The decay rate of radium is proportional to the amount present at any time t . If initially 60 gms was present and half life period of radium is 1600 years, then the amount of radium present after 3200 years is

Step-by-step solution

Let m be the mass of substance at time t . Then aligned & dm dt =- km , where k 0 & dm ~m =- kdt aligned Integrating on both sides, we get m=-k t+c when t =0, ~m =60 gms array ll & 60=-k(0)+c & c = 60 array array ll & c = 60 & ~m =- kt + 60 & when t =1600, ~m = 60 2 =30 gms & 30=-1600 k + 60 & 1600 k = 2 & k = 1 1600 2 array Equation (i) becomes m= -1 1600 ( 2) t+ 60 When t=3200 years aligned & m = -1 1600 2 3200+ 60 & ~m =- 4+ 60 & ~m = 60 4 & ~m =15 grams aligned