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Question
Show that half life period of radioactive material varies inversely to decay constant λ.
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Solution
From law of radioactive decay,
`"N" = "N"_0 "e"^{-lambda"t"}`
at t = `"T"_{1"/"2}, "N" = ("N"_0)/2`
∴ `("N"_0)/2 = "N"_0 "e"^{-lambda"T"_{1"/"2}}`
∴ `1/2 = "e"^{-lambda"T"_{1"/"2}}`
∴ `"e"^{lambda"T"_{1"/"2}} = 2`
∴ `lambda"T"_{1"/"2} = log_"e"2 = 0.693`
∴ `"T"_{1"/"2} = 0.693/lambda`
⇒ `"T"_{1"/"2} ∝ 1/lambda`
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