Titanium metal has a density of 4.506 g>cm3 and an atomic radius of 144.8 pm. In what cubic unit cell does titanium crystallize?

검증된 단계별 안내

Step 1: Convert the atomic radius from picometers to centimeters. 1 picometer (pm) is equal to 1e-10 centimeters (cm). So, the atomic radius in cm is 144.8 pm * 1e-10 cm/pm.

Step 2: Calculate the volume of one atom. Since the atoms in a metal are usually packed into a sphere, we can use the formula for the volume of a sphere, which is (4/3)πr^3, where r is the atomic radius.

Step 3: Calculate the volume of the unit cell. In a cubic unit cell, there are either 1, 2, or 4 atoms per unit cell for simple cubic (sc), body-centered cubic (bcc), or face-centered cubic (fcc) structures respectively. The volume of the unit cell is the volume of one atom multiplied by the number of atoms per unit cell.

Step 4: Calculate the mass of the unit cell. The mass of the unit cell is the mass of one atom (which can be calculated from the atomic weight of titanium and Avogadro's number) multiplied by the number of atoms per unit cell.

Step 5: Compare the calculated density (mass/volume) with the given density. The structure that gives a calculated density closest to the given density is the one in which titanium crystallizes.

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주요 개념

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Crystal Structure

Crystal structure refers to the orderly arrangement of atoms in a crystalline material. In metals, atoms are typically arranged in specific geometric patterns, which can be classified into different types of unit cells, such as face-centered cubic (FCC), body-centered cubic (BCC), or hexagonal close-packed (HCP). Understanding the crystal structure is essential for determining the properties of the metal, including its density and atomic packing.

Unit Cell

A unit cell is the smallest repeating unit in a crystal lattice that reflects the symmetry and structure of the entire crystal. It is defined by its dimensions and the arrangement of atoms within it. The type of unit cell influences the density and other physical properties of the material. Identifying the correct unit cell type is crucial for understanding how titanium crystallizes.

Density and Atomic Radius Relationship

The density of a material is related to its atomic mass and the volume occupied by its atoms in the crystal lattice. The atomic radius helps determine the volume of the unit cell, which is essential for calculating density. For metals like titanium, knowing the atomic radius allows us to infer the type of unit cell and how closely packed the atoms are, which is vital for answering questions about crystallization.