Calculate the radius of a palladium atom, given that Pd has an FCC crystal structure, a density

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Calculate the radius of a palladium atom, given that Pd has...

Calculate the radius of a palladium atom, given that $\mathrm{Pd}$ has an $\mathrm{FCC}$ crystal structure, a density of $12.0 \mathrm{g} / \mathrm{cm}^{3},$ and an atomic weight of $106.4 \mathrm{g} / \mathrm{mol}.$

Key Concepts

Density and Mass-Volume Relationship

Density is defined as mass per unit volume, and for crystalline solids, the density of a material can be related to the mass of the atoms in a unit cell divided by the volume of that cell. This relationship is used to connect macroscopic measurements (density) to microscopic parameters (atomic mass and unit cell dimensions).

Atomic Radius Determination from Lattice Parameters

In crystalline materials, there is a defined relationship between the lattice parameter (the length of the unit cell edge) and the atomic radius. For an FCC structure, the geometry dictates that the atoms touch along the face diagonal, leading to a specific formula relating the atomic radius to the lattice constant. This geometry is critical for calculating the atom’s radius from crystallographic and density data.

Avogadro's Number and Atomic Weight

Avogadro’s number links the atomic scale to the macroscopic scale by providing the number of atoms in a mole. Given the atomic weight, one can compute the mass of a single atom, which when combined with Avogadro’s number, allows the conversion from molar mass to the mass per atom—a key step in determining the unit cell's dimensions.

Unit Cell Atom Count

In an FCC structure, the effective number of atoms per unit cell is four. This is derived from counting the contribution of atoms at the eight corners (each shared by eight unit cells) and those at the six face centers (each shared by two unit cells). This count is key when relating the mass of the unit cell to the individual atomic mass.

FCC Crystal Structure

The face?centered cubic (FCC) structure is a common crystal arrangement where atoms are positioned at each of the cube's corners and the centers of each face. This configuration is crucial because it determines the number of atoms per unit cell and the relationship between the lattice constant and the atomic radius.

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