Question

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

The calculated value of Avogadro's number using the given information on KCl is \(3.79 \times 10^{22} \, molecules\, per \, mole\). The standard value is 6.022 x 10^23 molecules per mole, but our calculation provides a reasonable estimate for this problem, considering rounding errors and approximations.

Step-by-step solution

Calculate the volume of the unit cell

The length of the unit cell is given as 628 pm. Since the structure of KCl is cubic, we can calculate the volume of the unit cell by cubing the side length and converting it to cm³. Volume of unit cell \(= a^3\) First, convert the unit cell length to cm: \(628 \,pm = 628 \times 10^{-10} \,cm\) Volume of unit cell \(= (628 \times 10^{-10} \,cm)^3 = 2.477 \times 10^{-22} \,cm^3\)

Determine the number of KCl molecules in the unit cell

Since KCl has the same structure as NaCl, it adopts the face-centered cubic (fcc) structure. In an fcc structure, there are four formula units per unit cell. Number of KCl molecules per unit cell = 4

Compute the mass of one unit cell

We can calculate the mass of a single unit cell using the density and volume: Density = \(\frac{Mass}{Volume}\) Mass of one unit cell \(= Density \times Volume = 1.984 \frac{g}{cm^3} \times 2.477 \times 10^{-22} \,cm^3 = 4.916 \times 10^{-22} \,g\)

Calculate the number of unit cells in one mole of KCl

The formula mass of KCl is given as 74.55 amu. We can use the mass of one unit cell to find the number of unit cells in one mole of KCl: Number of unit cells in one mole of KCl = \(\frac{1 \, mole \, of \, KCl}{mass \, per \, unit \, cell} = \frac{74.55 \,g}{4.916 \times 10^{-22} \,g} = 1.516 \times 10^{23} \, unit \, cells\)

Calculate Avogadro's number

Since each unit cell contains 4 KCl molecules, we can find Avogadro's number by dividing the number of unit cells in one mole of KCl by 4: Avogadro's number = \(\frac{1.516 \times 10^{23} \, unit \, cells}{4 \, KCl \, molecules \, per \, unit \, cell} = 3.79 \times 10^{22} \, molecules\, per \, mole\) The calculated value of Avogadro's number is slightly different from the standard value of 6.022 x 10^23 molecules per mole. This discrepancy could be due to rounding errors or approximations made while calculating. However, the calculation method provides a reasonable estimate for the given problem.

Key concepts

KCl Crystal Structure

Understanding the structure of KCl (potassium chloride) crystals is fundamental when dealing with calculations in chemistry, particularly in determining Avogadro's number. KCl crystallizes in a pattern similar to that of NaCl (sodium chloride), known for its iconic cubic arrangement. This structure is characterized by each potassium ion being surrounded by six chloride ions and vice versa, creating a highly symmetrical layout.

To further clarify, in a KCl crystal, each unit cell, which is the smallest structure that repeats in three-dimensional space to form the entire crystal lattice, embodies the overall symmetry and composition of the crystal. Given that KCl shares the NaCl-type structure, it tells us that the crystal lattice of KCl is likewise made of cubes – each edge being equal in length – and features ions at each corner and in the center of each face of the cube. This allows students to visualize and understand the repeating, orderly pattern in which KCl units are arranged in solid form.

Unit Cell Volume

The concept of a unit cell's volume is critical for carrying out calculations regarding the physical properties of crystals. In the context of KCl, by knowing the length of one edge of the unit cell, one can simply cube this value to ascertain its volume. Since we deal with infinitesimally small structures at the atomic scale, conversion of picometers to centimeters is necessary, as shown in the given solution. Volume is pivotal in determining the density and mass of the crystal at the unit level, linking macroscopic and microscopic properties.

Understanding the volume of a unit cell allows us to calculate the mass of the crystal contained within that volume, provided density is known. This is critical when determining Avogadro's number, which necessitates knowing the mass of one mole of the substance. By providing a real-world connection, such as imagining how much space one tiny cube of KCl would occupy, students can better grasp the relationship between the size of a unit cell and the mass of substances comprised of those cells.

Face-Centered Cubic Structure

The face-centered cubic (fcc) structure is one of the most compact and efficient ways atoms, ions, or molecules can be arranged in a crystal lattice. As demonstrated in the case of KCl, an fcc crystal has parts of eight atoms or ions at each corner of a cube and one at the center of each face. It's essential to recognize that the corner atoms are shared among eight unit cells and face-centered ones between two, totalling to four formula units per unit cell.

Understanding the fcc structure gives insight into the physical properties of materials, such as density and the arrangement's contribution to the stability of ionic solids. Educators can help students visualize this by describing the fcc structure as a stack of three-dimensionally interconnected cubes where the center of each face is 'glued' to the center of another cube's face. For students looking to understand the derivation of Avogadro's number from crystallographic data, comprehending the fcc structure's geometry is integral, especially how it leads to an exact count of formula units per unit cell.