A material has 32 atoms in all; 24 decayed and the rest undecayed. If the half-life of the radioactive material is 1000 years, what is the age of the material?
a. 1000 years
b. 2000 years
c. 3000 years
d. 4000 years
The number of radioactive atoms has decayed to 8/32 = 1/4 of the original value. That takes two half lives.
Why two half-lives?
Angie, you can also work the problem as I showed you on the 30 mCi problem you had earlier.
k = 0.693/1000 years = 6.93 x 10^-4
ln(No/N) = kt
[No is the number of atoms we started with; N = number of atoms today]
ln(32/8) = 6.93 x 10^-4*t
ln 4 = 6.93 x 10^-4*t
1.386/6.93 x 10^-4 = t = 2,000 years.
ok thanks
To solve this question, we need to understand the concept of half-life in radioactive decay.
The half-life of a radioactive material is the time it takes for half of the atoms in a sample to decay. In this case, the half-life is given as 1000 years.
We are told that there are 32 atoms in total, with 24 atoms already decayed. This means that 32 - 24 = 8 atoms are still undecayed.
Since the half-life is 1000 years, we know that after each 1000-year interval, half of the remaining undecayed atoms will decay, leaving the other half undecayed.
Using this information, we can determine the number of half-lives it took for the 8 atoms to decay.
1st half-life: 4 undecayed atoms remaining (32 atoms / 2 = 16 atoms / 2 = 8 atoms)
2nd half-life: 2 undecayed atoms remaining (4 atoms / 2 = 2 atoms)
3rd half-life: 1 undecayed atom remaining (2 atoms / 2 = 1 atom)
It took three half-lifes for the 8 atoms to decay.
Therefore, the age of the material is equal to the number of half-lives multiplied by the length of each half-life. In this case, 3 half-lives x 1000 years = 3000 years.
So, the correct answer is:
c. 3000 years