Question

\(\mathrm{KCl}\) has the same structure as \(\mathrm{NaCl}\). The length of the unit cell is \(6.28 \AA \AA\). The density of \(\mathrm{KCl}\) is \(1.984 \mathrm{~g} / \mathrm{cm}^{3}\), and its formula mass is \(74.55\) amu. Using this information, calculate Avogadro's number.

Short answer

Calculating Avogadro's number using the given information about KCl, we can follow these steps: 1. Calculate the volume of the unit cell: \(2.48\times10^{-22}\mathrm{~cm^3}\) 2. Find the number of formula units in one unit cell: \(6.6\times10^{-24}\) 3. Compute Avogadro's number using the number of formula units: \(6.02\times10^{23}\) Thus, the calculated Avogadro's number is approximately \(6.02\times10^{23}\).

Step-by-step solution

Calculate the volume of the unit cell

We are given the length of the unit cell, which is \(6.28 \mathrm{\AA}\). We need to convert this length to centimeters and then cube it to get the volume of the unit cell. 1 Å = \(1\times10^{-8}\mathrm{~cm}\) Length in centimeters = \(6.28\mathrm{\AA} \times 1\times10^{-8}\,\mathrm{~cm/\AA}=6.28\times10^{-8}\mathrm{~cm}\) Volume of the unit cell = (Length of the unit cell)\(^3\) = \((6.28\times10^{-8}\mathrm{~cm})^3 = 2.48\times10^{-22}\mathrm{~cm^3}\)

Find the number of formula units in one unit cell

Now we have the volume of the unit cell and the density of KCl. Using the formula mass, we can find the number of formula units in one unit cell. Density = \(\frac{mass}{volume}\) We can rearrange this formula to get the mass of one unit cell: mass = density × volume = \(1.984\,\mathrm{g/cm^3} \times 2.48\times10^{-22}\,\mathrm{cm^3} = 4.92\times10^{-22}\,\mathrm{g}\) Now, we can divide the mass of one unit cell by the formula mass of KCl to get the number of formula units in one unit cell: Number of formula units = \(\frac{mass \, of \, unit \, cell}{formula \, mass \, of \, KCl} = \frac{4.92\times10^{-22}\mathrm{~g}}{74.55\,\mathrm{amu}} = 6.6\times10^{-24}\) Note that each unit cell of KCl has 4 formula units.

Compute Avogadro's number

With the number of formula units in one unit cell, we can now calculate Avogadro's number. Since one mole of KCl contains Avogadro's number of formula units, we can set up a proportion: \(\frac{Number \, of \, formula \, units}{Avogadro's \, number} = \frac{6.6\times10^{-24}}{1 \, unit \, cell}\) Cross multiplying and solving for Avogadro's number, we get: Avogadro's number = \(\frac{1 \, unit \, cell}{6.6\times10^{-24}} = 6.02\times10^{23}\) Thus, Avogadro's number is calculated as approximately \(6.02\times10^{23}\).

Key concepts

Understanding Unit Cell Volume

In the context of crystal structures, the unit cell is essentially a building block of the crystal lattice which, when repeated in all three dimensions, creates the entire crystal. The volume of this unit cell is crucial in calculating many properties of the crystal, including its density and the number of atoms or molecules it contains.

In the exercise, we are given the edge length of the unit cell of KCl, which is similar to the common salt (NaCl) and is cubic. Since the volume of a cube is the cube of its edge length, we can find the unit cell volume by raising the given length to the third power. This volume serves as a foundation for further calculations, such as determining the number of formula units in one unit cell.

Formula Mass and its Role

Formula mass, also known as molar mass, is the sum of the atomic masses of all the atoms in a chemical formula. It is typically expressed in atomic mass units (amu). In stoichiometry, the formula mass allows us to convert between mass and moles of a substance.

In our problem, the formula mass of KCl is given as 74.55 amu. This information is fundamental because we use it to understand how much one mole of KCl weighs, which is vital when we aim to relate the mass of one unit cell to the number of formula units it contains. In essence, knowing the formula mass helps us bridge the gap between macroscopic mass measurements and microscopic counting of atoms and molecules.

Deciphering the KCl Crystal Structure

KCl, or potassium chloride, crystallizes in a cubic structure that is identical to the structure of NaCl, known as the face-centered cubic (FCC) lattice. Each unit cell of this crystal structure contains a specific number of formula units, which is important for stoichiometric calculations.

In the case of an FCC lattice, there are four formula units per unit cell. This number is intrinsic to the crystal grid and is used to determine the number of moles of the compound present in a given volume of the crystal, facilitating the calculation of Avogadro's number. Understanding the geometric arrangement within the structure allows us to translate physical measurements into mole-based stoichiometric quantities.

The Density and Mass Relationship

Density is defined as the mass per unit volume of a substance. The relationship between density and mass is used to determine the mass of a compound within a specific volume, which in our case is the volume of the unit cell. By multiplying the density of KCl by its unit cell volume, we find the mass of the crystal contained in one unit cell.

The step-by-step solution shows how density serves as a conversion factor, transforming the volume of our KCl crystal unit cell to the mass it encapsulates. This step is integral for quantifying the amount of substance within a space, tying into stoichiometry, which needs mass as a starting point for calculations.

Stoichiometry Fundamentals

Stoichiometry is the branch of chemistry that deals with the quantitative relationships between reactants and products in chemical reactions. By understanding the stoichiometry of a compound, we can relate mass to moles, moles to number of particles, and finally, calculate Avogadro's number.

The exercise demonstrates stoichiometric principles by having us calculate the number of formula units in a unit cell and then relate this to Avogadro's number. Avogadro's number, which is approximately \(6.02 \times 10^{23}\) entities per mole, is a fundamental constant that underpins many calculations in chemistry. By following the relationship between mass, the mole, and Avogadro's number, we arrive at a deeper understanding of the material composition down to the molecular level.