(a) Derive planar density expressions for \(\mathrm{BCC}(100)\) and (110) planes in terms of the atomic radius \(R.\) (b) Compute and compare planar density values for these same two planes for molybdenum.

Understanding planar density is crucial in materials science as it relates to the arrangement of atoms on a crystal plane. Planar density is defined as the number of atoms per unit area on a specific crystallographic plane.

For calculations, the process usually starts with identifying the total number of atoms associated with the plane in concern - considering the entire geometry of the crystal lattice and how atoms are shared across unit cells. In a BCC structure, this requires a unique approach as atoms are located at the corners and the center of the cube.

Next, the side lengths of the unit cell on that plane must be determined to calculate the area. These lengths are directly related to the atomic radius. With these values, the area of the plane can be found and subsequently, the planar density can be expressed as an equation in terms of the atomic radius, reflecting the specific crystal orientation (like BCC(100) or BCC(110)).

This leads to the final step where the planar density for a specific material is computed by inserting the atomic radius into the derived formula. This numerical outcome gives a clear comparison between different planes regarding how densely packed they are with atoms.

The body-centered cubic (BCC) is one of the several crystal structures metals can adopt. In this lattice type, each unit cell is defined by eight corner atoms and a single atom at the very center of the cube.

When exploring the BCC structure, it is important to focus on attributes like coordination number, which is the number of nearest neighbors to an atom; for BCC, it's eight. Moreover, the atomic packing factor (APF) - measuring the fraction of space taken by atoms in a unit cell - can also be insightful. The BCC structure is less densely packed than its close-packed counterparts.

Various planes within the BCC structure, like the (100) and (110) planes in question, have different geometries and, therefore, different planar densities. The atoms' arrangement on these planes significantly influences material properties such as slip and hardness, which are pivotal in defining a material's mechanical behavior.

Molybdenum is a transition metal with unique properties that allow it to have a wide range of applications, especially in alloys. With its high melting point, molybdenum is ideal for high-temperature environments.

From an atomic perspective, molybdenum atoms pack in a BCC crystalline structure, leading to distinctive mechanical and chemical properties. When considering molybdenum for planar density calculations, its atomic radius of 137 picometers is essential. This measurement plays a key role when plugging into formulas derived from the BCC structure to calculate planar densities for various planes.

Understanding these properties of molybdenum enables materials scientists and engineers to predict how the substance behaves under various conditions and to tailor its use in applications ranging from electronics to aerospace components, where robust and high-performance materials are requisite.