Question

The density of argon (face centered cubic cell) is \(1.83 \mathrm{~g} / \mathrm{cm}^{3}\) at \(20^{\circ} \mathrm{C}\). What is the length of an edge a unit cell? (Atomic weight: \(\mathrm{Ar}=40\) ) (a) \(0.599 \mathrm{~nm}\) (b) \(0.569 \mathrm{~nm}\) (c) \(0.525 \mathrm{~nm}\) (d) \(0.551 \mathrm{~nm}\)

Short answer

The length of an edge of a unit cell is approx. 0.525 nm, so the answer is (c) 0.525 nm.

Step-by-step solution

- Recall the formula for density and the relationship with unit cell

The density of a crystal can be calculated using the formula: \(\rho = \frac{z \times M}{a^3 \times N_A}\), where \(\rho\) is the density, \(z\) is the number of atoms per unit cell, \(M\) is the molar mass, \(a\) is the edge length of the unit cell, and \(N_A\) is Avogadro's number. For a face-centered cubic (FCC) cell, \(z = 4\) because there are 4 atoms per unit cell (1/8th of an atom at each corner and 1/2 of an atom at each face).

- Calculate the volume of the unit cell

Re-arrange the density formula to solve for \(a^3\): \(a^3 = \frac{z \times M}{\rho \times N_A}\). Plugging in the known values, \(a^3 = \frac{4 \times 40 \mathrm{~g/mol}}{1.83 \mathrm{~g/cm}^3 \times 6.022 \times 10^{23} \mathrm{~atoms/mol}}\). Note that \(M\) for argon is 40 g/mol.

- Convert molar mass and density to the same units

Make sure that the molar mass and density are in the same units so that they will cancel out properly. Here, the molar mass is given in grams per mole and density in grams per cubic centimeter. No conversion is necessary.

- Calculate the edge length

Carry out the calculation: \(a^3 = \frac{4 \times 40}{1.83 \times 6.022 \times 10^{23}} = 1.539 \times 10^{-22} \mathrm{~cm}^3\), and then take the cube root to find \(a\): \(a = \sqrt[3]{1.539 \times 10^{-22}}\).

- Convert Units (if necessary)

After calculating \(a\), the result is in centimeters. If necessary, convert the result to nanometers by multiplying by \(10^7\) (since 1 cm = 10^7 nm).

- Match the calculated edge length with the options given

Compare the calculated edge length in nanometers with the given options to find the correct answer.

Key concepts

Face Centered Cubic Cell

Understanding the structure of a face centered cubic (FCC) cell is crucial when it comes to calculating properties like the density of a crystal. In simple terms, a face centered cubic cell is a type of unit cell, a small repeating unit that makes up the crystal lattice. An FCC unit cell is characterized by atoms located at each of the corners of a cube and at the centers of each face of the cube.

When we say an atom is at a corner, it's actually shared among eight surrounding unit cells, which means that each corner atom contributes only an eighth of an atom to the unit cell in question. Likewise, an atom at the center of a face is shared between two unit cells, contributing half an atom to each. Combining these contributions, an FCC cell contains a total of 4 atoms (from eight corners giving 1 atom and six faces giving 3 atoms).

When solving problems involving the calculation of a crystal's density or edge length of an FCC cell, it's important to know the number of atoms per unit cell in order to correctly apply the density formula.

Molar Mass

The molar mass of a chemical element is fundamental when performing calculations in chemistry and physics. Defined as the mass of one mole of that element, the molar mass typically is expressed in grams per mole (g/mol). It represents the mass of Avogadro's number worth of atoms of the substance.

Molar mass can be found on the periodic table as the atomic weight for each element. For instance, in the case of argon (Ar), the atomic weight, and thus the molar mass, is approximately 40 g/mol. Knowing the molar mass allows us to translate between the mass of a substance and the quantity of atoms or molecules it contains, which is especially useful in density calculations of crystals where we consider the number of moles contained within a unit cell.

Avogadro's Number

Avogadro's number, symbolized by N_A, is a pivotal constant in chemistry. It is defined as the number of constituent particles, usually atoms or molecules, that are contained in one mole of a substance. Its exact value is 6.022 x 10²³ particles per mole.

This incredible number allows chemists and physicists to move between the macroscopic world that we can see and the microscopic world of atoms and molecules. By using Avogadro's number, we can calculate the number of atoms in a given molar mass of a substance; it is the bridge between the molar mass (in grams) and the actual number of atoms present. When calculating the density of a crystal, Avogadro's number is crucial because it lets us work with the number of atoms in each unit cell to find the mass of a given volume of the crystal.