Isotopes are atoms having

$A$ nucleus disintegrates into two nuclear parts,in such a way that the ratio of their nuclear radii is $1: 2^{1/3}$. Their respective speeds have a ratio of $n: 1$. The value of $n$ is $.....$.

The ratio of the density of oxygen nucleus $\left({ }_{8}^{16} O \right)$ and helium nucleus $\left({ }_{2}^{4} He \right)$ is (in $:1$)

$(a)$ Two stable isotopes of lithium $_{3}^{6} Li$ and $_{3}^{7} Li$ have respective abundances of $7.5 \%$ and $92.5 \%$. These isotopes have masses $6.01512 \; u$ and $7.01600 \; u$,respectively. Find the atomic mass of lithium.
$(b)$ Boron has two stable isotopes,$_{5}^{10} B$ and $_{5}^{11} B$. Their respective masses are $10.01294 \; u$ and $11.00931 \; u$,and the atomic mass of boron is $10.811 \; u$. Find the abundances of $_{5}^{10} B$ and $_{5}^{11} B$.

$A$ radioactive nucleus of mass $M$ emits a photon of frequency $f$ and the nucleus recoils. The recoil energy will be

The nearest distance between two atoms in case of a $bcc$ lattice is equal to