# How do I find the diameter of an aluminum atom from its density and atomic mass? [closed]

I am supposed to find the diameter of an aluminum atom using data given by my teacher. Here is the data: \begin{array}{cc} \text{Density of Aluminum}& \pu{2.70 g/cm3}\\ \text{Atomic Mass of Aluminum}& \pu{26.98 g} \end{array}

Here is the theoretical value of the diameter of the aluminum atom (this is what I am supposed to get): \begin{array}{cc} \text{Diameter Al atom}& \pu{2.86*10^-8cm} \end{array}

I don't understand how I am supposed to do this.

When we are assuming that aluminium is a sphere. Notice that $$\text{Density} = \frac{\text{mass}}{\text{volume}}.$$ Also notice that the atomic mass given is for one mole of aluminium. Therefore, you must convert the mass to a single atom (use mole). The Volume is $$\text{Volume}=\frac{\text{mass}}{\text{density}}$$ and the Volume of a sphere is $$\text{Volume}=\frac{4}{3}\pi r^3.$$ Set $\frac{4}{3}\pi r^3$ equal to $\displaystyle\frac{\text{mass}}{\text{density}}$, so that $$\frac{4}{3}\pi r^3=\frac{\text{mass}}{\text{density}},$$ isolate $r$ (radius) and since the diameter is twice the radius, multiply the value for radius by 2. Doing this you get $\pu{3.16\times10^-8 cm}$.

• See, we can't quite fill 100% of space with spheres. Now if you'll introduce a correction factor for that, you'll probably arrive at the desired answer. – Ivan Neretin Sep 17 '15 at 7:44