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Question
The rate of disintegration of a radio active element at time t is proportional to its mass, at the time. Then the time during which the original mass of 1.5 gm. Will disintegrate into its mass of 0.5 gm. is proportional to ______.
Options
log 5
log 3
log 2
log 4
Solution
The rate of disintegration of a radioactive element at time t is proportional to its mass, at the time. Then the time during which the original mass of 1.5 gm. Will disintegrate into its mass of 0.5 gm. is proportional to log 3.
Explanation:
Let M be the mass of radioactive element at anytime 't'
Given, `"dM"/"dt"` = kM
`"dM"/"M"` = kdt
On Integrating, we get
log M = kt + C
When, t = 0, M = 1.5 gm
∴ C = log(1.5)
∴ log M = kt + log (1.5)
When, t = t, M = 0.5 gm
log (0.5) = kt + log(1.5)
log`(0.5/1.5)` = kt
`=> log 1/3` = kt
⇒ t = (log 3) kt
∴ Time is proportional to log 3.
