Question

\(\mathrm{CsCl}\) has bce structure with \(\mathrm{Cs}^{+}\) at the centre and \(\mathrm{Cl}^{-}\) ion at each corner. If \(r_{\mathrm{Cs}^{+}}\) is \(1.69 \AA\) and \(r_{\mathrm{Cl}}\) is \(1.81 \AA\), what is the edge length of the cube? (a) \(3.50 \AA\) (b) \(3.80 \AA\) (c) \(4.04 \AA\) (d) \(4.50 \AA\)

Short answer

The edge length of the cube is approximately \(4.04 \AA\), which corresponds to option (c).

Step-by-step solution

Understand the Crystal Structure

Cesium chloride (CsCl) has a body-centered cubic (bcc) structure. This means that a Cs+ ion is at the center of the cube and a Cl- ion is at each corner of the cube. The edge length of the cube is the distance between two adjacent corners.

Calculating the Edge Length of the Cube

In a bcc structure, the body-centered ion touches the corner ions along the body diagonal of the cube. The body diagonal's length is four times the ion radius for a bcc structure because the central ion is in contact with the corner ions. The body diagonal d can be calculated using the Pythagorean theorem in three dimensions, where d is the diagonal, and a is the edge length: \(d = \sqrt{3}a\).

Finding the Body Diagonal Length

The body diagonal is also equal to the sum of the radii of the cesium and chlorine ions along the diagonal, considering there is one Cl- at each corner and one Cs+ at the center. Thus, \(d = 2r_{\mathrm{Cs}^{+}} + 2r_{\mathrm{Cl}^{-}} = 2(1.69 \AA) + 2(1.81 \AA) = 3.38 \AA + 3.62 \AA = 7.00 \AA\).

Calculating the Edge Length

We can find the edge length a by rearranging the body diagonal equation: \(a = \frac{d}{\sqrt{3}}\). Substituting the value of d, we get \(a = \frac{7.00 \AA}{\sqrt{3}} \approx 4.04 \AA\).

Key concepts

Understanding the Body-Centered Cubic Structure

In the exploration of crystal structures, it's fundamental to understand the body-centered cubic (bcc) layout. This specific arrangement is characterized by an atom positioned at the center of a cube with additional atoms at each of the eight corners. The positioning within a bcc crystal lattice is pivotal when it comes to properties like density and packing efficiency, which are influential factors for the material's physical characteristics.

When visualizing the bcc structure, it's useful to imagine a 3D chessboard where one piece is confidently placed in the middle of the board, representing the body-centered ion, and pawns occupy the corners of the board. This setup ensures direct contact along the body diagonal, a line stretching from one corner of the cube, passing through the center, and reaching the opposite corner. The body diagonal is crucial as it's the key to unlocking the cube's edge length, which is a vital dimension for calculations in solid-state physics and chemistry.

Crystal Lattice Calculations

Crystal lattice calculations are at the heart of understanding the structure-property relationship in materials science. To dissect the intricacies of a crystal like CsCl, one employs geometrical and mathematical tools. Knowing that the cesium ion lies at the bcc's heart, and chloride ions fill the corners, the task is to understand how these ions are positioned relative to one another.

Diving deeper into the geometry, the Pythagorean theorem paves the way for calculating the body diagonal in three dimensions, exploiting the inherent symmetry of the cube. This method hinges on the relationship between the cube's edge length and its diagonal, giving rise to the formula involving the square root of three, a reflection of the cube's three equal dimensions. Mastery of these calculations not only reveals the edge length but also offers a window into other properties such as the crystal's density and ionic packing efficiency.

Ionic Radii in Crystal Structures

The sizes of ions, commonly referred to as ionic radii, play a monumental role in determining the architecture of crystal structures. In the CsCl lattice, the ionic radii of cesium and chloride are like puzzle pieces that fit perfectly within the space of the cube. The ionic radii govern how closely ions can pack together, influencing not only the crystal's stability but also its bulk properties, such as melting point and hardness.

In practice, to calculate the edge length of a cube from ionic radii, one must consider the line connecting the ions along the cube's body diagonal. For body-centered cubic structures, this length is twice the sum of the radii of the central and corner ions. Such precision in measuring ionic radii is conclusive to deducing the correct spacing between ions, thereby enabling material scientists to predict a multitude of crystal characteristics with accuracy.