Question

Estimate the average mass density of a sodium atom assuming its size to be about 2.5 (Use the known values of Avogadro’s number and the atomic mass of sodium). Compare it with the density of sodium in its crystalline phase: 970 kg m^–3. Are the two densities of the same order of magnitude? If so, why?

Answer 1

Diameter of sodium atom = Size of sodium atom = 2.5 A°
Radius of sodium atom = 2.5A°/2 = 1.25 A°

= 1.25x10^-10m
Volume of sodium atom V= 4/3 πr³ 
= 4/3 × 22/7 × (1.25A°)³ 
= 8.177 × 10^-30 m³ [ 1A° = 10^-10m]
According to the Avogadro hypothesis, one mole of sodium contains 6.023 × 10²³ atoms and has a mass of 23 g or 23 × 10^–3 kg.
Mass of sodium atom = mass of one mole of sodium atom/Avogadro constant 
= 23 × 10^-³/6.023 × 10²³ Kg 
= 3.818 × 10^-26 Kg 

Now, 
Density of sodium atom, ρ = 3.818× 10^-26/8.177 × 10^-30 
= 4.67 × 10³Kg/m³

It is given that the density of sodium in crystalline phase is 970 kg m^–3.
Hence, the density of sodium atom and the density of sodium in its crystalline phase are not in the same order. This is because in solid phase, atoms are closely packed. Thus, the inter-atomic separation is very small in the crystalline phase.