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 loge0.989 = 0.01106 and loge2 = 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