The radioactive isotope of lead Pb209 decays at a rate propo

The radioactive isotope of lead, Pb-209, decays at a rate proportional to the amount present at time t and has a half-life of 3.3 hours. If 1 gram of this isotope is present initially, how long will it take for 90% of the lead to decay? (Round your answer to two decimal places.)

Solution

Dear Student

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Since the decay rate is proportional to the amoutn present at time t

P\' = kP ..........................where k is a constant

Integrating Both sides we get

P = ce^(kt) .............................Eq(1)

Given

P(0) = 1 gram ..............................Eq(2)

P(3.3) = 0.5 grams .........................Eq(3)

Let x be the time for 90% decay then,

P(x) = 0.1 gram ............................Eq(4)

Using Eq(1) & Eq(2) we get C=1.

Using Eq(1) & Eq(3) we get value of k i.e.

P= e^(kt)

0.5 = e^(3.3k)

Taking Natural lagorithm on both sides

ln (0.5) = 3.3k

k = -0.21

P = e^(-0.21t) .........................(Radioactive decay equation)

0.1 = e^(-0.21t)

Gives t = 10.96 years ..........Solution