The density of a sample of metal was measured to be 6.84 g>cm3. An X-ray diffraction experiment measures the edge of a face-centered cubic cell as 350.7 pm. What is the atomic weight, atomic radius, and identity of the metal?

검증된 단계별 안내

Calculate the volume of the cubic unit cell using the edge length provided. The volume (V) of a cube is given by the formula V = a^3, where 'a' is the edge length of the cube.

Convert the edge length from picometers (pm) to centimeters (cm) to match the units of the density given. Remember that 1 cm = 10^10 pm.

Calculate the mass of the unit cell using the density formula, Density = Mass/Volume. Rearrange the formula to find the mass (Mass = Density × Volume).

Determine the number of atoms per unit cell for a face-centered cubic (FCC) lattice. In an FCC lattice, there are 4 atoms per unit cell.

Calculate the atomic weight of the metal. The atomic weight can be found by dividing the mass of the unit cell by the number of atoms per unit cell. Finally, use the atomic weight to identify the metal by comparing it with known atomic weights of elements.

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주요 개념

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Density and its Calculation

Density is defined as mass per unit volume, typically expressed in grams per cubic centimeter (g/cm³). In this context, the density of the metal sample is given as 6.84 g/cm³, which can be used alongside the volume of the unit cell to determine the mass of the metal atoms within that cell. Understanding how to relate density to atomic weight and volume is crucial for solving the problem.

Face-Centered Cubic (FCC) Structure

A face-centered cubic (FCC) structure is a type of crystal lattice where atoms are located at each of the corners and the centers of all the faces of the cube. The edge length of the unit cell is critical for calculating the atomic radius, which can be derived from the relationship between the edge length and the atomic radius in an FCC lattice. This structure is common in metals and influences their properties.

X-ray Diffraction and Unit Cell Parameters

X-ray diffraction is a technique used to determine the atomic structure of a crystal by measuring the angles and intensities of scattered X-rays. The edge length of the unit cell, measured as 350.7 pm (picometers), is essential for calculating the volume of the unit cell and subsequently the number of atoms it contains. This information, combined with density, allows for the determination of atomic weight and the identity of the metal.