An unknown metal is found to have a density of 7.8748 g/cm³ and to crystallize in a body-centered cubic lattice. The edge of the unit cell is 0.28664 nm. Calculate the atomic mass of the metal.

검증된 단계별 안내

Convert the edge length of the unit cell from nanometers to centimeters by using the conversion factor: 1 nm = 1 x 10^{-7} cm.

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 in centimeters.

Determine the mass of the unit cell by multiplying the volume of the unit cell by the density of the metal: mass = density x volume.

In a body-centered cubic (BCC) lattice, there are 2 atoms per unit cell. Use this information to find the mass of a single atom by dividing the mass of the unit cell by 2.

Calculate the atomic mass of the metal by converting the mass of a single atom from grams to atomic mass units (amu) using the conversion factor: 1 amu = 1.66053906660 x 10^{-24} g.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Density and its Calculation

Density is defined as mass per unit volume, typically expressed in grams per cubic centimeter (g/cm³). To calculate the density of a substance, one can use the formula: density = mass/volume. In this context, knowing the density of the metal allows us to relate it to the atomic mass and the volume of the unit cell.

Body-Centered Cubic (BCC) Lattice

A body-centered cubic lattice is a type of crystal structure where atoms are located at each corner of a cube and one atom is positioned at the center of the cube. This arrangement affects the packing efficiency and the volume occupied by the atoms, which is crucial for calculating the atomic mass based on the unit cell dimensions.

Unit Cell Volume and Atomic Mass Relationship

The volume of a unit cell can be calculated using the edge length of the cube, raised to the third power (V = a³). In a BCC structure, there are two atoms per unit cell, and the relationship between the mass of the atoms, the density, and the volume of the unit cell can be used to derive the atomic mass of the metal using the formula: atomic mass = (density × unit cell volume) / number of atoms per unit cell.

An unknown metal is found to have a density of 7.8748 g/cm3 and to crystallize in a body-centered cubic lattice. The edge of the unit cell is 0.28664 nm. Calculate the atomic mass of the metal.