Question

The nuclear radius is of the order
\[\begin{align}
 & A{{.10}^{^{-10}}}m \\
 & B{{.10}^{-6}}m \\
 & C{{.10}^{^{-15}}}m \\
 & D{{.10}^{^{-14}}}m \\
\end{align}\]

Answer 1

Hint: We know that the nucleus is present inside the atom, and then its radius must be smaller than the radius of the atom. We know that the radius of atom is around $10^{-10}m$ then the radius of nucleus cannot be $10^{-10}m$, also it cannot be $10^{-6} m$ as $10^{-6} m$ is greater than $10^{-10} m$.
Then the radius of the nucleus is either $10^{-15} m$ or $10^{-14} m$.

Complete step-by-step answer:
Since we know that the radii of the nucleus must be smaller than the radii of the atom. For the nucleus to be contained in the atom. Then the radius of the nucleus is either $10^{-15} m$ or $10^{-14} m$ .
Rutherford qualitatively determined the radius of the nucleus using the famous gold foil experiment or the alpha scattering experiment.
This was later tested experimentally, by Geiger-Marsden.
It was found that, the potential energy of the atom is given by: $E_{a}=\dfrac{2Ze^{2}}{4\pi \epsilon_{0}}$
For gold, atomic no $Z=79$ and substituting the other constants, we get, $E_{a}=7.7MeV$.
It was found that the radius of nuclei was about $10^{-14} m$

Then the radius of the nucleus is $10^{-14} m$

So, the correct answer is “Option D”.

Additional Information: Other important findings of the Geiger-Marsden alpha scattering experiments are:
1. Since most of the alpha particles could pass through the gold foil, it was concluded that the gold foil was mostly empty.
2. Since a very small number of alpha particles get deflected by about $180^{\circ}$ from the gold foil, it was concluded that there was a small amount of +ve charge which was concentrated in the centre of the atom.
Later they coined the term nucleus to denote the positive centre of the atom, which contributes majorly to the mass of the atom.

Note: Since the powers are negative, greater the power implies the radius is smaller and vice-versa. Also note that Rutherford proposed the idea of alpha scattering due to gold foil, but Geiger-Marsden proved the idea experimentally. The proof behind the radius of the nucleus is $10^{-14} m$, is long and complicated. Hence it is suggested to remember this as a fact.