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Asked by Prashant DIGHE | 31 Jan, 2020, 10:25: PM

Expert Answer

Radioactivity is proportional to number of undecayed atoms. Hence activity (dN/dt)₁ at time t = t₁ is given as

begin mathsize 14px style open parentheses fraction numerator d N over denominator d t end fraction close parentheses subscript 1 space equals space lambda space N subscript 1 space equals space lambda space N subscript o space e to the power of negative lambda space t subscript 1 end exponent end style

.......................(1)

Similarly at time t= t₂ , activity (dN/dt)₂ is given by

begin mathsize 14px style open parentheses fraction numerator d N over denominator d t end fraction close parentheses subscript 2 space equals space lambda space N subscript 2 space equals space lambda space N subscript o space e to the power of negative lambda space t subscript 2 end exponent end style

.........................(2)

By dividing eqn.(1) using eqn.(2), we get

begin mathsize 14px style left parenthesis fraction numerator d N over denominator d t end fraction right parenthesis subscript 1 over left parenthesis fraction numerator d N over denominator d t end fraction right parenthesis subscript 2 space equals 700 over 500 space space equals space space e to the power of lambda left parenthesis space t subscript 2 minus t subscript 1 right parenthesis end exponent end style

................................(3)

Hence , we get, λ ( t₂ - t₁ ) = ln(1.4) = 0.3365 ....................(4)

we have , λ = 0.693 / T_1/2 and (t₂ - t₁ ) = 30 min

using the above substitutions in eqn.(4), we get , T_1/2 = ( 0.693 × 30 ) / 0.3365 ≈ 62 min

Answered by Thiyagarajan K | 01 Feb, 2020, 05:06: AM

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