Iron crystallizes in a body-centered cubic unit cell with an edge length of 287 pm. Iron metal has a density of 7.86 g>cm3 and a molar mass of 55.85 g. Calculate a value for Avogadro's number.

검증된 단계별 안내

Convert the edge length of the unit cell from picometers to centimeters to ensure consistency in units for density calculations.

Calculate the volume of the unit cell using the formula for the volume of a cube, V = a^3, where 'a' is the edge length of the cube.

Determine the number of atoms per unit cell for a body-centered cubic structure, which is 2 atoms per unit cell.

Calculate the mass of the unit cell by using the density formula, \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \). Rearrange to find the mass: \( \text{Mass} = \text{Density} \times \text{Volume} \).

Use the molar mass and the mass of the unit cell to find the number of moles of iron in the unit cell. Then, use Avogadro's number formula, \( N_A = \frac{\text{Number of atoms in the unit cell}}{\text{Moles of atoms in the unit cell}} \), to find Avogadro's number.

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주요 개념

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Body-Centered Cubic (BCC) Structure

The body-centered cubic (BCC) structure is a type of crystal lattice where atoms are located at each corner of a cube and a single atom is positioned at the center of the cube. This arrangement affects the packing efficiency and density of the material. In BCC, each unit cell contains two atoms, which is crucial for calculating properties like density and molar volume.

Density and Molar Mass Relationship

Density is defined as mass per unit volume, and it can be used to relate the mass of a substance to its volume in a given crystal structure. The molar mass indicates the mass of one mole of a substance, and by using the density and the volume of the unit cell, one can derive the number of atoms per unit cell and ultimately calculate Avogadro's number.

Avogadro's Number

Avogadro's number is a fundamental constant that represents the number of atoms, ions, or molecules in one mole of a substance, approximately 6.022 x 10^23. It is essential for converting between the macroscopic scale of substances (grams, liters) and the microscopic scale (individual particles). Calculating Avogadro's number from the properties of a crystal structure involves understanding the relationship between the unit cell dimensions, density, and molar mass.