Use the exponential growth model #P(t) = P_0e^(kt)#. The half-life of thorium-229 is 7,340 years. How long will it take for a sample of this substance to decay to 20% of its original amount?

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Answer 1 · 2016-08-02T04:08:59

#t=7340(ln0.2/ln0.5)~=3,161# yrs.

We first find the original amt. of thorium-229, i.e., its amt. #P(0)#at

time #t=0#.

Taking #t=0# in, #P(t)=P_0e^(kt).........(1)#, we get, #P(0)=P_0#.

#:.# The Original Amt. of thorium-229 is #P_0#.

We are given that, its, Half-life is #7,340# yrs, which means that, for

#t=7340, P(t)=1/2P_0#.

Using this data in #(1)#, we get, #1/2P_0=P_0*e^(7340k)#

#rArr 0.5=e^(7340k) rArr ln0.5=7340k, or, k=ln0.5/7340....(2)#.

What we require, now, is time #t=?# for #P(t)=20%of P_0=1/5P_0#

By #(1), then, 1/5P_0=P_0e^(kt) rArr 1/5=e^(kt), i.e., ln0.2=kt#.

But, from #(2), k=ln0.5/7340, so, ln0.2=(ln0.5/7340)t#

Therefore, #t=7340(ln0.2/ln0.5)#

#=7340{log_(10)0.2/log_10e*log_10e/log_(10)0.5}#

#=7340(0.3010/0.6990)~=3,161#.

Thus, it will take (approx.) #3,161# yrs. for thorium-229 to decay to

20% of its original amt.