Q11 of 462 Page 22

The decay rate of radium at any time t is proportional to its mass at that time. Find the time when the mass will be halved of its initial mass.

Let the quantity of mass 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, the Initial quantity of masss be A0 when the 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)


Let the mass becomes half at time t1, A =


From equation(3),we have


∴ – kt = log (


⇒ – k×t1 = log (


⇒ – k×t1 =


⇒ – k×t1 = – log 2


⇒ t1 =


∴ Required time = where k is the constant of proportionality.


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