Iron has a density of 7.87 g / cm^3, and the mass of an iron atom is 9.27 ×10^-26 kg . If the at

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Iron has a density of 7.87 g / cm^3, and the mass of an iron...

Iron has a density of $7.87 \mathrm{g} / \mathrm{cm}^{3},$ and the mass of an iron atom is $9.27 \times 10^{-26} \mathrm{kg}$ . If the atoms are spherical and tightly packed, (a) what is the volume of an iron atom and (b) what is the distance between the centers of adjacent atoms?

Key Concepts

Density and Mass Relationship

This concept relates to the definition of density as mass divided by volume. It is fundamental for determining the volume occupied by a single atom when the mass of the atom and the bulk density of the material are known, thus linking macroscopic properties with microscopic structure.

Atomic Volume Estimation

Atomic volume estimation involves calculating the volume of an individual atom by dividing its mass by the material’s density. This technique is useful for understanding how much space an atom occupies within a solid, which is a key aspect of material science and atomic physics.

Sphere Geometry

The geometry of a sphere is essential when atoms are approximated as spherical objects. Using the formula for the volume of a sphere (V = 4/3?r^3), one can determine the atom's radius from its calculated volume and then deduce the distance between the centers of adjacent atoms in a closely packed arrangement.

Close Packing in Crystalline Structures

Close packing refers to the arrangement of atoms in a solid where they are as tightly packed as possible, often following specific packing geometries such as face-centered cubic or hexagonal close-packed. This concept helps in understanding how the physical distance between atoms (center-to-center distance) is determined by the atomic radius in a crystalline lattice.

Unit Conversion

Unit conversion is a critical step in calculations involving physical quantities. It ensures consistency when working with different measurement systems (for example, converting grams to kilograms or cubic centimeters to cubic meters), which is necessary for accurate computation of atomic dimensions.

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