but if somebody could point me in the right direction that would be great)

 

[denverdoc]

This might be better posted on the chemistry, other sciences, forum for homework help.

My first blush impression is that you need to be using the same units, in one case you're given the measured breakdown per second, and comparing it to a rate constant given in years. I don't believe that the age of the universe enters it. Instead, you have a number of events/sec, from the half life should be able to predict number of events on normalized basis, the fraction should give some idea about the percentage.

In other words say we have 1,000,000 carbon 14 atoms per kg.
In 5730 years, we would expect 500,000 to decay. In a second we would expect how many to decay? We know we have 255. That proportion of actual/expected is proportional to a million. What I don't understand is if its a Kg of atmosphere or Kg of carbon 14 that's being cited.

,

 

[m2003]

It seems like you are on the right track with your attempted solution, but there are a few things to consider. First, the activity of atmospheric carbon due to Carbon 14 (255 Bq/kg) is not the same as the proportion of atmospheric carbon atoms that are Carbon 14. Activity is a measure of the rate of decay of a substance, while proportion is a measure of the amount of a substance relative to the total amount of the substance present.

To find the proportion of atmospheric carbon atoms that are Carbon 14, we need to know the total amount of carbon in the atmosphere and the amount of Carbon 14 specifically. This can be estimated using the atomic mass of Carbon 14 (14 amu) and the average atomic mass of carbon in the atmosphere (12.01 amu). Using these values, we can calculate the number of Carbon 14 atoms in 1 kg of carbon in the atmosphere.

Number of Carbon 14 atoms in 1 kg of carbon = (255 Bq/kg) * (1 Ci/2.2*10^10 Bq) * (1 mole/6.02*10^23 atoms) * (14 amu/12.01 amu) = 1.18 * 10^13 atoms

Next, we need to find the total number of carbon atoms in 1 kg of carbon in the atmosphere. This can be estimated using the mass of the atmosphere (5.15*10^18 kg) and the percentage of carbon in the atmosphere (0.04%). This gives us a total of 2.06*10^15 carbon atoms in 1 kg of carbon in the atmosphere.

Therefore, the proportion of atmospheric carbon atoms that are Carbon 14 is: (1.18 * 10^13 atoms)/(2.06*10^15 atoms) = 5.73 * 10^-3 or 0.573%. This means that about 0.573% of the carbon atoms in the atmosphere are Carbon 14.

I hope this helps guide you in the right direction. Keep in mind that these calculations are based on estimates and assumptions, so the actual proportion may vary slightly. It's also important to consider that the proportion of atmospheric Carbon 14 can change over time due to factors such as human activities and natural processes.