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Question: 1.2 mg of \( ^{239}Pu \) emits \( 1.4 \times {10^7} \) particles per minute. What is its half-life? ...

1.2 mg of 239Pu^{239}Pu emits 1.4×1071.4 \times {10^7} particles per minute. What is its half-life?
(A) 2.372 years
(B) 3.372 years
(C) 4.472 years
(D) None of the above

Explanation

Solution

The half-life of a quantity is the time it takes for it to decrease to half of its original value. In nuclear physics, the phrase is used to explain how rapidly unstable atoms disintegrate radioactively and how long stable atoms survive. The word is also used to describe any sort of exponential or non-exponential decay in general. The biological half-life of medicines and other substances in the human body, for example, is discussed in medical research. Doubling time is the inverse of half-life.

Complete answer:
The half-life of an exponentially decaying quantity is constant during its lifespan, and it is a characteristic unit for the exponential decay equation. The reduction of a quantity as a function of the number of half-lives elapsed is shown in the table below. The decay of discrete things, such as radioactive atoms, is generally described by a half-life. In such a situation, the notion that "half-life is the time necessary for exactly half of the entities to decay" does not apply. If just one radioactive atom exists and its half-life is one second, there will be no "half of an atom" remaining after one second. The half-life is instead described in terms of probability: "Half-life is the average time necessary for exactly half of the entities to decay." In other words, a radioactive atom has a 50% chance of disintegrating within its half-life. The half-life of radioactive decay is the time after which there is a 50% probability that an atom will have experienced nuclear decay. It is generally calculated empirically and varies based on the atom type and isotope.
Now we know that
−dNdt=λN- \dfrac{{{\text{dN}}}}{{{\text{dt}}}} = \lambda {\text{N}}
Hence
0.1mg239Pu=0.1×10−3  g=0.1×10−3239  mol0.1{\text{m}}{{\text{g}}^{239}}{\text{Pu}} = 0.1 \times {10^{ - 3}}\;{\text{g}} = \dfrac{{0.1 \times {{10}^{ - 3}}}}{{239}}\;{\text{mol}}
We know that
∴N=0.1×10−3×6.02×1023239atoms\therefore {\text{N}} = \dfrac{{0.1 \times {{10}^{ - 3}} \times 6.02 \times {{10}^{23}}}}{{239}}{\text{atoms}}
So
∴1.4×107=2×0.1×10−3×6.02×1023239\therefore 1.4 \times {10^7} = \dfrac{{2 \times 0.1 \times {{10}^{ - 3}} \times 6.02 \times {{10}^{23}}}}{{239}}
Hence
∴λ=5.56×10−11  min−1\therefore \lambda = 5.56 \times {10^{ - 11}}\;{\text{mi}}{{\text{n}}^{ - 1}}
Also
∴t50=0.698i=7.481×1011  s\therefore {{\text{t}}_{50}} = \dfrac{{0.698}}{i} = 7.481 \times {10^{11}}\;{\text{s}}
To conclude
⇒t12=2.372years\Rightarrow {t_{\dfrac{1}{2}}} = 2.372{\text{years}}
Hence option a is correct.

Note:
The word "half-life" is almost exclusively applied to exponential or roughly exponential decay processes. The half-life of a degradation process that is not even close to exponential will fluctuate drastically while it is occurring. In this situation, it is uncommon to speak of half-life in the first place, but people will occasionally refer to the decay as having a "first half-life," "second half-life," and so on, where the first half-life is defined as the time required to decay from the initial value to 50%, the second half-life is defined as the time required to decay from 50% to 25%, and so on.