Question

Argon has a cubic closest packed structure as a solid. Assuming that argon has a radius of \(190 . \mathrm{pm}\), calculate the density of solid argon.

Short answer

The density of solid argon can be calculated using the given radius and the cubic closest packed structure formula. The steps are as follows: 1. Calculate the edge length of the unit cell: \(a = 2\sqrt2 (190 \, \text{pm})\). 2. Calculate the volume of the unit cell: \(V = (2\sqrt2 (190 \, \text{pm}))^3\). 3. Determine the number of argon atoms in one unit cell: 4 atoms. 4. Calculate the mass of one unit cell: \(m = \frac{4 \times 39.95 \, \text{g/mol}}{6.022 \times 10^{23} \, \text{atoms/mol}}\). 5. Calculate the density of solid argon: \(\rho = \frac{4 \times 39.95 \, \text{g/mol}}{(6.022 \times 10^{23} \, \text{atoms/mol}) \times (2\sqrt2 (1.90 \times 10^{-8} \, \text{cm}))^3}\). After evaluating the final expression, we obtain the density of solid argon in g/cm³.

Step-by-step solution

Calculate the Edge Length of the Unit Cell

In a cubic closest packed structure, the edge length (a) can be determined using the radius (r) of the atoms by the following formula: \[a = 2\sqrt2 r\] Given the radius of argon atom is 190 pm, we will substitute this value in the above formula and calculate the edge length: \[a = 2\sqrt2 (190 \, \text{pm})\]

Calculate the Volume of the Unit Cell

Now that we have the edge length of the unit cell, we can determine the volume (V) using the formula for the volume of a cube: \[V = a^3\] Substitute the value of edge length obtained in step 1: \[V = (2\sqrt2 (190 \, \text{pm}))^3\]

Calculate the Number of Argon Atoms in One Unit Cell

In a ccp structure, each unit cell contains 4 whole atoms. Therefore, there are 4 argon atoms in one unit cell.

Calculate the Mass of One Unit Cell

To calculate the mass of one unit cell, first, we need to find the molar mass of argon. The molar mass of argon is approximately 39.95 g/mol. Since we have 4 argon atoms in one unit cell, the mass of one unit cell (m) can be calculated using the following formula: \[m = \frac{\text{number of atoms} \times \text{molar mass}}{\text{Avogadro's number}}\] \[m = \frac{4 \times 39.95 \, \text{g/mol}}{6.022 \times 10^{23} \, \text{atoms/mol}}\]

Calculate the Density of Solid Argon

Now that we have the mass and volume of one unit cell, we can calculate the density (ρ) of solid argon using the following formula: \[\rho = \frac{m}{V}\] Substitute the values of mass and volume obtained in step 4 and step 2, respectively: \[\rho = \frac{4 \times 39.95 \, \text{g/mol}}{(6.022 \times 10^{23} \, \text{atoms/mol}) \times (2\sqrt2 (190 \, \text{pm}))^3}\] Don't forget to convert pm to cm while calculating the density in g/cm³: \[\rho = \frac{4 \times 39.95 \, \text{g/mol}}{(6.022 \times 10^{23} \, \text{atoms/mol}) \times (2\sqrt2 (1.90 \times 10^{-8} \, \text{cm}))^3}\] After calculating the above expression, we will get the density of solid argon.

Key concepts

Cubic Closest Packed Structure

In crystallography, one common arrangement of atoms is the cubic closest packed structure (ccp). It is a highly efficient arrangement that minimizes empty space between atoms, allowing for a higher density. This structure is also known as face-centered cubic (fcc) and is characterized by atoms being present at each corner and the center of each face of the cube.
Each unit cell in a ccp structure contains the equivalent of 4 whole atoms. This is an important factor when calculating properties like density.
Materials with ccp structures generally have a high packing efficiency, around 74%. This makes them very compact and strong, properties that are important for many materials found in both nature and industry.

Argon

Argon is a noble gas, meaning it is typically chemically non-reactive due to its full valence electron shell. As a solid, argon adopts the cubic closest packed structure (ccp), benefiting from its efficient packing.
Argon atoms are spherical under this arrangement and maintain a set distance from each other. This distance is characterized by the atomic radius, which for argon is approximately 190 picometers (pm).
This radius is significant when calculating density, as it helps determine the dimensions of the unit cell, a cube formed by the arrangement of its atoms. By understanding these atomic arrangements, scientists can predict physical properties like density, melting points, and more.

Unit Cell

In solid-state chemistry, a unit cell is the smallest repeating unit of a crystal lattice. It acts as the building block for the entire structure, with its repeat creating the crystal.
For a cubic closest packed structure, the unit cell is a cube. The length of its edges, or edge length, can be calculated if you know the atomic radius. The formula to find the edge length in a ccp structure is given by: \[ a = 2\sqrt{2} \times r \] where \( r \) is the atomic radius.
The volume of the unit cell, which is essential for density calculations, is simply the cube of the edge length: \[ V = a^3 \]
Knowing the unit cell's volume and the number of atoms it contains allows for precise calculation of material density.

Molar Mass

Molar mass is the mass of one mole of a given substance, often expressed in grams per mole (g/mol). It is a critical factor in chemistry for converting between mass and amount of substance.
For argon, the molar mass is about 39.95 g/mol. This value is crucial when calculating the mass of substances at the atomic level in problems involving solid-phase argon.
In density calculations, knowing the molar mass allows you to determine the mass of the atoms within a unit cell. Combined with Avogadro's number, which is the number of atoms in a mole (\[ 6.022 \times 10^{23} \text{ atoms/mol} \]), you can calculate the mass contained in a small, measurable volume of the substance.