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# The mean life of a radioactive material for " $\alpha$ " and " $\beta$ " decay are 1620 years and520 years. The half-life of the sample is nearly:(A) 490 years(B) 126 years(C) 273 years(D) 888 years

Last updated date: 20th Sep 2024
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Hint
We know that alpha decay or α-decay is a type of radioactive decay in which the atomic nucleus emits an alpha particle thereby transforming or decaying into a new atomic nucleus. Here the atomic mass number of the newly formed atom will be reduced by four and the atomic number will be reduced by two. Alpha decay occurs when a nucleus is unstable because it has too many protons. The nucleus emits an alpha particle and energy. An alpha particle consists of two protons and two neutrons, which is actually a helium nucleus. Losing the protons and neutrons makes the nucleus more stable. Based on this we have to solve this question.

Given in the question that:
$\tau_{\mathrm{a}}=1620$ years
${{\tau }_{\text{b}}}=520$ years
Mean life of the sample,
$\tau=\dfrac{\tau_{\mathrm{a}} \times \tau_{\mathrm{b}}}{\tau_{\mathrm{a}}+\tau_{\mathrm{b}}}$
So,
$\tau =\dfrac{1620\times 520}{1620+520}=393.6$ years
Half life $\mathrm{T}=\dfrac{\tau}{11.44}=\dfrac{393.6}{1.44}=273 \mathrm{years}$
Therefore, the correct answer is Option (C).

Note
It is required to be known as in nuclear physics, beta decay (β-decay) is a type of radioactive decay in which a beta particle (fast energetic electron or positron) is emitted from an atomic nucleus, transforming the original nuclide to an isobar. Beta decay occurs when, in a nucleus with too many protons or too many neutrons, one of the protons or neutrons is transformed into the other. Since a neutron is converted into a proton, the atomic number of the element will increase by 1. Another conserved quantity is electric charge. And since the electron is the lightest particle that has electric charge, there is nothing that it can decay to; only neutrinos, photons, gluons and gravitons are lighter, but they are all electrically neutral, so any combination of them would have zero electric charge.