Answer to Question #236523 in Physics for Arif

Question #236523
Calculate the packing efficiency and density of sodium chloride from the following data: Radius of sodium ion = 0.98 Angstrom; Radius of chloride ion = Angstrom ; Atomic mass of sodium = 22.99 amu Atomic mass of chlorine = 35.45 amu
1
Expert's answer
2021-09-14T09:36:13-0400

Sodium chloride has a face-centered cubic (FCC) crustal lattice. First, find the packing efficiency.

The volume of the cubic unit cell:


V=a3=(8r)3=88r3.V=a^3=(\sqrt8r)^3=8\sqrt8r^3.

The volume of space that the atoms occupy:


v=443πr3=163πr3.v=4·\frac43\pi r^3=\frac{16}{3}\pi r^3.

The packing efficiency:


ϵ=vV=16πr3/388r3=0.74.\epsilon=\frac vV=\frac{16\pi r^3/3}{8\sqrt8 r^3}=0.74.

Find the density:


ρ=nMVNA=n(MNa+MCl)(88r3)(NA)= =4(23103+35.5103)(0.563109)3(6.0231023)=2177 kg/m3.\rho=\frac{n\Mu}{VN_A}=\frac{n(\Mu_\text{Na}+\Mu_\text{Cl})}{(8\sqrt8r^3)(N_A)}=\\\space\\ =\frac{4(23·10^{-3}+35.5·10^{-3})}{(0.563·10^{-9})^3·(6.023·10^{23})}=2177\text{ kg/m}^3.


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