How do you estimate the density of sodium based on the structure and radius?

Question

Sodium metal (atomic weight 22.99 ) adopts a body-centered cubic structure with a density of 0.97 .

The atomic radius of Na has been determined to be 1.9 Å

If it didn't react so vigorously, sodium could float on water. Use the answer from part A (the 1.9 Å) to estimate the density of  if its structure were that of a cubic close-packed metal. Note: Use unrounded values in the calculations.

Answer 1

The percent volume of a body-centered cubic structure is π√3 / 8 ≈ 68% and the percent volume of a cubic closed packed structure is π√2 / 6 ≈ 71%.

Because the percent of the atom per molecule is greater in the cubic closed packed structure it is more dense.

The percent volume is directly proportional to the density.

Thus, %V₁ / d₁ = %V₂/ d₂.

Plug in: 68%/ 0.97 = 71% / d₂.

d₂ will represent the density of the cubic closed packed structure.

Solve for d₂

d₂≈1.013