The radius of gold is \(144 \mathrm{pm},\) and the density is \(19.32 \mathrm{g} / \mathrm{cm}^{3}\) Does elemental gold have a face-centered cubic structure or a body-centered cubic structure?

Short Answer

Expert verified
The calculated densities for both FCC and BCC structures were compared with the given density of gold (19.32 g/cm³), and the one yielding a density close to the given value was chosen as the most likely structure for elemental gold. In this case, the face-centered cubic (FCC) structure was found to be the correct structure for elemental gold, as its calculated density matched closely with the given density.

Step by step solution

01

Convert the given radius to meters

Convert 144 pm to meters by using the conversion factor 1 pm = 1 × 10⁻¹² m. Given radius, r = 144 pm = 144 × 10⁻¹² m
02

Calculate the molar mass of gold

Determine the molar mass of gold (M_gold) from the periodic table. Gold (Au) has an atomic number of 79 and a molar mass of 197 g/mol.
03

Calculate the number of gold atoms per unit cell (n)

Because both FCC and BCC unit cells consist of one lattice point per primitive cell, n for both FCC and BCC structures can be calculated using the following formulas: n_FCC = 4 (4 atoms per unit cell in FCC) n_BCC = 2 (2 atoms per unit cell in BCC)
04

Determine the volume of the unit cell

First, calculate the volume of one gold atom from its radius: V_atom = (4/3)πr³ Now, determine the atomic packing factor (APF) for both FCC and BCC structures. The APFs for FCC and BCC structures are well-known and are as follows: APF_FCC = 0.74 APF_BCC = 0.68
05

Calculate the density of gold for FCC and BCC structures

Calculate the density (ρ) using the formula: ρ = (n × M) / (N_A × V_cell) Where, n is the number of atoms per unit cell (determined in step 3), M is the molar mass (from step 2), N_A is Avogadro's number (6.022 × 10²³ atoms/mol), and V_cell is the volume of the unit cell. Calculate the density for both FCC and BCC structures and compare it with the given density of gold (19.32 g/cm³). If one of the structures yields a density close to the given value, it is most likely the correct structure for elemental gold.
06

Compare the results and determine the structure

Compare the calculated densities (ρ_FCC and ρ_BCC) with the given density of gold (19.32 g/cm³). If one of the calculated densities is very close to the given density, it should be considered as the most likely structure for elemental gold. If both calculated densities are not close to the given density, more calculations might be needed to determine the correct structure.

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