A radioactive element has rate of disintegration 16 , 000 disintegrations per minute at a particular instant. After four minutes, it becomes 2000 disintegrations per minute. The decay constant per min

Question

A radioactive element has rate of disintegration 16 , 000 disintegrations per minute at a particular instant. After four minutes, it becomes 2000 disintegrations per minute. The decay constant per minute is

TestHub · Accepted Answer

The rate of disintegration (activity) of a radioactive element follows the exponential decay law:

$A = A_{0} e^{- λ t}$

Given initial activity $A_{0} = 16000$ disintegrations per minute, final activity $A = 2000$ disintegrations per minute, and time $t = 4$ minutes.

Substituting these values into the equation:

$2000 = 16000 e^{- λ (4)}$

$\frac{2000}{16000} = e^{- 4 λ}$

$\frac{1}{8} = e^{- 4 λ}$

Taking the natural logarithm on both sides:

$lo g_{e} (\frac{1}{8}) = - 4 λ$

$lo g_{e} (2^{- 3}) = - 4 λ$

$- 3 lo g_{e} 2 = - 4 λ$

$λ = \frac{3}{4} lo g_{e} 2$

$λ = 0.75 lo g_{e} 2$

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