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Question
There is more than one isotope of natural uranium. If a researcher isolates 1.00 mg of the relatively scarce $^{235}\textrm{U}$ and finds this mass to have an activity of 80.0 Bq, what is its half-life in years?
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Final Answer
$7.03\times 10^{8}\textrm{ y}$
Solution Video

OpenStax College Physics Solution, Chapter 31, Problem 51 (Problems & Exercises) (3:14)

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Video Transcript

This is College Physics Answers with Shaun Dychko. A 1 milligram sample of uranium-235 has an activity of 80 becquerels and the question is what is the half-life of this material? So to solve this, we'll need to convert this mass into number of atoms by knowing the molar mass so we look that up in appendix A and it's 235.044 grams per mol. And so we'll begin with this formula for activity it's logarithm of 2 times the number of atoms divided by the half-life and then we'll solve this for half-life by multiplying both sides by <i>t</i> one-half and dividing it by the activity <i>R</i>. So we have half-life is natural logarithm of 2 times the number of uranium-235 atoms divided by the activity. Now the number of atoms we don't know but we do know the mass and so we'll take that mass which is in grams and then—we'll write it in grams anyhow— and then we'll divide that by the molar mass which is units of grams per mol and this works out to units of mols and then we'll multiply by Avogadro's number which is number of atoms per mol and then we'll have the number of atoms and I explain these units more clearly down in the next line here. So we substitute <i>N</i> in here and we do that in red on this line here. So the half-life is natural logarithm of 2 times the mass times Avogadro's number divided by the molar mass of uranium-235 times the activity. So it's natural log of 2 times 1 milligram which is 1 times 10 to the minus 3 grams times Avogadro's number—6.0221 times 10 to the 23 atoms per mol— and divide by 235.044 grams per mol times 80 decays per second— that's becquerels written in more base units— and so these grams cancel and this is reciprocal mols in the denominator which works out to mols in the numerator and these mols cancel with these mols leaving us with number of atoms. And then this reciprocal seconds works out to seconds in the numerator so that's why seconds are the units of our answer and atoms are not really a unit, it's just a counting so the units in the end are seconds. So this is 2.219904 times 10 to the 16 seconds which is easier to understand if we convert it into years so we times by 1 hour for every 3600 seconds times 1 day for every 24 hours times 1 year for every 365 and a quarter days giving us 7.03 times 10 to the 8 years and if you look up in the appendix A what the accepted half-life is for uranium-235, it's 7.04 times 10 to the 8 years and so this experimentalist has done a good job in calculating or measuring what the half-life is.