Question

By X-ray diffraction it is found that nickel (at mass \(=59 \mathrm{~g} \mathrm{~mol}^{-1}\) ), crystallizes with ccp. The edge length of the unit cell is \(3.5 \AA\). If density of Ni crystal is \(9.0 \mathrm{~g} / \mathrm{cm}^{3}\). Then value of Avogadro's number from the data is : (a) \(6.05 \times 10^{23}\) (b) \(6.11 \times 10^{23}\) (c) \(6.02 \times 10^{23}\) (d) \(6.023 \times 10^{23}\)

Short answer

Avogadro's number is approximately \(6.02 \times 10^{23}\) based on the calculation.

Step-by-step solution

Understanding the ccp structure

The ccp (Cubic Close-Packed) structure is also known as fcc (Face-Centered Cubic). In this type of arrangement, each unit cell contains 4 atoms.

Calculate the volume of the unit cell

The volume of the unit cell can be calculated using the edge length. Since the edge length of the unit cell is given as \(3.5 \text{Å}\), we convert it to cm by multiplying with \(10^{-8}\), as \(1 \text{Å} = 10^{-8} \text{cm}\). The volume, \(V\), is then \(V = a^3 = (3.5 \times 10^{-8} \text{ cm})^3\).

Calculate density using the unit cell

The density \(\rho\) of nickel can be calculated by the formula \(\rho = \frac{m}{V}\). Substituting the mass of the unit cell and its volume in this formula will allow us to solve for Avogadro's number.

Solving for Avogadro's number

Rearranging the density formula to solve for Avogadro's number gives us \(N_A = \frac{4 \times 59}{\rho \times V}\). Plug in the values for \(\rho\) (density of nickel), mass of a nickel atom, and volume of the unit cell, and solve for \(N_A\).

Key concepts

Cubic Close-Packed (ccp) structure

The cubic close-packed (ccp) structure, also commonly referred to as face-centered cubic (fcc), is one of the most efficient ways atoms, ions, or molecules can be packed in a three-dimensional space. This arrangement is characterized by layers of particles stacked over each other, with each layer offset so that the particles nestle in the gaps of the layer below.

In a ccp structure, each unit cell is imagined as a cube with atoms at each corner and in the center of each face. This means there are a total of 8 corner atoms and 6 face atoms within a cube. However, each corner atom is shared by 8 adjacent cubes and each face atom is shared by 2 cubes, so the CCP unit cell effectively contains 4 atoms (8 corners atoms × 1/8 + 6 face atoms × 1/2).

Let's apply this to a real substance, like nickel. If nickel forms a ccp crystal structure, it reflects the efficient packing and high density of atoms within the material. The ccp structure profoundly impacts the physical properties of the substance, including its density, as well as optical and electrical behaviors.

Avogadro's number calculation

Avogadro's number, symbolized as N_A, is a fundamental constant in chemistry. It represents the number of constituent particles, usually atoms or molecules, that are contained in one mole of a substance. The accepted value of Avogadro's number is approximately 6.022 × 10²³ particles per mole.

When considering a crystal structure like ccp, you can derive Avogadro's number if you know the atomic mass and density of the substance, and the volume of the crystal's unit cell. The number of atoms per unit cell in a ccp structure is 4, as established. Knowing this and the mass of the individual atoms, you can determine the mass of a mole of atoms and, subsequently, the number of atoms in a mole.

For example, with nickel, knowing the mass of one mole of nickel atoms (its molar mass) and the density of the nickel crystal, you can establish the volume of a mole of nickel atoms. With this volume, you can calculate the number of moles of nickel atoms per cubic centimeter, leading to Avogadro's number by dividing the number of atoms in the unit cell by the volume of the cell multiplied by the density.

Density and molar mass relationship

The density of a substance is an intrinsic property that indicates how much mass is contained in a given volume of the substance. In the context of a crystal such as one with a ccp structure, you can use the density to find the molar mass, which is the mass of one mole of a chemical element or chemical compound.

The relationship between density (ρ), molar mass (M), and the volume of the unit cell (V) is expressed in the formula ρ = M / (N_A × V), where N_A is Avogadro's number. Given a constant density for a substance like nickel, as the molar mass increases, the volume of the unit cell must also increase to maintain that constant density.

In practical terms, if you know the density of a substance and the volume of its unit cell, you can rearrange the formula to solve for the molar mass. This relationship is particularly important in X-ray crystallography, the method used to determine the structure of crystals, as it can provide insights into the types of atoms present and their arrangements within the crystal.