Question

If $2.60g$ of cobalt-$60$ (half-life =$5.30$ y) are allowed to decay. How many grams would be left after $1.00$ y and after $10.0$ y?

Answer 1

We need to know that chemical kinetics is one of the topics used to study the kinetics nature of the chemical reaction. It is used to optimize the chemical reaction for industrial purposes. The reactant to the product so many parameters are required. All are optimised by using this chemical kinetics. Chemical kinetics is used as a mechanism of reactant to product in the chemical reaction.
Formula used:
The remaining amount of the substance depends on the half-life period and initial amount of the substance.
${\text{remaining amount = }}\dfrac{{{\text{initial amount}}}}{{{{\text{2}}^{\text{n}}}}}$
Here, n is dependent on the half-life period.

Complete answer:
The given data is
The half-life period of cobal-60 for $2.60g$ is $5.30$ y.
The half-life period means the time required to reduce the half of the amount of substance from initial concentration.
The remaining amount of the substance depends on the half-life period and initial amount of the substance.
${\text{remaining amount = }}\dfrac{{{\text{initial amount}}}}{{{{\text{2}}^{\text{n}}}}}$
Here, n is dependent on the half-life period.
We calculate the number of half-life period in one year as,
${\text{n = }}\dfrac{{\text{1}}}{{{\text{5}}{\text{.30}}}}$
On division we get,
$\Rightarrow {\text{n}} = 0.18868$
We calculate the remaining amount of the substance in one year as,
${\text{remaining amount = }}\dfrac{{{\text{initial amount}}}}{{{{\text{2}}^{\text{n}}}}}$
${\text{n}} = 0.18868$
The initial amount of the substance cobal-60 is ${\text{2}}{\text{.60 g}}$
${\text{remaining amount}} = \dfrac{{2.60}}{{{2^{0.18868}}}}$
On simplification we get,
$= 2.28$
The remaining amount of the substance in one year is $2.28$ g.
We calculate the number of half-life period in ten year as,
${\text{n = }}\dfrac{{{\text{10}}}}{{{\text{5}}{\text{.30}}}}$
On simplification we get,
${\text{n}} = 1.8868$
We calculate the remaining amount of the substance in ten year as,
${\text{remaining amount = }}\dfrac{{{\text{initial amount}}}}{{{{\text{2}}^{\text{n}}}}}$
${\text{n}} = 1.8868$
The initial amount of the substance cobal-60 is ${\text{2}}{\text{.60 g}}$
${\text{remaining amount}} = \dfrac{{2.60}}{{{2^{1.8868}}}}$
On simplification we get,
$= 0.703$
The remaining amount of the substance in ten year is $0.703$ g.

Note:
We need to know that the rate of the reaction is an important factor for the study of reaction. The rate of reaction is an important concept for chemical kinetics. Rate of the reaction depends on the concentration of the reactant. The rate of reaction is also calculated by using the concentration of the product in the chemical reaction. Depending on the concentration, the sign of the rate will change.
The rate of the reaction is directly proportional to the concentration of reactant and inversely proportional to the time of the reaction.
The rate of the reaction is equal to the product of the concentration of the reactant with respect to that order of the reaction.
${\text{rate = k[A}}{{\text{]}}^{\text{m}}}{{\text{[B]}}^{\text{n}}}$
Here, k is proportionality constant, known as rate constant.
A and B are reactants of the reaction.
m and n are the order of the reaction of A and B respectively.
The time period calculation,
${{\text{t}}_{{\text{1/2}}}}{\text{ = }}\dfrac{{{\text{0}}{\text{.6932}}}}{{\text{k}}}$
Here, the time period is t.