Question

the half life of a radioactive element is.3.8 days.the fraction left after 19 days will be?

Answer 1

plz see the upload. hope it helps.

Answer 2

Concept: The period of time it takes for one-half of a radioactive isotope to decay is known as the half-life. A given radioactive isotope's half-life is constant; it is unaffected by external factors and independent of the isotope's starting concentration.

Given: half life of a radioactive element is.3.8 days

To find: the fraction left after 19 days

Solution:

In time t = T,

$N = \frac{N_{0} }{2}$

in another half-life (i.e., after 2 half-lives)

$N = \frac{1}{2} \frac{N_0}{2} = \frac{N_0}{4} = N_0(\frac{1}{2} )^2$

after yet another half-life (i.e., after 3 half-lives)

$N = \frac{1}{2} \frac{N_0}{4} = \frac{N_0}{8} = N_0(\frac{1}{2} )^3$ and so on.

Hence, after n half-lives

$N = N_0(\frac{1}{2} )^n\\N = N_0 (\frac{1}{2} )^\frac{t}{T}$

where t = n × T = total time of n half lives

Now, $n = \frac{t}{T} = \frac{19}{3.8} = 5$

The fraction left

$\frac{N}{N_0} =( \frac{1}{2} )^n = ( \frac{1}{2} )^5 = \frac{1}{32} = 0.031$

Hence, the fraction left after 19 days will be 0.031

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