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Home / Doubts and Solutions/NEET/Physics

establish a relation between half life and mean life

Asked by jyotishbaraik85 7th August 2018, 9:38 AM
Answered by Expert
Answer:
 
We know that,
N(t)= N0e-λt ...(1)
 
R is defined as,
 
R= -dN/dt  
 
R= λN0e-λt
R=R0e-λt  ...(2)
 
Thus, from (1)
R = R0e-λt 
 
R- total decay rate
R0 - decay rate at time, t=0
 
Half life of radioactive substance is average time it takes for sample N0 to reduce to N0/2.
 
t = t1/2 
t subscript 1 divided by 2 end subscript space equals space fraction numerator ln left parenthesis 2 right parenthesis over denominator lambda end fraction equals fraction numerator 0.693 over denominator lambda end fraction A v e r a g e space o r space m e a n space l i f e space tau space c a n space b e space o b t a i n e d space f r o m space e q u a t i o n space left parenthesis 1 right parenthesis T h e space n u m b e r space o f space n u c l e i space w h i c h space d e c a y space i n space t i m e space i n t e r v a l space t space t o space t plus increment t space i s comma R left parenthesis t right parenthesis increment space t space equals space lambda N subscript 0 e to the power of negative lambda t end exponent tau space equals space fraction numerator lambda N subscript 0 integral subscript 0 superscript infinity t e to the power of negative lambda t end exponent d t over denominator N subscript 0 end fraction equals space lambda integral subscript 0 superscript infinity t e to the power of negative lambda t end exponent B y space s o l v i n g space t h i s space i n t e g r a l comma tau space equals space 1 over lambda t subscript 1 divided by 2 space end subscript equals fraction numerator space ln left parenthesis 2 right parenthesis over denominator lambda end fraction equals space tau space ln 2 space T h i s space i s space t h e space r e l a t i o n s h i p space b e t w e n space t h e space a v e r a g e space l i f e space a n d space h a l f space l i f e. space space  
 
 
 
 
Answered by Expert 7th August 2018, 12:24 PM
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