Question

There are three cubic unit cells $A,B$ and $C$. $A$ is $FCC$ and all of its tetrahedral voids are also occupied. $B$ is also $FCC$ and all of its octahedral voids are also occupied. $C$ is simple cubic and all of its cubic voids are also occupied. If voids in all unit cells are occupied by the spheres exactly at their limiting radius, the order of packing efficiency would be:

• A. $A<B<C$
• B. $C<A<B$
• C. $C<B<A$
• D. $A<C<B$

Answer 1

The correct option is B $C<A<B$

$(P.E.{)}_A=\frac{4\times \frac{4}{3}\pi r^3+8\times \frac{4}{3}\pi (0.225r{)}^3}{{\left(\frac{4r}{\sqrt{2}}\right)}^3}=0.76$

$(P.E.{)}_B=\frac{4\times \frac{4}{3}\pi r^3+4\times \frac{4}{3}\pi (0.414r{)}^3}{{\left(\frac{4r}{\sqrt{2}}\right)}^3}=0.79$

$(P.E.{)}_C=\frac{1\times \frac{4}{3}\pi r^3+1\times \frac{4}{3}\pi (0.732r{)}^3}{{\left(2r\right)}^3}=0.72$