Adams C.
asked • 10/17/16The half-life of francium-212 is 19 minutes. How do I determine the number of minutes it takes for 8.0 grams of this isotope to decay to 0.250 grams
the answer to this question is what i am looking for and shown work as to ho you got the answer
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1 Expert Answer
Arturo O. answered • 10/17/16
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Set the problem up as a decaying exponential for the mass.
m(t) = m(0)e-kt, t in minutes, k in minutes-1
0.5 = 1e-k(19)
k = -ln(0.5) / 19 = 0.036481 min-1
m(t) = m(0)e-0.036481t
Let T be the time when 8.0 g decays to 0.250 g
0.250 = 8.0e-0.036481T
T = ln(0.250/8.0) /(-0.036481) minutes = 95.00 minutes
Test the solution:
m(95.00) = 8.0e-0.036481(95) = 0.250 [good]
Adams C.
where and how did you get the number "0.036481(95)"
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10/17/16
Arturo O.
Adams,
Please look at the first 4 steps of the solution, leading up to k = 0.036481 min-1. It comes from algebraic operations involving exponential functions, combined with the given information.
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10/17/16
Adams C.
is there another way you can get the answer because the way your showing me is too confusing
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10/17/16
Arturo O.
Please explain what you find confusing. We know these kinds of problems involve exponential decay, so the amounts present as a function of time must obey a law like
m(t) = m(0)e-kt
where m(t) is the amount present at time t, m(0) is the initial amount present (i.e. at t = 0), t is time elapsed from the moment there was m(0) amount present, and k is a positive constant with units of 1/time. You can find k by substituting given data into the law above. You can find half-life from
1/2 = 1*e-kT1/2
Note that implies the relationship
k = -ln(1/2) / T1/2
It all comes down to algebraically manipulating the exponential decay law, and substituting given values to find the unknowns.
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10/17/16
Adams C.
i think you are making it more difficult than it actually is. its already givin the half life of francium which is 19. and to determine the number of minutes it take for an 8.0 gram of this isotope to decay to 0.250 wouldnt all i need to do is divide the 0.250 by the 8.0 grams and the answer of that multiple it by 19?
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10/17/16
Adams C.
to determine the number of minutes it takes?
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10/17/16
Arturo O.
That is not correct. If you do that, you are assuming that the decay obeys a linear law, but it does not. It obeys an exponential law, like I showed you. Now to prove that the law is exponential, we would have to solve a differential equation, which entails math that is probably beyond the scope of the prerequisites to your chemistry class. But the law is exponential, NOT linear. A simple ratio will not produce a correct answer.
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10/17/16
Adams C.
ohhh ok i see
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10/17/16
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Adams C.
10/17/16