Question

The half-life of a radioactive element is 8 years. How much amount will be present after 32 years?
\[\begin{align}
  & \text{A) }\dfrac{1}{4} \\
 & \text{B) }\dfrac{1}{8} \\
 & \text{C) }\dfrac{1}{16} \\
 & \text{D) }\dfrac{1}{32} \\
\end{align}\]

Answer 1

Hint: We can very easily find the substance left over after years if we know the half-life time of the radioactive element. The key idea is that after each half-life time half of the initial sample gets converted or gets decayed. We can just do the sum even without the formulae.

Complete answer:
Radioactive elements are those substances which decay into other elements due to its inability to continue in its form or due to its atomic instability. These elements have a specific time period in which half of the total sample gets decayed. This is called the half-life time. This is derived from the radioactive decay law as –
\[R={{R}_{0}}{{e}^{-\lambda t}}\]
Where R is the rate of radioactivity,
\[{{R}_{0}}\] is the initial rate of radioactivity,
\[\lambda \] is the disintegration constant or decay constant,
t is the time taken
Now, we can write this in terms of the number of radioactive samples as –
\[N={{N}_{0}}{{e}^{-\lambda t}}\]
Where, N is the number of radioactive elements left over after ‘t’ time,
\[{{N}_{0}}\]is the initial number of radioactive elements.
The disintegration constant is the reciprocal of the mean time period of decay of particles.
Also, the half-life time of the element is related to the disintegration constant as –
\[\begin{align}
  & {{T}_{1/2}}=\dfrac{\ln 2}{\lambda }=\dfrac{0.693}{\lambda } \\
 & \Rightarrow \text{ }\lambda =\dfrac{0.693}{{{T}_{1/2}}} \\
\end{align}\]
Now, let us substitute this in the decay law –
\[\begin{align}
  & N={{N}_{0}}{{e}^{-\lambda t}} \\
 & \Rightarrow \text{ }N={{N}_{0}}{{e}^{-\dfrac{0.693}{{{T}_{1/2}}}t}} \\
 & \text{here, } \\
 & {{T}_{1/2}}=8\text{years} \\
 & t=32\text{years} \\
 & \Rightarrow \text{ }N={{N}_{0}}{{e}^{-\dfrac{0.693}{8}\times 32}} \\
 & \Rightarrow \text{ }N={{N}_{0}}{{e}^{-2.772}} \\
 & \Rightarrow \text{ }N=\dfrac{{{N}_{0}}}{16} \\
\end{align}\]
From the above calculations, we understand that the remaining number of radioactive elements is one-sixteenth of the initial number.

So, the correct answer is “Option C”.

Note:
We can find this without doing the calculations as we did here. We can reduce one-half of the initial quantity each time we pass a half-life time. At first 8 years it becomes half, the next 8 years, i.e., the 16 years afterwards, it becomes one-quarter and this continues up to 32 years when it finally becomes one-sixteenth.