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?

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Solution

Diameter of sodium atom = Size of sodium atom = 2.5

Radius of sodium atom, r =

= 1.25 × 10^–10 m

Volume of sodium atom, V =

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 one atom =

Density of sodium atom, ρ =

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.

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

The unit of length convenient on the nuclear scale is a fermi : 1 f = 10^{– 15}m. Nuclear sizes obey roughly the following empirical relation :

where r is the radius of the nucleus, A its mass number, and r₀ is a constant equal to about, 1.2 f. Show that the rule implies that nuclear mass density is nearly constant for different nuclei. Estimate the mass density of sodium nucleus. Compare it with the average mass density of a sodium atom obtained in Exercise. 2.27.

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Substance	Atomic Mass (u)	Density (10 $^{3}$ kg m $^{- 3}$
Carbon (diamond)	12.01	2.22
Gold	197.00	19.32
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[Hint: Assume the atoms to be 'tightly packed' in a solid or liquid phase, and use the known value of Avogadro's number. You should, however, not take the actual numbers you obtain for various atomic sizes too literally because of the crudeness of the tight packing approximation, the results only indicate that atomic sizes are in the range of a few

˚ A

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