Except helium, all the noble gases crystallize with cubic close packed structures. Derive an equation relating atomic radius to the density of cubic close packed solid of given mass and apply it to deduce the relative size of Ar and Xe if their densities are 1.4 g/cm and 2.83 g/cm^(3) respectively.

Except helium, all the noble gases crystallize with cubic close packed

1.	Except helium, all the noble gases crystallize with cubic close packed structures. Derive an equation relating atomic radius to the density of cubic close packed solid of given mass and apply it to deduce the relative size of Ar and Xe if their densities are 1.4 g/cm and 2.83 g/cm^(3) respectively.
Answer» Solution : For a CCP solid : `(rho)=(nM)/(n_(A)a^(3))` Where, `rho`=Density,n=No. of atoms/unit cell M = Molar MASS, `N_(A)` = Avogadro constant, and a=Edge LENGTH `implies rho=(4M)/(N_(A)(2sqrt(2R))^(3))=(M)/(4sqrt(2)N_(A)R^(3))` `implies(rho_(x_(e)))/(rho_(Ar))=(2.83)/(1.40)((r_(Ar))/(r_(Xe)))^(3)` `implies(r_(Xe))/(R_(Ar))=((131xx1.4)/(40xx2.83))^(1//3)=1.174`