Question

The half-life of Uranium-233 is 160000 years i.e., Uranium-233 decays at a constant rate in such a way that it reduces to 50% in 160000 years. In how many years will it reduce to 25%?
(A) 80000 years
(B) 240000 years
(C) 320000 years
(D) 40000 years

Answer 1

Hint: Assume the initial quantity of the uranium be x. Now, calculate the quantity of Uranium after 160000 years. Then, again calculate the amount of Uranium after the next 160000 years. We can see that the final amount of uranium is \[\dfrac{x}{4}\] . We can see that the final quantity \[\dfrac{x}{4}\] , is 25% of the initial quantity. Now, calculate total time taken to reach the quantity \[\dfrac{x}{4}\].

Complete step by step solution:
According to the question, it is given that the half life of Uranium-233 is 160000 years. We have that after how many years the uranium will be reduced to its 25% of the original quantity.
The half-life of Uranium-233 is the time after which the quantity of Uranium reduces to 50% of its original quantity.
First of all, let us assume the original quantity of the Uranium-233 be x …………………(1)
Now, after the first half life that is, after 160000 years we have,
The quantity of Uranium reduces to its 50% of the original quantity.
From equation (1), we have the original quantity.
Now, after the first 160000 years, Uranium has reduced to its 50% = x-50% of x =
\[x-\dfrac{50}{100}x=\dfrac{100x-50x}{100}=\dfrac{50x}{100}=\dfrac{x}{2}\] ………………………(2)
The remaining quantity of Uranium after 160000 years = \[\dfrac{x}{2}\] ………………………….(3)
Again, after the second 160000 years, the quantity of uranium will be reduced to 50% of \[\dfrac{x}{2}\].
The remaining quantity of Uranium = \[\dfrac{x}{2}\] - 50% of \[\dfrac{x}{2}\] =\[\dfrac{x}{2}-\dfrac{50}{100}.\dfrac{x}{2}=\left( \dfrac{x}{2} \right)\left( \dfrac{100x-50x}{100} \right)=\left( \dfrac{x}{2} \right)\times \dfrac{50}{100}=\dfrac{x}{4}\] …………………………..(4)
The quantity \[\dfrac{x}{4}\] is 25% of the original quantity.
So, the total number of years in which the Uranium reduces to 25% = 160000+160000=320000 years
Therefore, after 320000 years the uranium will be reduced to its 25%.
Hence, the correct option is (C).

Note: We can also solve this question, by using the formula of radioactivity decay.
\[Final\,amount=\left( Initial\,amount \right){{e}^{-\lambda t}}\] …………………………..(1)
 Here, \[\lambda \] is constant and t is the time.
Let the initial amount of Uranium be x ……………………………..(2)
In case \[{{1}^{st}}\] , we have the final amount reduced to 50% of the initial amount.
After 160000 years, the amount reduces to 50%.
Time = 160000 years ……………………….(3)
The final amount of Uranium = 50% of x = \[\dfrac{50x}{100}=\dfrac{x}{2}\] ………………..(4)
Now, from equation (1), equation (2), and equation (3), we get
\[\begin{align}
  & \Rightarrow \dfrac{x}{2}=x.{{e}^{-160000\lambda }} \\
 & \Rightarrow \dfrac{1}{2}=\dfrac{1}{{{e}^{160000\lambda }}} \\
 & \Rightarrow {{e}^{160000\lambda }}=2 \\
\end{align}\]
Taking \[\ln \] in LHS and RHS of the above equation, we get
\[\begin{align}
  & \Rightarrow \ln \left( {{e}^{160000\lambda }} \right)=\ln 2 \\
 & \Rightarrow 160000\lambda =\ln 2 \\
\end{align}\]
\[\Rightarrow \lambda =\dfrac{\ln 2}{160000}\] …………………….(5)
In case \[{{2}^{nd}}\] , we have the final amount reduces to 25% of the initial amount.
The final amount of Uranium = 25% of x = \[\dfrac{25x}{100}=\dfrac{x}{4}\] ………………………….(6)
Now, from equation (1), equation (2), and equation (6), we get
\[\begin{align}
  & \Rightarrow \dfrac{x}{4}=x.{{e}^{-\lambda t}} \\
 & \Rightarrow \dfrac{1}{4}=\dfrac{1}{{{e}^{\lambda t}}} \\
 & \Rightarrow {{e}^{\lambda t}}=4 \\
\end{align}\]
Taking \[\ln \] in LHS and RHS of the above equation, we get
\[\begin{align}
  & \Rightarrow \ln \left( {{e}^{\lambda t}} \right)=\ln 4 \\
 & \Rightarrow \lambda t=\ln {{2}^{2}} \\
\end{align}\]
\[\Rightarrow t=\dfrac{2\ln 2}{\lambda }\] ……………………………(7)
Now, from equation (5) and equation (7), we get
\[\begin{align}
  & \Rightarrow t=\dfrac{2\ln 2}{\dfrac{\ln 2}{160000}} \\
 & \Rightarrow t=\dfrac{2\ln 2}{\ln 2}\times 160000 \\
 & \Rightarrow t=2\times 160000 \\
 & \Rightarrow t=320000 \\
\end{align}\]
Therefore, after 320000 years the uranium will be reduced to its 25%.
Hence, the correct option is (C).