The Avogadro Project aims to define the kilogram with respect to a precise value of the Avogadro constant, $L$. For this task, spheres of ultrapure silicon are being made. Let the mass of a sphere be $M,$ and its volume be $V .$ (a) Write down an expression for the density of silicon. (b) What is the mass of one atom of silicon if the molar mass is $M_{\mathrm{r}} ?$
(c) The structure of elemental silicon is described in terms of a repeating unit called the unit cell. If the volume of the unit cell is $V_{c}$ and it contains $n$ atoms, how many atoms are present in the entire sphere?
(d) What is the mass of the sphere of silicon in terms of $M_{r}, L, V, V_{c}$ and $n ?$ (e) Rearrange the equation from part (d) to show how a value of the Avogadro constant can be determined.