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 ______.

पर्याय

• log 5

• log 3

• log 2

• log 4

Answer 1

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.