Q2.27:

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 ?

Step 1: Identifying Given Constants and Variables

• Atomic mass of sodium ($M$): $23.0 \, \text{g/mol} = 0.023 \, \text{kg/mol}$
• Avogadro’s number ($N_A$): $6.022 \times 10^{23} \, \text{atoms/mol}$
• Atomic radius of sodium ($r$): $2.5 \, \text{Å} = 2.5 \times 10^{-10} \, \text{m}$
• Density of crystalline sodium ($\rho_{\text{crystal}}$): $970 \, \text{kg/m}^3$

Step 2: Calculating Mass of a Single Sodium Atom

The mass of a single atom ($m$) is obtained by dividing the molar mass by Avogadro's number [Per the definition of the mole]:

$m = \frac{M}{N_A} = \frac{0.023 \, \text{kg/mol}}{6.022 \times 10^{23} \, \text{atoms/mol}}$

$m \approx 3.819 \times 10^{-26} \, \text{kg}$

Step 3: Calculating the Volume of the Sodium Atom

Assuming the atom is a sphere [Geometric approximation for atomic volume]:

$V = \frac{4}{3} \pi r^3$

$V = \frac{4}{3} \times 3.14159 \times (2.5 \times 10^{-10} \, \text{m})^3$

$V = \frac{4}{3} \times 3.14159 \times 15.625 \times 10^{-30} \, \text{m}^3 \approx 6.545 \times 10^{-29} \, \text{m}^3$

Step 4: Calculating Average Atomic Mass Density

The density ($\rho_{\text{atom}}$) is defined as mass per unit volume [By definition of density]:

$\rho_{\text{atom}} = \frac{m}{V} = \frac{3.819 \times 10^{-26} \, \text{kg}}{6.545 \times 10^{-29} \, \text{m}^3} \approx 583.5 \, \text{kg/m}^3$

Visual Representation of Atomic Model

Step 5: Comparison and Analytical Conclusion

Comparing the values:

$\frac{\rho_{\text{crystal}}}{\rho_{\text{atom}}} = \frac{970}{583.5} \approx 1.66$

Both densities are of the same order of magnitude ($10^2 \, \text{kg/m}^3$). This occurs because in the crystalline phase, atoms are packed closely together, and the interatomic spacing is not significantly larger than the atomic dimensions, meaning the volume occupied by the atom is representative of the volume it occupies in the crystal lattice.

Final Solution: The average mass density of a sodium atom is approximately 583.5 kg/m³. Yes, both densities are of the same order of magnitude, $10^2$ kg/m³, because the physical state of the solid is primarily determined by the tight packing of these spherical atomic volumes.