Let the quantity of radium at any time t be A.

According to the question,  where k is a constant  Integrating both sides, we have = – k∫dt

log|A| = – kt + c……(1)

Given, Initial quantity of radium be A0 when t = 0 sec

Putting the value in equation (1)

log|A| = – kt + c

log| A0| = 0 + c

c = log| A0| ……(2)

Putting the value of c in equation (1) we have,

log|A| = – kt + log| A0|

log|A| – log| A0| = – k t [ ]

log ( = – kt ……(3)

Given that the radium decomposes 1.1% in 25 years,

A = (100 – 1.1)% = 98.9% = 0.989 A0 at t = 25 years

From equation(3),we have

– kt = log ( – k×25 = log ( k = – The equation becomes

log ( = – t

Now, log ( = – t

log ( = – t = – t (log 2 = 0.6931 and log 0.989 = 0.01106)  t = 1567 years

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