so you have 67.0/83.5 = 0.8024 of the original U238
You need to solve
(1/2)^(t/(4.5*10^9)) = 0.8024
t = 1.42923*10^9 years
That's about 1/3 of a half-life
The half-life for the process 238U→206Pb is 4.5×109yr. A mineral sample contains 67.0 mg of 238U and 16.5 mg of 206Pb.
What is the age of the mineral? Express your answer using two significant figures.
5 answers
ln (No/N) = kt where
k = 0.693/4.5E9 = ? Solve for k.
The sample started out as x mg U 238 and 0 mg Pb 206. Pb 206 now is 16.5 mg. What amount of U 238 converted to Pb 206 during the sample's life? That is 16.5 mg x (238/206) = 19.1 mg U. Starting amount of U 238 was 19.1 mg + 67.0 mg = 86.1mg. Plug all of this into
ln (No/N) = kt
ln(86.1/67.0) = kt. k is from above but I didn't calculate it.
Solve for t in years and round as per the problem to two significant figures.
Post your work if you get stuck.
k = 0.693/4.5E9 = ? Solve for k.
The sample started out as x mg U 238 and 0 mg Pb 206. Pb 206 now is 16.5 mg. What amount of U 238 converted to Pb 206 during the sample's life? That is 16.5 mg x (238/206) = 19.1 mg U. Starting amount of U 238 was 19.1 mg + 67.0 mg = 86.1mg. Plug all of this into
ln (No/N) = kt
ln(86.1/67.0) = kt. k is from above but I didn't calculate it.
Solve for t in years and round as per the problem to two significant figures.
Post your work if you get stuck.
I get about 1.6E9 years as the age of the mineral. The 0.2E9 yrs difference between my answer and the one by oobleck is that I corrected for the mass of Pb @ 206 converting to mass U 238 so the ratio (if I had used a ratio) would be about 0.78.
ahh - nice catch on the mass. I hadn't thought of that. Is that due to all those missing alpha particles?
Partly, yes, and mostly yes. The difference is the sum of the masses of alpha particles, the slight difference in mass of the electrons, the number of protons and neutrons individually and as alpha particles and the mass defect effect. All of that adds up to 206 relative for Pb vs 238 relative for U238. What I didn't take into account is the mass of all of those intermediate elements since U 238 doesn't go to Pb 206 in one fell swoop. There are 14 intermediate disintegrations between 238 and 206 and two of those intermediates have chains of their own. What we do when we convert the mass of U238 to Pb 206 is we assume that all of the intermediate elements have ended up as U238 but I don't think that is so unless the half lives are such that all end up as Pb 206 (or 207 or 208) by the the time U238 has done its thing and ended up as Pb 206. And even then we assume the mass of Pb is the same as 206 even though the isotope may not be 206. AAARGH !!
3 answers