A radioactive element decays by following three different parallel paths:A⟶λ1B;λ1=5×10−2sec−12A⟶λ2C;λ2=3×10−2sec−13A⟶λ3D;λ3=5×10−3sec−1Calculate average lifetime of element (in sec.)λ1,λ2,λ3are decay· constants for respective reactions]

−d[A]dt=dxdt+dydt+dzdt=d[B]dt+12d[C]dt+13d[D]dt=λ1[A]+2λ2[A]+3λ3[A]λ=50×10−3+2×30×10−3+5×10−3×3=125×10−3tavg =1λ=8 sec

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A radioactive element decays by following three different parallel paths:
$A \overset{λ_{1}}{⟶} B; λ_{1} = 5 \times 10^{- 2} \sec^{- 1}$
$2 A \overset{λ_{2}}{⟶} C; λ_{2} = 3 \times 10^{- 2} \sec^{- 1}$
$3 A \overset{λ_{3}}{⟶} D; λ_{3} = 5 \times 10^{- 3} \sec^{- 1}$
Calculate average lifetime of element (in sec.)
$[λ_{1}, λ_{2}, λ_{3}$ are decay· constants for respective reactions]

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$- \frac{d [A]}{d t} = \frac{d x}{d t} + \frac{d y}{d t} + \frac{d z}{d t}$

$= \frac{d [B]}{d t} + \frac{1}{2} \frac{d [C]}{d t} + \frac{1}{3} \frac{d [D]}{d t}$

$= λ_{1} [A] + 2 λ_{2} [A] + 3 λ_{3} [A]$

$\begin{array}{l} λ = 50 \times 10^{- 3} + 2 \times 30 \times 10^{- 3} + 5 \times 10^{- 3} \times 3 \\ = 125 \times 10^{- 3} \end{array}$

$t_{avg} = \frac{1}{λ} = 8 \sec$

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