Question

The radius of tungsten is \(137 \mathrm{pm}\) and the density is \(19.3 \mathrm{~g} / \mathrm{cm}^{3}\). Does elemental tungsten have a face-centered cubic structure or a body-centered cubic structure?

Short answer

Elemental tungsten has a face-centered cubic (FCC) structure. This is determined by comparing the given density of tungsten \(19.3 \mathrm{~g} / \mathrm{cm}^{3}\) to the theoretical densities of FCC and BCC structures. The theoretical density of the FCC structure is approximately \(19.546 \mathrm{~g} / \mathrm{cm}^{3}\) and the theoretical density of the BCC structure is approximately \(9.773 \mathrm{~g} / \mathrm{cm}^{3}\). Since the difference between the given density and the theoretical density of the FCC structure is smaller, we can conclude that elemental tungsten has a face-centered cubic (FCC) structure.

Step-by-step solution

Find the atomic mass of tungsten in grams

Using a periodic table, we find that the atomic mass of tungsten is 183.84 amu. To convert this to grams, we multiply by the atomic mass constant: Atomic mass of tungsten (in g/mol) = 183.84 amu * (1.660539 x 10^(-24) g/amu) = 305.43 x 10^(-24)g

Find the unit cell lattice parameter for tungsten

The unit cell lattice parameter is the distance between the centers of two atoms along the edge of the cube. For a cubic structure, we'll use the radius of the tungsten atom to calculate the lattice parameter. Radius of tungsten = 137 pm = 137 x 10^(-12) m For a cubic structure, unit_cell_edge_length = 2 * radius Unit cell lattice parameter (a) for tungsten = 2 * 137 x 10^(-12) m = 274 x 10^(-12) m

Calculate theoretical densities for FCC and BCC structures

The formula for theoretical density is: Theoretical Density = (n * atomic mass) / (volume of the unit cell * Avogadro's number) For FCC, n = 4 and for BCC, n = 2. Volume of the unit cell is given by: Volume = a^3 Where a is the unit cell lattice parameter. Theoretical density (FCC) = (4 * 305.43 x 10^(-24) g) / ((274 x 10^(-12) m)^3 * 6.022 x 10^23 atoms/mol * 10^6 cm^3/m^3) Theoretical density (BCC) = (2 * 305.43 x 10^(-24) g) / ((274 x 10^(-12) m)^3 * 6.022 x 10^23 atoms/mol * 10^6 cm^3/m^3) Theoretical density (FCC) ≈ 19.546 g/cm³ Theoretical density (BCC) ≈ 9.773 g/cm³

Compare theoretical densities with given density

Given density = 19.3 g/cm³ Comparing with the theoretical densities of FCC and BCC structures: Difference (FCC) = |19.546 - 19.3| ≈ 0.246 g/cm³ Difference (BCC) = |9.773 - 19.3| ≈ 9.527 g/cm³ Since the difference between the given density and the theoretical density of the FCC structure is smaller, we can conclude that elemental tungsten has a face-centered cubic (FCC) structure.