A certain metal has a specific gravity of 10.200 at 25^∘ C. It crystallizes in a body-centered c

step 1 So in this question, we want to identify the identity of this metal. It has a body -centered cub

A certain metal has a specific gravity of 10.200 at 25^∘ C....

Key Concepts

Unit Cell Volume and Unit Conversions

The total volume of the unit cell is determined from the cube of its edge length. Conversion of units (for example, from angstroms to centimeters) is necessary to ensure consistency in units when applying physical formulas, particularly when calculating properties like density and atomic weight.

Density Formula in Crystallography

The density of a crystalline material can be calculated using the formula ? = (n × M) / (N_A × V_cell), where ? is the density, n is the number of atoms per unit cell, M is the atomic weight, N_A is Avogadro’s number, and V_cell is the volume of the unit cell. This relationship connects the microscopic structure of the material to its macroscopic density, allowing determination of one property if the others are known.

Atomic Weight Calculation and Metal Identification

By rearranging the density equation and solving for the atomic weight, one can determine the atomic weight of a metal from its measured density, unit cell dimensions, and crystal structure. Comparing the calculated atomic weight to known values in the periodic table allows for the identification of the metal.

Crystal Structure and Unit Cell

A crystal structure describes the ordered arrangement of atoms in a solid, and the unit cell is the smallest repeating unit that fully describes this arrangement. The unit cell, characterized by its edge lengths and angles, is fundamental in relating the microscopic atomic arrangement to macroscopic properties such as density.

Specific Gravity and Density

Specific gravity is a dimensionless quantity that compares the density of a material to that of a reference substance (typically water at 4°C). When the specific gravity is known, it directly gives the density in g/cm³ for materials when water is taken as the reference, which is useful in converting experimental measurements into mass per unit volume for use in further calculations.

Body-Centered Cubic Arrangement

In a body-centered cubic (BCC) crystal structure, atoms are positioned at each of the eight corners of the cube with an additional atom at the center of the cube. This configuration results in a specific number of atoms per unit cell (2 atoms for BCC), which is critical for calculating the relationship between the measured density, the atomic weight, and the dimensions of the unit cell.

Transcript

00:01 So in this question, we want to identify the identity of this metal.

00:08 It has a body -centered cubic variation.

00:10 So let's draw what that would look like.

00:14 So in a bcc unit cell, we have the atoms at the corners.

00:21 And then there's going to be one atom in the center.

00:24 So within the unit cell, each of the atoms in the corner contribute one -eighth of an atom, and the one in the center contributes a whole atom.

00:33 So there are two atoms total in this unit cell.

00:37 So we know the edge length of unit cell and angstroms, and we can solve for the volume.

00:43 We know the specific gravity, which is the ratio of the density of a substance to the density of a standard, which is usually water, is what we're going to use.

00:56 So therefore, we can find the mass of the unit cells since we know density and we can get volume.

01:02 Then we're going to divide by the number of atoms to get the mass of each atom, and then convert to grams per cell.

01:07 Mole to get the atomic mass.

01:10 And then we'll use a periodic table to identify the metal.

01:13 So the density of the metal divided by density of pure water at 25 to pure celsius is equal to the ratio that is given as a specific gravity.

01:26 So that's 10 .200.

01:29 Then we are going to plug in the density of water.

01:33 So you can use the more exact value of 0 .997 grams per millimeter, or you can also try the estimated value of, 1 .00 grams per milliliter...

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