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10 gram of radioactive material of half-life 15 years is kept in a box for 20 years. The disintegrated material is:
A. 10.2 g
B. 6.03 g
C. 4.03 g
D. 12.6 g

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Last updated date: 25th Feb 2024
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Answer
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Hint: The disintegrated material refers to the material left after the completion of the half-life of the radioactive material, that is, the disintegrated material is the difference between the initial amount of the material and the substance remaining.
Formulae used:
\[N={{N}_{0}}{{\left( \dfrac{1}{2} \right)}^{\dfrac{t}{{{t}_{{1}/{2}\;}}}}}\]

Complete step-by-step solution
The formula for computing the half-life of the radioactive material is given as follows.
\[N={{N}_{0}}{{\left( \dfrac{1}{2} \right)}^{\dfrac{t}{{{t}_{{1}/{2}\;}}}}}\]
Where N is the quantity of the substance remaining, \[{{N}_{0}}\] is the initial quantity of the substance, t is the time elapsed and \[{{t}_{{1}/{2}\;}}\] is the half-life of the substance.
From the given data, we have the data as follows.
10 grams of radioactive material of half-life 15 years are kept in a box for 20 years.
Therefore, we have, the initial quantity of the radioactive material is 10 grams, that is, \[{{N}_{0}}\]= 10 gram.
The time elapsed is 20 years, that is, t = 20 years.
The half-life of the radioactive material is 15 years, that is, \[{{t}_{{1}/{2}\;}}\]= 15 years.
Now substitute these values in the above formula to find the quantity of the radioactive material remaining.
\[\begin{align}
  & N=10{{\left( \dfrac{1}{2} \right)}^{\dfrac{20}{15}}} \\
 &\Rightarrow N=10{{\left( \dfrac{1}{2} \right)}^{\dfrac{4}{3}}} \\
 &\Rightarrow N=3.968 \\
\end{align}\]
Therefore, the amount of radioactive material remaining after its half-life is 3.968 g.
The amount of the disintegrated material is calculated as the difference between the initial amount of the material and the substance remaining. Thus, we get,
The disintegrated material is,
\[\begin{align}
  & =10-3.968 \\
 & =6.03 \\
\end{align}\]
\[ \therefore \]The disintegrated material is 6.03 g.
As the value of the disintegrated material is 6.03 g, thus, option (B) is correct.

Note: The units of the physical parameters should be known. The formula for computing the half-life of radioactive material should be known. The meaning of the disintegrated material should be known. Even, they can only ask for the amount of the material remaining after the completion of the half-life.
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