Question 43.3: How Many Nuclei Are Left? The isotope carbon-14, 6^14C, is r...
How Many Nuclei Are Left?
The isotope carbon-14, { }_6^{14} C, is radioactive and has a half-life of 5 730 years. If you start with a sample of 1 000 carbon-14 nuclei, how many nuclei will still be undecayed in 25 000 years?
Conceptualize The time interval of 25 000 years is much longer than the half-life, so only a small fraction of the originally undecayed nuclei will remain.
Categorize The text of the problem allows us to categorize this example as a substitution problem involving radioactive decay.
Analyze Divide the time interval by the half-life to determine the number of half-lives:
n=\frac{25 000 yr}{5 730 yr}=4.363Determine how many undecayed nuclei are left after this many half-lives using Equation 43.9:
N=N_0\left(\frac{1}{2}\right)^n=1 000\left(\frac{1}{2}\right)^{4.363}=49Finalize As we have mentioned, radioactive decay is a probabilistic process and accurate statistical predictions are possible only with a very large number of atoms. The original sample in this example contains only 1 000 nuclei, which is certainly not a very large number. Therefore, if you counted the number of undecayed nuclei remaining after 25 000 years, it might not be exactly 49.