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Question
The rate of disintegration of a radioactive element at any time t is proportional to its mass at that time. Find the time during which the original mass of 1.5 gm will disintegrate into its mass of 0.5 gm.
Solution
Let m be the mass of the radioactive element at time t.
Then the rate of disintegration is `"dm"/"dt"` which is proportional to m.
∴ `"dm"/"dt" prop "m"`
∴ `"dm"/"dt"` = - km, where k > 0
∴ `"dm"/"m"` = - k dt
On integrating, we get
`int 1/"m" "dm" = - "k" int "dt" + "c"`
∴ log m = - kt + c
Initially, i.e. when t = 0, m = 1.5
∴ log(1.5) = - k × 0 + c ∴ c = log`(3/2)`
∴ log m = - kt + log`(3/2)`
∴ log m - log`3/2` = - kt
∴ `log("2m"/3)` = - kt
When m = 0.5 = `1/2`, then
`log ((2 xx 1/2)/3) = - "kt"`
∴ `log (1/3)` = - kt
∴ log(3)-1 = - kt
∴ - log 3 = - kt
∴ t = `1/"k" log 3`
∴ the original mass will disintegrate to 0.5 gm when t = `1/"k" log 3`
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