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the half life of U234 is 245.5 years. It decays by α emmission to Th230 whos half life is 75.38. there are 10^6 atoms of U234 and 0 atoms of Th230 at t=0.
At what time will the number of atoms of each type be equal?
the half life of U234 is 245.5 years and half life of Th230 is 75.38
λa for U is 2.82e^-3 and λb for Th is 9.2e^-3
Na = No*exp(-λut)
Nb = No*(λa/(λb-λa))*(exp(-λa*t)-exp(-λb*t))
They Na and Nb are equal when Na = Nb
therefore:
No*exp(-λat) = No*(λa/(λb-λa))*(exp(-λa*t)-exp(-λb*t))
Divide by No and exp(-λat)
1 = (λa/(λb-λa))*(1-exp-(λa + λb)t)
Divide by (λb-λa)/a
1 - ((λb-λa)/λa) = exp-(λa + λb)t
getting t on its own
t = ln(1-((λb-λa)/λa)) / -(λb + λb)
t = ln(1-((9.2e-3-2.82e-3)/2.82e-3)) / -(2.82e-3 + 9.2e-3)
t = ln(1-(0.00638/2.82e-3)) / (- 0.01202)
t = ln(1-2.26) / (-0.01202)
t = ln(-1.26)/-.01202
here is where the problem is as you can not get the ln of a negitave number
Can anyone figure out where i have gone wrong?
or is there a problem with the question itself
Homework Statement
the half life of U234 is 245.5 years. It decays by α emmission to Th230 whos half life is 75.38. there are 10^6 atoms of U234 and 0 atoms of Th230 at t=0.
At what time will the number of atoms of each type be equal?
the half life of U234 is 245.5 years and half life of Th230 is 75.38
λa for U is 2.82e^-3 and λb for Th is 9.2e^-3
Homework Equations
Na = No*exp(-λut)
Nb = No*(λa/(λb-λa))*(exp(-λa*t)-exp(-λb*t))
The Attempt at a Solution
They Na and Nb are equal when Na = Nb
therefore:
No*exp(-λat) = No*(λa/(λb-λa))*(exp(-λa*t)-exp(-λb*t))
Divide by No and exp(-λat)
1 = (λa/(λb-λa))*(1-exp-(λa + λb)t)
Divide by (λb-λa)/a
1 - ((λb-λa)/λa) = exp-(λa + λb)t
getting t on its own
t = ln(1-((λb-λa)/λa)) / -(λb + λb)
t = ln(1-((9.2e-3-2.82e-3)/2.82e-3)) / -(2.82e-3 + 9.2e-3)
t = ln(1-(0.00638/2.82e-3)) / (- 0.01202)
t = ln(1-2.26) / (-0.01202)
t = ln(-1.26)/-.01202
here is where the problem is as you can not get the ln of a negitave number
Can anyone figure out where i have gone wrong?
or is there a problem with the question itself
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