Radium decomposes at a rate proportional to the quantity of radium present. It is found that in 25 years, approximately 1.1% of a certain quantity of radium has decomposed. Determine approximately how long it will take for one - half of the original amount of radium to decompose?

[Given log_e0.989 = 0.01106 and log_e2 = 0.6931]

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