Question

By \(x\) -ray diffraction it is possible to determine the geometric pattern in which atoms are arranged in a crystal and the distances between atoms. In a crystal of silver, four atoms effectively occupy the volume of a cube \(0.409 \mathrm{~nm}\) on an edge. Taking the density of silver to be \(10.5 \mathrm{~g} / \mathrm{cm}^{3}\), calculate the number of atoms in one mole of silver.

Short answer

Answer: There are approximately 6.009 × 10^{23} silver atoms in one mole of silver.

Step-by-step solution

Find the volume of the cube occupied by four silver atoms

We are given that four silver atoms occupy the volume of a cube with edge length 0.409 nm. To find the volume of this cube, we need to cube the edge length. Volume of a single cube = (edge length)^3 = (0.409 nm)^3 = 6.838 × 10^{-2} nm^3.

Convert the volume to cm^3

To be consistent with the units of the density, we need to convert the volume from nm^3 to cm^3. The conversion factor is 1 nm = 10^{-7} cm. 6.838 × 10^{-2} nm^3 × (10^{-7} cm/nm)^3 = 6.838 × 10^{-23} cm^3.

Find the mass of four silver atoms using density

The density of silver is given as 10.5 g/cm^3. We can use the formula of density, which is: density = mass/volume mass = density × volume mass of four silver atoms = (10.5 g/cm^3) × (6.838 × 10^{-23} cm^3) = 7.1799 × 10^{-22} g.

Find the mass of one mole of silver atoms

We need to find the mass of one mole of silver atoms, which is the molar mass. Silver is represented by the symbol Ag, and its atomic mass is approximately 107.87 amu (atomic mass units). When we convert atomic mass units to grams, we get the molar mass. Molar mass of Ag = 107.87 g/mol.

Calculate the number of silver atom sets in one mole of silver

Since four silver atoms have a mass of 7.1799 × 10^{-22} g, we can find out how many sets of four atoms are present in one mole of silver: Number of sets of four silver atoms = (molar mass of Ag) / (mass of four silver atoms) = (107.87 g/mol) / (7.1799 × 10^{-22} g) = 1.50233 × 10^{23} sets.

Calculate the number of silver atoms in one mole

Since one set contains four silver atoms, we can now find the total number of silver atoms in one mole by multiplying the number of sets by four: Number of silver atoms in one mole of silver = 4 × (1.50233 × 10^{23} sets) = 6.009 × 10^{23} silver atoms. The number of silver atoms in one mole of silver is approximately 6.009 × 10^{23}.

Key concepts

X-Ray Diffraction

X-ray diffraction is a powerful technique used to investigate the structural arrangement of atoms in crystalline materials. When X-rays are directed onto a crystal, they are diffracted at specific angles, revealing a pattern that can be analyzed to determine the crystal's atomic structure. By examining these diffraction patterns, scientists can map out how atoms are spaced in a crystal lattice. This technique is crucial for understanding the geometric arrangement of atoms in various materials, such as metals, minerals, and even biological molecules. In the case of silver, x-ray diffraction helps determine that the atoms are arranged in a specific cubic pattern, which is vital for further calculations concerning the material's properties.

Crystal Structure

The term "crystal structure" refers to the orderly and repeating pattern in which atoms are arranged in a crystalline solid. For example, in a crystal of silver, we note that four atoms effectively occupy a cubic volume. This regular and repeating pattern is what gives crystalline materials their unique properties, like strength, and also their ability to diffract X-rays predictably. A cubic crystal structure is one of the simplest forms and is widely studied because of its symmetries and mathematical simplicity in calculations. Understanding the crystal structure provides the backbone for calculating densities, masses, and other derivative properties.

Density Calculations

Density calculations are a crucial part of determining the mass and volume relationship in materials. Density is defined as mass per unit volume. Given the density of silver as 10.5 g/cm³, we can use this information to calculate the mass of a given volume. First, we find the volume occupied by silver atoms, convert it to the necessary unit (cm³ in this case), and use the formula: \( \text{mass} = \text{density} \times \text{volume} \). This approach was used to find the mass of the four silver atoms in the given volume. Being consistent with units is key, so always ensure volume and density are in compatible units when performing such calculations.

Molar Mass

Molar mass is a fundamental concept in chemistry, representing the mass of one mole of a substance. For silver (Ag), the molar mass is approximately 107.87 grams per mole. This is determined by adding up the atomic masses of all isotopes of silver as they occur naturally. By using molar mass, chemists can relate the mass of a sample of a substance to the number of atoms or molecules it contains, which ties back directly to Avogadro's number. In our silver example, by understanding the molar mass, we could find out how many sets of atoms fit within one mole, eventually leading to the number of atoms in a mole of silver. This relationship is vital for converting between mass and atoms or molecules in chemical calculations.