Some Properties of Atoms — Question 4

Question 4

Atomic number density supplies an interatomic length scale

A monatomic sample has mass density $\rho=2.40\times10^3\,\mathrm{kg/m^3}$ and atomic mass $m_a=4.00\times10^{-26}\,\mathrm{kg}$. Define a characteristic spacing $d=n^{-1/3}$, where n is number density.

1. Find n and d.

2. For an assigned atomic radius $R=.100\,\mathrm{nm}$, find the geometric occupied-volume fraction $n(4\pi R^3/3)$.

Question 4 - Solution

Atomic number density supplies an interatomic length scale

(a) Method and calculation

$$n=\rho/m_a=\boxed{6.00\times10^{28}\,\mathrm{m^{-3}}}.$$$$\boxed{d=n^{-1/3}=2.55\times10^{-10}\,\mathrm m}.$$

(b) Interpretation and further calculation

$$V_a=\frac{4\pi}{3}(10^{-10}\,\mathrm m)^3.$$$$\boxed{nV_a=.0800\pi=.2513}.$$

Physical check

The spacing is a volume-per-atom estimate, not necessarily a nearest-neighbor distance. Atomic radii and hard-sphere filling fractions depend on the adopted model.