The half-life of a neutron is $693 \ s$. What fraction of neutrons will decay when a beam of neutrons, having kinetic energy of $0.084 \ eV$, travels a distance of $1 \ km$? (mass of neutron $= 1.68 \times 10^{-27} \ kg$, and $\ln 2 = 0.693$)

Obtain the amount of $_{27}^{60} Co$ necessary to provide a radioactive source of $8.0 \; mCi$ strength. The half-life of $_{27}^{60} Co$ is $5.3$ years.

$A$ parent nucleus $X$ undergoes $\alpha$-decay with a half-life of $75000 \text{ yrs}$. The daughter nucleus $Y$ undergoes $\beta$-decay with a half-life of $9 \text{ months}$. In a particular sample, it is found that the rate of emission of $\beta$-particles is nearly constant (over several months) at $10^{7} / \text{h}$. What will be the number of $\alpha$-particles emitted in an hour?

$A$ sample initially contains $10^{20}$ radioactive atoms. The number of $\alpha$-particles emitted in the third year is $0.3$ times the number of $\alpha$-particles emitted in the second year. How many $\alpha$-particles are emitted in the first year?

Substance $A$ has an atomic mass number of $16$ and a half-life of $1$ day. Another substance $B$ has an atomic mass number of $32$ and a half-life of $0.5$ day. If both $A$ and $B$ start undergoing radioactivity simultaneously with an initial mass of $320 \, g$ each,how many total atoms of $A$ and $B$ combined would be left after $2$ days? (Answer in $......... \times 10^{24}$)

Radon $({Rn})$ decays into Polonium $({Po})$ by emitting an $\alpha$-particle with a half-life of $4\, days$. $A$ sample contains $6.4 \times 10^{10}$ atoms of $Rn$. After $12\, days$,the number of atoms of $Rn$ left in the sample will be: