 Solution

Closest approach. In the heavy-target approximation, a head-on alpha particle of charge $2e$ converts its initial kinetic energy to repulsive Coulomb potential energy at its turning point. The nuclear radius bound from a Coulomb turning point is thereforeThus the point-projectile interpretation of the radius bound isIt is an upper limit from Coulomb scattering, not a measurement of the actual nuclear radius. If “size” denotes diameter, the associated bound is twice this value. Including nuclear recoil, $E_{\mathrm{cm}}\simeq [207/(207+4)]E_{\alpha }$, increases the closest separation by about two per cent to $4.4\times 10^{-14}\mathrm{m}$.

Count-rate estimate and normalization. The graph has approximate slope $\Delta \ln S/\Delta \ln A\simeq 15/10=3/2$. It therefore supplies the empirical log-log plot relation $S\propto A^{3/2}$. Calibrating its normalization with the lead observation givesThe Rutherford differential cross-section at fixed beam energy and scattering angle instead has a $Z^2$ dependence per target nucleus. More generally, the target normalization in Rutherford scattering rates iswhere $\mathcal{N}$ is the target nuclei per unit area. If the foils have equal $\mathcal{N}$, the theoretical estimate is $19.7$, or about $20$ particles per minute, close to the graphical estimate at its rough level. Equal foil mass per area would instead give $37.8$ per minute because $\mathcal{N}\propto 1/A$.

The source does not specify that target normalization. Thus $23$ per minute is the estimate from its supplied graph; an exact rate prediction from Rutherford theory alone is underdetermined without comparable foil areal densities. The empirical $A^{3/2}$ slope and an exact equal-$\mathcal{N}$ $Z^2$ law should not be presented as identical assumptions.