Problem 19 - Chemistry Central Science

Question

The distance between \(\mathrm{W}\) atoms in tungsten metal is \(2.74 \mathrm{~A}\). What is the atomic radius of a tungsten atom in this environment? (This radius is called the metallic radius)

Step-by-Step Solution

Answer

The metallic radius of a tungsten atom, which is half the distance between neighboring atoms in a metallic crystal lattice, can be calculated as follows: \( Metallic \, radius = \frac{Distance \, between \, neighboring \, atoms}{2} = \frac{2.74 \mathrm{~A}}{2} = 1.37 \mathrm{~A} \). Therefore, the atomic radius of a tungsten atom in this environment is 1.37 angstroms.

Step 1: Understand the concept of metallic radius.

The metallic radius is defined as half the distance between two neighboring atoms in a metallic crystal lattice. This is because the metallic bond, which holds the atoms together in the lattice, is non-directional and the atoms are considered to be touching each other.

Step 2: Calculate the distance between neighboring atoms in angstroms.

We are given that the distance between tungsten atoms in the metallic lattice is 2.74 angstroms.

Step 3: Calculate the metallic radius of a tungsten atom.

To find the metallic radius, we will simply divide the distance between neighboring atoms by 2. In this case, the metallic radius of a tungsten atom in the lattice is: \[ Metallic \, radius = \frac{Distance \, between \, neighboring \, atoms}{2} = \frac{2.74 \mathrm{~A}}{2} \]

Step 4: Find the atomic radius of a tungsten atom.

Now, we can use the formula from Step 3 to calculate the metallic radius of a tungsten atom: \[ Metallic \, radius = \frac{2.74 \mathrm{~A}}{2} = 1.37 \mathrm{~A} \] So, the metallic radius of a tungsten atom (which is also its atomic radius in this environment) is 1.37 angstroms.

Key Concepts

Atomic Radius

Understanding the atomic radius is crucial when delving into the world of chemistry and physics. It refers to the distance from the center of an atom's nucleus to the outermost boundary of its electrons. Imagine it as the 'sphere of influence' an atom has among its neighbors. In the context of a metallic radius, as in the textbook exercise with tungsten, the atomic radius represents half the distance between two adjacent atoms in a metal.

When metals bond, they form a sea of electrons that allows them to pack closely together, so the atomic radius in a metallic lattice can actually vary slightly from the radius in an isolated state. Therefore, understanding the concept of atomic radius not only answers how large an atom is but also reveals the intricacies of atomic interactions within different states of matter.

Crystal Lattice

The crystal lattice forms the backbone of solid materials, particularly metals. It is a highly ordered, repeating pattern of atoms that extends in all three spatial dimensions, providing metals with their characteristic properties, such as strength and conductivity. Tungsten, like other metals, packs its atoms in a lattice that determines how they bond and interact.

In metallic crystals, each atom is situated at a lattice point and is surrounded by other atoms, with which it shares the 'sea of electrons'. This arrangement allows metals to maintain a structure that is both stable and flexible enough to be shaped and used for various purposes. By examining a crystal lattice, scientists and engineers can predict many properties of the material, from how it might conduct electricity to its potential melting point.

Angstrom Measurement

When dealing with atoms and molecules, standard units of measure like meters and inches are immensely large and impractical. That's where the angstrom,(Å), comes into play. It is a unit typically used to express atomic and molecular dimensions. One angstrom is equivalent to one ten-billionth of a meter (10^-10 meters), which makes it ideal for articulating the exceptionally small sizes of atoms and the distances between them in a crystal lattice.

In practical terms, using angstroms provides a more comprehendible number to work with, such as the 2.74 Å distance between tungsten atoms. Having a standard unit like the angstrom allows chemists and physicists to share information and compare measurements with precision and ease, making science more interconnected and comprehensible across the globe.