Radioactive Decay: Calculate % Decayed, Atoms Remaining, Half-Life

In summary: For part (c), the half-life is approximately 40.71 days.In summary, for a radioactive sample with 3.25 x 10^18 atoms of a nuclide decaying at a rate of 3.4 x 10^13 disintegrations per 26 min, after 159 days, approximately 8.46% of the nuclide will have decayed and there will be 2.976 x 10^18 atoms remaining. The half-life of the nuclide is approximately 40.71 days.
  • #1
JstCallMeTrix
1
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Homework Statement



A radioactive sample contains 3.25 1018 atoms of a nuclide that decays at a rate of 3.4 1013 disintegrations per 26 min.
(a) What percentage of the nuclide will have decayed after 159 d?
%

(b) How many atoms of the nuclide will remain in the sample?
atoms

(c) What is the half-life of the nuclide?
days

Homework Equations



The equations I used were
t1/2 = ln2/k to find the half life
and N'=Ne-kt

The Attempt at a Solution



I used the equations above and solved for part b, the number of atoms and found this to be 2.976E18 --which IS correct. I converted the 159 days into minutes, and found k by taking the rate (now in days) and dividing by the original number (N) and getting 3.836E-7 for k

I then plugged this into N*e-kt with t now in minutes and got my answer for part B. (2.976E18 atoms).

So my problem is with part 1 and 3...I thought it would be pretty straight forward, subtracting hte remaining atoms from the original to get the amount that decayed. Then taking that amount, dividing it by the original to get hte percent decayed. I keep getting 8.43% for this...but it's incorrect.

Finally for C, I thought I would just convert everything back to days, then take ln2/k (in days now) to get the half life...but I guess something is wrong here too. Can someone please explain how to do this? Thanks!
 
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  • #2
Your methods are correct. You could be making errors in arithmetic. For what it's worth, I get 8.46% for part (a).
 

1. What is radioactive decay?

Radioactive decay is the process by which an unstable atom releases energy in the form of radiation in order to become more stable. This process is random and cannot be controlled or sped up.

2. How do you calculate the percentage of atoms that have decayed?

To calculate the percentage of atoms that have decayed, you would divide the number of atoms that have decayed by the total number of atoms in the sample and then multiply by 100. This will give you the percentage of atoms that have decayed.

3. How do you calculate the number of atoms remaining after a certain amount of time?

To calculate the number of atoms remaining after a certain amount of time, you would use the equation N = N0(1/2)t/h, where N is the number of atoms remaining, N0 is the initial number of atoms, t is the time elapsed, and h is the half-life of the isotope.

4. What is the half-life of an isotope?

The half-life of an isotope is the amount of time it takes for half of the atoms in a sample to decay. This value is constant and unique to each isotope.

5. How does radioactive decay affect the stability of an atom?

Radioactive decay can change the number of protons and neutrons in an atom, thus changing its atomic mass. This can make the atom more stable by bringing it closer to the most stable isotope of that element.

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