Question

In a radioactive sample ${}_{19}^{40}K$ nuclei either decay into stable ${}_{20}^{40}Ca$ nuclei with decay constant \[(4.5)({{10}^{-10}})\] per year or into stable ${}_{18}^{40}Ar$ nuclei with decay constant \[(0.5)({{10}^{-10}})\] per year. Given that in this sample all the stable ${}_{20}^{40}Ca$ and ${}_{18}^{40}Ar$ nuclei are produced by the${}_{19}^{40}K$ nuclei only. In time $tX{{10}^{9}}$years, if the ratio of the sum of stable ${}_{20}^{40}Ca$ and ${}_{18}^{40}Ar$nuclei to the radioactive ${}_{19}^{40}K$ nuclei is 99, the value of t will be? [Given: In 10 = 2.3]
(A) $(9.2)({{10}^{9}})$
(B) $(1.5)({{10}^{6}})$
(C) None
(D) Both

Answer 1

Hint: The dissociation of a radioactive sample follows the first order kinetics. An unstable nuclei dissociates into stable nuclei and releases particle i.e. alpha particle or beta particle or gamma particle or neutrons or positron.

Complete step by step solution:
Given in the question:
Radioactive sample ${}_{19}^{40}K$ nuclei decay into stable ${}_{20}^{40}Ca$ nuclei
The decay constant for the reaction = \[(4.5)({{10}^{-10}})\] per year
Radioactive sample ${}_{19}^{40}K$ nuclei decay into stable ${}_{18}^{40}Ar$nuclei
The decay constant for the reaction = \[(0.5)({{10}^{-10}})\] per year
At time $tX{{10}^{9}}$years, if the ratio of the sum of stable ${}_{20}^{40}Ca$ and ${}_{18}^{40}Ar$nuclei to the radioactive ${}_{19}^{40}K$ nuclei is 99
Let ${{\lambda }_{1}}and\text{ }{{\lambda }_{2}}$be the decay constant for ${}_{19}^{40}K$ nuclei decay into stable ${}_{20}^{40}Ca$ nuclei and sample ${}_{19}^{40}K$ nuclei decay into stable ${}_{18}^{40}Ar$nuclei respectively
\[\begin{align}
& \dfrac{dN}{dt}=-{{\lambda }_{1}}N-{{\lambda }_{2}}N \\
& \dfrac{dN}{dt}=-({{\lambda }_{1}}+{{\lambda }_{2}})N \\
\end{align}\]
And we know that,
$N={{N}_{o}}{{e}^{-({{\lambda }_{1}}-{{\lambda }_{2}})t}}$
And given in the question N=${{N}_{o}}-99%of\text{ }{{\text{N}}_{0}}=0.01{{N}_{0}}$
We get
$Time = t =$$\dfrac{\ln 100}{{{\lambda }_{1}}+{{\lambda }_{2}}}=\dfrac{(2.3)(2)}{(5)({{10}^{-10}})}=(9.2)({{10}^{9}})$years

Hence the correct answer is option (A).

Note: Rate is defined as the speed at which a chemical reaction occurs. Rate is generally expressed in the terms of concentration of reactant which is consumed during the reaction in a unit of time or the concentration of product which is produced during the reaction in a unit of time. If the reaction is a first order reaction, the unit for first order reaction is ${{S}^{-1}}$. The negative and positive sign in the expression of the rate or reaction only means the change in concentration. A negative charge indicates that the concentration of the reactant is decreasing, similarly a positive charge means that the concentration of product is increasing.