Question

A nuclide has a half life of 2.41 x104 years. How long will it take a sample to reach a level of radioactivity that is 2.5% the level it had when it was first made?

Answer #1

Radio active decay is a first order reaction.

For first order recation,

half life t_{1/2} = 0.693 /k where k is rate
constant

k = 0.693/ t_{1/2 --- Eq (1)}

k = 1/t ln { [A]_{o}/[A]_{t}} -----Eq (2)

From Eqs (1) and (2),

0.693/ t_{1/2} = (1/t) ln {[A]_{o/}
[A]_{t}} ------Eq (3)

Given that

half life t_{1/2}= 2.41 x10^{4} yrs = 24100
yrs

time t = ? hrs

Initial radioactivity [A]_{o} = 100 %

Final radioactivity [A]_{t} = 2.5 %

Substitute all the values in Eq (3),

0.693/ t_{1/2} = (1/t) ln {[A]_{o/}
[A]_{t}}

[(0.693)/ 24100 yrs)] = (1/t ) ln{100/2.5}

t = ln {100/2.5} x ( 24100 yrs/ 0.693)

= 128285 yrs

t = 128285 yrs

Therefore, 128285 yrs required to reach a level of radioactivity that is 2.5% the level it had when it was first made.

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