The density of an unknown metal is 12.3 g/cm³, and its atomic radius is 0.134 nm. It has a face-centered cubic lattice. Find the atomic mass of this metal

Verified step by step guidance

1. Convert the atomic radius from nanometers to centimeters. Since 1 nm = 1 x 10^{-7} cm, multiply the atomic radius by this conversion factor.
2. Calculate the volume of the unit cell. For a face-centered cubic (FCC) lattice, the edge length (a) of the unit cell is related to the atomic radius (r) by the formula: a = 2\sqrt{2}r.
3. Determine the volume of the unit cell using the formula: Volume = a^3, where a is the edge length calculated in the previous step.
4. Calculate the mass of the unit cell using the density formula: Density = Mass/Volume. Rearrange to find Mass = Density x Volume, using the given density and the volume from the previous step.
5. In a face-centered cubic lattice, there are 4 atoms per unit cell. Use the mass of the unit cell to find the atomic mass by dividing the mass of the unit cell by the number of atoms per unit cell.

Key Concepts

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 is given, which can be used alongside its atomic structure to derive its atomic mass. The formula for density (density = mass/volume) is crucial for solving the problem.

Face-Centered Cubic (FCC) Structure

A face-centered cubic lattice is a type of crystal structure where atoms are located at each corner and the centers of all the cube faces. This arrangement affects the packing efficiency and the volume occupied by the atoms. Understanding the FCC structure is essential for calculating the volume of the unit cell, which is necessary for determining the atomic mass from the given density.

Atomic Mass and Molar Volume Relationship

The atomic mass of a substance can be related to its density and the volume of its unit cell. For an FCC structure, the volume of the unit cell can be calculated using the atomic radius, and the number of atoms per unit cell (4 for FCC) allows for the calculation of atomic mass using the formula: atomic mass = (density × molar volume). This relationship is key to solving the problem.