Problem 27 - Chemistry: An Atoms-Focused Approach

Question

Europium is a lanthanide element that is used in the phosphors in fluorescent lamps. Metallic Eu forms bcc unit cells with an edge length of \(240.6 \mathrm{pm}\) Use this information to calculate the radius of a europium atom.

Step-by-Step Solution

Answer

Answer: The radius of a europium atom is approximately 104.13 pm.

Step 1: Write down the given information and the formula for calculating the radius of an atom in a bcc lattice

We are given the edge length of the bcc unit cell, \(a = 240.6 \ \mathrm{pm}\). The formula relating the radius (\(r\)) and the edge length (\(a\)) in a bcc lattice is: \(r = \dfrac{\sqrt{3} a}{4}\)

Step 2: Calculate the radius of a europium atom

Plug in the given edge length into the formula: \(r = \dfrac{\sqrt{3}(240.6 \ \mathrm{pm})}{4}\) \(r \approx 104.13 \ \mathrm{pm}\) So, the radius of a europium atom is approximately \(104.13 \ \mathrm{pm}\).

Key Concepts

Europium

Europium is a fascinating element, belonging to the lanthanide series. It has a bright silver appearance and is known for its reactivity, especially with water and air. Although it is relatively soft compared to other metals, it finds diverse applications in several fields. This is mainly because of its unique properties.

Key uses of Europium include:

• Phosphors in television and fluorescent lamps due to its luminescent properties.
• Neutron absorber in nuclear reactors, contributing to efficiency and safety.
• Various research and industrial applications, where its unique characteristics are crucial.

Understanding europium’s positions among elements involves diving into its atomic structure, particularly in how it forms crystalline structures like the body-centered cubic (bcc) lattice. This arrangement, along with its atomic size, plays a critical role in its behavior and applications.

Atomic Radius Calculation

Calculating the atomic radius is essential in understanding the space an atom occupies. It is particularly interesting when dealing with crystalline structures like the bcc lattice. For metals forming this structure, the radius of the atom can be precisely estimated if the edge length of the cube is known.

In body-centered cubic (bcc) lattices:

• The unit cell consists of a cube where an atom sits at each corner and one atom is in the center of the cube.
• The atoms at the corners only contribute partially to the unit cell, while the middle atom is counted fully.

This unique configuration gives rise to the formula for atomic radius in a bcc lattice, described as \[ r = \dfrac{\sqrt{3} a}{4} \] where \( r \) is the atomic radius and \( a \) is the edge length of the cube. Substituting the known edge length, such as in the europium lattice with \( a = 240.6 \ \mathrm{pm} \), results in an atomic radius of approximately 104.13 \( \mathrm{pm} \). This precise calculation allows scientists and engineers to predict how europium will behave in different settings.

Lanthanide Elements

The lanthanide series consists of 15 elements, spanning from Lanthanum (\( Z = 57 \)) to Lutetium (\( Z = 71 \)). Europium is part of this group, known for filling their 4f electron orbitals.Key characteristics of lanthanide elements include:

• They exhibit similar chemical properties, often appearing together in mineral compounds.
• Commonly, these elements have high magnetic permeability, making them useful in various technological applications.
• With relatively large atomic and ionic sizes, their compounds are often vividly colored.

Europium, like its lanthanide counterparts, has special electronic configurations that contribute to its intriguing properties. These properties are crucial in its applications in electronics, medicine, and materials science. Understanding europium as part of the lanthanide series helps illustrate why these elements are indispensable in modern technology and industrial processes.