For a radioactive material, its activity A and rate of change of its activity R are defined as A = - d N d t and R = - d A d t , where N ( t ) is the number of nuclei at time t . Two radioactive sources P (mean life τ ) and Q (mean life 2 τ ) have the same activity at t = 0 . Their rates…

Question

For a radioactive material, its activity A and rate of change of its activity R are defined as A = - d N d t and R = - d A d t , where N ( t ) is the number of nuclei at time t . Two radioactive sources P (mean life &#964; ) and Q (mean life 2 &#964; ) have the same activity at t = 0 . Their rates of change of activities at t = 2 &#964; are R P and R Q , respectively. If R P R Q = n e , then the value of n is

Aicharya · Accepted Answer

N = N 0 e - λ t A = λ N 0 e - λ t R = λ 2 N 0 e - λ t A P = A Q at t = 0 λ P N P e - λ P t = λ Q N Q e - λ Q t at t = 0 λ P N P = λ Q N Q ...(i) R P R Q = λ P λ Q 2 N P N Q e - λ P 2 τ e - λ Q 2 τ R P R Q = λ P λ Q 2 N P N Q 1 e ...(ii) λ P = 1 m e a n l i f e From equation (i) and equation (ii) R P R Q = 2 e ⇒ n = 2

For a radioactive material, its activity A and rate of change of its activity R are defined as A = - d N d t…

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