Question

Clausthalite is a mineral composed of lead selenide (PbSe). The mineral adopts a NaCl-type structure. The density of PbSe at \(25^{\circ} \mathrm{C}\) is \(8.27 \mathrm{~g} / \mathrm{cm}^{3}\). Calculate the length of an edge of the PbSe unit cell.

Short answer

The edge length of the PbSe unit cell is approximately 3.943 x 10⁻⁸ cm.

Step-by-step solution

Calculate the molar mass of PbSe

First, we need to find out the molar masses of Pb and Se, then sum them to obtain the molar mass of PbSe. - Molar mass of Pb = 207.2 g/mol - Molar mass of Se = 78.971 g/mol Now, let's add the molar masses: M = M_Pb + M_Se = 207.2 g/mol + 78.971 g/mol = 286.171 g/mol So, the molar mass of PbSe is 286.171 g/mol.

Use the formula for density and solve for the unit cell edge length (L)

Now we will use the relationship between the density, edge length, and the molar mass of the mineral to find the unit cell edge length: \( density = \frac{(n_a * M)}{(L^3 * N_A)} \) We know that: - density = 8.27 g/cm³, - \(n_a\) = 4 (since it's an fcc lattice), - M = 286.171 g/mol, - \(N_A\) = 6.022 x 10²³ mol⁻¹. Now, we solve the equation for L: \( L^3 = \frac{(n_a * M)}{(density * N_A)} \) Then, we plug in the values: \( L^3 = \frac{(4 * 286.171)}{(8.27 * 6.022 * 10^{23})} \) Calculate L³: L³ = 6.137 x 10⁻²³ cm³ Next, we'll calculate the value for L by taking the cube root of L³: L = (6.137 x 10⁻²³)^(1/3) = 3.943 x 10⁻⁸ cm So the edge length of the PbSe unit cell is approximately 3.943 x 10⁻⁸ cm.

Key concepts

NaCl-type structure

The NaCl-type structure is a commonly found crystalline arrangement in minerals and compounds. In this structure, each sodium (Na) ion is surrounded by six chlorine (Cl) ions and vice versa. This creates a highly symmetrical, cubic lattice where each ion assists in forming an octahedral shape around its counterparts. This property significantly influences the stability and properties of the compound. The structure can be visualized as two interpenetrating face-centered cubic (fcc) sub-lattices. One is entirely sodium, and the other entirely chlorine. Both are offset, forming a shared three-dimensional grid. This type of arrangement is found in many binary compounds like lead selenide (PbSe), where lead and selenium take the respective positions of sodium and chlorine, maintaining the structural integrity found in NaCl.

molar mass calculation

A molar mass calculation is essential for determining the mass of one mole of a compound or element. For a compound like PbSe, it's important first to find the molar masses of the constituent elements. Lead (Pb) has a molar mass of 207.2 g/mol, while selenium (Se) is 78.971 g/mol. To find the molar mass of PbSe, we simply add the molar masses of these two elements together:

• Molar mass of PbSe = Molar mass of Pb + Molar mass of Se

Thus, the molar mass = 207.2 g/mol + 78.971 g/mol = 286.171 g/mol.
This calculation provides the necessary foundation for further calculations related to the compound like density and molecular quantities.

density formula

The density of a material is always a key parameter to understand when studying materials. It is defined as the mass per unit volume. In this context, it's vital in finding the dimensions of the unit cell of a compound. The density formula can be rearranged and used in conjunction with other properties to solve for unknown parameters. Here’s an important equation rewritten to find the unit cell's edge length:

• \[ L^3 = \frac{(n_a * M)}{(density * N_A)} \]

In this formula:

• \( n_a \) is the number of atoms per unit cell, which equals 4 in an fcc lattice.
• \( M \) is the molar mass of the material.
• \( N_A \) is Avogadro's number (\(6.022 \times 10^{23} \text{mol}^{-1}\)).

This formula's application helps us trace back from known densities to dimensions of the unit cell.

fcc lattice

The fcc (face-centered cubic) lattice is a particular arrangement that maximizes efficiency in packing. This structure forms when atoms occupy the corners and the centers of each face of the cube. It's an efficient structure that leads to a high packing density, making it common in metallic and ionic crystals. An fcc lattice is notable in crystals like NaCl, where each unit cell contains four atoms entirely within its structure. This is calculated considering the contribution from corner atoms (each contributing \(1/8\)th of their volume) and face-centered atoms (each contributing \(1/2\) their volume). Understanding this packing sets the stage for further exploration of more complex properties and behaviors, such as those seen in lead selenide's crystalline structure.