Question

\(\mathrm{KCl}\) has the same structure as \(\mathrm{NaCl}\). The length of the unit cell is \(628 \mathrm{pm}\). 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

Avogadro's number for KCl, calculated using the provided information, is approximately \(6.074 × 10^{23}\mathrm{mol^{-1}}\).

Step-by-step solution

Find the number of KCl units in one unit cell

Since KCl has the same structure as NaCl, it has a face-centered cubic (fcc) structure. In an fcc structure, there are 4 formula units in one unit cell. So, there are 4 KCl units in one unit cell.

Calculate the volume of one unit cell

The length of the unit cell is given as 628 pm, which must be converted to cm. Recall that 1 pm = 1 × 10⁻¹² m and 1 m = 100 cm. So, 1 pm = 1 × 10⁻¹² × 100 cm = 1 × 10⁻¹⁰ cm. Thus, the length of the unit cell in cm is 628 × 10⁻¹⁰ cm. Now we can find the volume (V) of one unit cell: \[V = (\text{length})^3 = (628 × 10^{-10} \mathrm{cm})^3 = 2.474 × 10^{-24}\mathrm{cm^3}\]

Calculate the total mass of one unit cell

The density of KCl is given as \(1.984\mathrm{~g/cm^3}\). We can use this density along with the volume found in step 2 to find the total mass (M) of KCl in one unit cell: \[M = \text{density} × \text{volume} = 1.984 \mathrm{g/cm^3} × 2.474 × 10^{-24}\mathrm{cm^3} = 4.913 × 10^{-24}\mathrm{g}\]

Calculate the mass of one KCl unit

Recall that there are 4 KCl units in one unit cell. We can find the mass of one KCl unit by dividing the total mass of one unit cell by the number of KCl units per unit cell: \[\text{mass of one KCl unit} = \dfrac{\text{total mass of one unit cell}}{\text{number of KCl units per unit cell}} = \dfrac{4.913 × 10^{-24}\mathrm{g}}{4\text{ KCl units}} = 1.228 × 10^{-24}\mathrm{g}\]

Calculate Avogadro's number

To find Avogadro's number, we need to find the mass of one mole of KCl, which is the formula mass given as 74.55 amu. This formula mass can be converted to grams by multiplying with the conversion factor (1 mol = 1 g/amu): \[\text{mass of one mole of KCl} = 74.55\text{ amu} × \dfrac{1\text{ g}}{1\text{ amu}} = 74.55\mathrm{g/mol}\] Now, we can find Avogadro's number (N_A) by dividing the mass of one mole of KCl by the mass of one KCl unit: \[N_A = \dfrac{\text{mass of one mole of KCl}}{\text{mass of one KCl unit}} = \dfrac{74.55\mathrm{g/mol}}{1.228 × 10^{-24}\mathrm{g}} = 6.074 × 10^{23}\mathrm{mol^{-1}}\] So, Avogadro's number is approximately \(6.074 × 10^{23}\mathrm{mol^{-1}}\).

Key concepts

Crystal Structure

Understanding the crystal structure of a substance like potassium chloride (KCl) is essential for various calculations in chemistry. The crystal structure relates to the orderly arrangement of atoms in a crystalline solid. Different substances can crystallize in different patterns or structures, such as cubic, tetragonal, orthorhombic, and more. KCl, like NaCl (table salt), crystallizes in a face-centered cubic (fcc) structure. This configuration significantly influences the properties of the crystal, including its density and how light or electricity passes through it.

Knowing KCl has an fcc structure allows us to determine that there are exactly four formula units (pairs of K+ and Cl- ions) within one unit cell of the crystal. These units repeat in a specific pattern in three-dimensional space, forming the solid crystal. The arrangement is not just a visual depiction but also a mathematical model that plays a fundamental role in calculations like density and Avogadro's number.

Density

The density of a substance is a measure of its mass per unit volume. It's a fundamental property that can be used to identify substances and to understand their structure and composition. The density formula is expressed as \( \text{density} = \frac{\text{mass}}{\text{volume}} \).

For crystalline solids like KCl, density not only accounts for the mass of the atoms but also the way they're packed within the crystal structure. If we know the density of KCl and the volume of the KCl's unit cell, we can calculate the mass of the unit cell. This mass relates directly to the amount of matter within the cell and is used as a stepping stone to determine Avogadro's number, which is a fundamental constant expressing the number of particles in one mole of a substance.

Unit Cell Volume

The unit cell volume represents the volume occupied by the smallest repeating unit—that is, the unit cell—of a crystal. This volume can be calculated if the length of the edges of the unit cell is known. For cubic cells like those of KCl, the volume is simply the cube of the cell's edge length (V = edge length^3).

In our exercise, we took the given edge length for KCl, which is quantified in picometers (pm), and first converted it to centimeters (cm) to ensure consistency with the density measurement. Once we have that volume, it can be used along with the substance's density to find the mass contained within a single unit cell. This is vital for further calculations, such as finding Avogadro's number.

Molar Mass

The molar mass, measured in grams per mole (g/mol), corresponds to the mass of 1 mole of a given substance. One mole contains Avogadro's number of particles, whether they're atoms, ions, molecules, or other entities. In the case of KCl, the molar mass tells us the combined mass of one mole of potassium ions (K+) and one mole of chloride ions (Cl-).

Calculating the molar mass is crucial for converting between the mass of a substance and the amount (in moles) of particles it contains. It serves as the bridge between the macroscopic world we can measure and the microscopic world of atoms and molecules. Once we determine the mass of an individual KCl unit from the unit cell, we can use the molar mass to figure out Avogadro's number by comparing the mass of a single unit to the mass of a mole of units.