Problem 21 - Chemistry The Central Science In Si Units

Question

Tungsten has the highest melting point of any metal in the periodic table: \(3422^{\circ} \mathrm{C}\). The distance between \(\mathrm{W}\) atoms in tungsten metal is \(274 \mathrm{pm}\). (a) What is the atomic radius of a tungsten atom in this environment? (This radius is called the metallic radius.) (b) If you put tungsten metal under high pressure, predict what would happen to the distance between \(\mathrm{W}\) atoms.

Step-by-Step Solution

Answer

The atomic radius of tungsten is 137 pm, and high pressure would decrease the distance between tungsten atoms.

Step 1: Understand the Structure

Tungsten atoms in a metallic structure are closely packed, typically in a simple cubic, body-centered cubic (bcc), or face-centered cubic (fcc) lattice. In this problem, assume a body-centered cubic structure, which is common for tungsten.

Step 2: Calculate Atomic Radius in bcc

In a body-centered cubic (bcc) lattice, each corner atom touches the atom in the body center. The body diagonal for the cube in a bcc lattice is equal to \(4r\), where \(r\) is the atomic radius. The relation for the body diagonal \(d\) can also be represented in terms of the lattice constant \(a\) as \[d=\sqrt{3}a\]. The distance between two \mathrm{W} atoms is given as 274 pm, which corresponds to half of the body diagonal (d/2). Since \(d = 4r\), we have \[2r = 274 pm\Rightarrow r = rac{274}{2} = 137 pm\].

Step 3: Predict Effect of High Pressure

When a metal like tungsten is subjected to high pressure, the atoms are forced closer together, reducing the interatomic distance. Thus, the distance between tungsten atoms will decrease under high pressure.

Key Concepts

Metallic Radius

The metallic radius is a term used to describe the radius of an atom when it is part of a metallic lattice. It is a measure of how far the outer layer of one atom is from the outer layer of an adjacent atom in a metal. This is slightly different from the atomic radius, as it takes the packing of atoms in metals into account. In metals, atoms are closely packed in a crystalline structure, and the metallic radius is typically half the distance between two adjacent atoms. When calculating the metallic radius within a body-centered cubic lattice, as is the case for tungsten, we assume that each atom is in contact with its neighbor across the diagonal of the cube. Understanding the concept of metallic radius helps explain various properties of metals, such as their malleability and electrical conductivity.

Body-Centered Cubic Lattice

A body-centered cubic (bcc) lattice is one of the most common crystalline structures found in metals. In this structure, each unit cell consists of an atom at each corner of a cube and another atom at the very center. This arrangement allows a higher packing density compared to simple cubic structures, which contributes to the strength and durability found in some metals.For tungsten, which crystallizes in a bcc lattice, each corner atom in the cube touches the center atom via the body diagonal. The relationship between the body diagonal and the atomic radius is represented by the equation \(d = 4r\), where \(d\) is the body diagonal, and \(r\) is the atomic radius. This configuration is responsible for tungsten's notable properties like its high melting point and hardness. As arguments about tungsten illustrated, finding the atomic radius involves geometric reasoning based on the structure of the bcc lattice itself.

Atomic Radius

The atomic radius is a measure of the size of an atom, usually defined as the distance from the center of the nucleus to the outermost electron cloud. It is a key parameter in understanding the chemical and physical properties of elements. In different contexts, this definition might slightly change. For example, in a metallic or crystalline environment, the atomic radius is often considered as half the distance between nuclei of two adjacent atoms. This becomes particularly important in structures like the body-centered cubic (bcc) lattice, where the spatial arrangement of atoms dictates the effective radius. When pressure is applied to metals, such as tungsten, the interatomic distances decrease. This compression might affect the perceived atomic radius as atoms move closer to one another, influencing many material properties, including electrical conductivity and hardness. Understanding how atomic radius changes under different conditions allows scientists and engineers to predict material behaviors in various applications.