Question

The selling rate of a radioactive isotope is decided by its activity. The second-hand rate (in rupees) of a month old $^{32}P$ sample having ${t_{\dfrac{1}{2}}} = 14.3$ day if it was originally purchased for 800 rupees will be:

Answer 1

Hint: The half of radioactive decay alternate by changing the state of electrons surrounding the nucleus. This is called electron capture, which absorbs nucleus one of the atoms electrons combined with a proton to make a neutron and neutrino. It is the set of various processes by which it is random.

Complete step by step answer:
Here we find out the value from given equation;
 $k = \dfrac{{0.693}}{{{t_{1/2}}}} = \dfrac{{2.303}}{t}\log \dfrac{a}{{a - x}}$
Here we have value of
 ${t_{\dfrac{1}{2}}} = 14.3$
 \[a = 800\]
Put the value of substitutes in above equation;
 $k = \dfrac{{0.693}}{{14.3}} = \dfrac{{2.303}}{{30}}\log \dfrac{{800}}{x}$

$\Rightarrow$ $ \log \dfrac{{800}}{{a - x}} = 0.6313 \\ $
$\Rightarrow$ $ \dfrac{{800}}{{a - x}} = 4.2785 \\ $
$\Rightarrow$ $a - x = 187 \\ $

So now we see that the original purchase for 800 will be 187.

Additional information:
The unstable nucleus of a radioisotope can occur naturally, or as a result of artificially alternating the atom. Radioactive isotopes have unstable nuclei that decays or emits excess energy or radiation until the nucleus becomes stable. They can be naturally occurring or artificial isotopes of an element. The atoms of a given amount of radioactive substance disintegrated which is the time required to one half.

Note: The process is depended on the value of half-life. Half-life in radioactive isotopes, the time is required for one half of atomic nuclei into the other nuclear species by emitting particles and energy. Here we have the given value of half-life and find out the values with the help of substituted values.