The rate of disintegration of a radioactive element at any time is proportional to its mass at that time, where k(k>0) is the constant of proportionality. The time during which an original mass of 1.5 gm will disintegrate to a mass of 0.5 gm is

Step-by-step solution

Let m be the mass of the radioactive element at time t . The rate of disintegration is given by dm dt = -km dm m = -k dt Integrating both sides with appropriate limits: _ 1.5 ^ 0.5 dm m = -k ₀^ t dt [ m]_ 1.5 ^ 0.5 = -kt 0.5 - 1.5 = -kt ( 0.5 1.5 ) = -kt ( 1 3 ) = -kt - 3 = -kt t = 1 k 3 Answer: 1 k 3