Some Properties of Atoms — Question 2

Question 2

A nuclear volume fraction is much smaller than its radius ratio

Model an atom and its nucleus as spheres of radii $R=1.20\times10^{-10}\,\mathrm m$ and $r=4.80\times10^{-15}\,\mathrm m$. This geometric model does not treat the electron cloud as a solid shell.

1. Find the radius ratio and nuclear fraction of the atomic volume.

2. If the atomic radius were represented by 30.0 m, what radius would represent the nucleus?

Question 2 - Solution

A nuclear volume fraction is much smaller than its radius ratio

(a) Method and calculation

$$\eta=r/R=\frac{4.80\times10^{-15}}{1.20\times10^{-10}}=\boxed{4.00\times10^{-5}}.$$$$\boxed{V_n/V_a=\eta^3=6.40\times10^{-14}}.$$

(b) Interpretation and further calculation

$$\frac{r_{\mathrm{model}}}{30.0\,\mathrm m}=\eta.$$$$\boxed{r_{\mathrm{model}}=1.20\times10^{-3}\,\mathrm m=1.20\,\mathrm{mm}}.$$

Physical check

A small radius fraction is cubed in a volume comparison. Most atomic volume lies outside the nucleus, but the electron probability density need not vanish there.