Question

The face centered cubic cell of platinum has a length of \(0.392 \mathrm{~nm}\). Calculate the density of platinum \(\left(\mathrm{g} / \mathrm{cm}^{3}\right):\) (Atomic weight : \(\mathrm{Pt}=195\) ) (a) \(20.9\) (b) \(20.4\) (c) \(19.6\) (d) \(21.5\)

Short answer

The density of platinum is approximately 21.45 g/cm³.

Step-by-step solution

Convert Cell Length to cm

Convert the given cell length from nanometers to centimeters to ensure consistency of units. There are 10^7 nanometers in a centimeter, so the conversion is as follows: 0.392 nm = 0.392 x 10^(-7) cm.

Calculate the Volume of the Unit Cell

Calculate the volume of the FCC unit cell by cubing the length of one side. Since the side length is 0.392 x 10^(-7) cm, the volume (V) is (V = (0.392 x 10^(-7) cm)^3).

Determine the Number of Atoms per Unit Cell for FCC

For a face-centered cubic (FCC) structure, there are 4 atoms per unit cell. This results from 8 corner atoms at 1/8 each and 6 face atoms at 1/2 each.

Calculate the Mass of Atoms in One Unit Cell

To find the mass, multiply the number of atoms in the unit cell by the atomic weight of platinum (Pt). The atomic mass unit (amu) weight of platinum is 195, so the mass of Pt atoms per unit cell is (4 atoms/unit cell x 195 amu/atom). However, the atomic mass unit must be converted to grams by using the fact that 1 amu = 1.66053906660 x 10^(-24) grams.

Calculate Density of Platinum

Density is mass per unit volume. Use the mass from Step 4 and the volume from Step 2 to calculate the density (\(\rho\)) in grams per cubic centimeter (\(\mathrm{g} / \mathrm{cm}^{3}\)).

Key concepts

Face Centered Cubic Unit Cell

At the heart of understanding the construction of certain materials at the atomic level is the face centered cubic (FCC) unit cell. It's one of the most common ways that atoms pack together in many metals, including precious metals like platinum. Imagine a cube where, in addition to the corners, each face of the cube hosts a single atom precisely at its center. This arrangement allows for a more efficient way for atoms to occupy space, leading to densely packed structures.

To visualize it, think of a Rubik's cube where instead of each small cube representing an atom, only the corners and the middles of each face count. These positions are significant because when FCC unit cells stack together, atoms at the corners and centers of faces are shared among adjacent unit cells, contributing to overall structural cohesiveness.

Atomic Weight

Atomic weight, or relative atomic mass, is essentially the 'weight' of an atom of a chemical element expressed in atomic mass units (amu). It's a dimensionless quantity (that is, no units) reflecting how much one atom of the element weighs relative to the mass of a carbon-12 atom, which is arbitrarily given the value of 12. For platinum (Pt), the atomic weight is 195.

This number doesn't tell you the actual mass of an atom in grams, unless you convert it using the fact that one amu corresponds to approximately 1.66053906660 x 10^{-24} grams. This figure is critical in calculating the mass of atoms when dealing with macroscopic quantities of material, as you'd need to do in a density calculation.

Unit Cell Volume

The unit cell volume is the amount of space that a single unit cell occupies, and it's a vital piece in the puzzle when calculating the density of a crystalline solid. A unit cell's volume is calculated by cubing the edge length of the unit cell. For instance, in our platinum example, where the edge length given is 0.392 nm, we must first convert this to centimeters to match common density units, and then cube the length to find the volume.

Understanding the volume of the unit cell is key as it plays directly into the calculation of density, providing the 'per unit volume' part of the 'mass per unit volume' ratio that defines density.

Atoms per Unit Cell

Knowing how many atoms are within each unit cell of a crystalline structure is crucial for calculations in chemistry and material science. In an FCC structure, each unit cell has 4 whole atoms. This is because each of the eight corner atoms is shared among eight unit cells and each of the six face-centered atoms is shared among two unit cells.

Here's the math: 8 corners x 1/8 contribution per corner atom + 6 faces x 1/2 contribution per face atom adds up to 4 atoms per unit cell. This count allows us to calculate the total mass of atoms in a unit cell when multiplied by the atomic weight, thus leading to the further determination of a material's density.