Spherical Aromaticity of Inorganic Cage Molecules

Spherical coordinate systems are preferred over Cartesian, polar, or cylindrical coordinates for systems with spherical symmetry. For example, to describe the surface of a sphere, Cartesian coordinates require all three coordinates. On the other hand, the spherical coordinate system requires only one parameter: the sphere's radius. As a result, the complicated mathematical calculations become simple. Spherical coordinates are used in science and engineering applications like electric and...

The thoracic or rib cage forms the body's thorax (chest) portion. Its primary function in the body is to protect vital organs in the thoracic cavity, such as the heart and the lungs. It consists of 12 pairs of ribs with their costal cartilages and the sternum. The ribs are anchored posteriorly to the 12 thoracic vertebrae (T1-T12).
The sternum is the elongated bony structure on the anterior side of the thoracic cage. It consists of three parts: the manubrium, the body, and the xiphoid...

Ribs are curved, flattened bones forming the thoracic cavity wall with the thoracic muscles. There are 12 pairs of thoracic ribs. The posterior ends of all the ribs articulate with the T1–T12 thoracic vertebrae. In contrast,the anterior ends of most ribs attach to the sternum via their costal cartilages.
Parts of a Typical Rib
A typical rib has a head, neck, and body. The posterior end of the rib is called the head, followed by a narrow neck. The head articulates primarily with the costal...

Newton's law of gravitation describes the gravitational force between any two point masses. However, for extended spherical objects like the Earth, the Moon, and other planets, the law holds with an assumption that masses of spherical objects are concentrated at their respective centers.
This assumption can be proved easily by showing that the expression for gravitational potential energy between a hollow sphere of mass (M) and a point mass (m) is the same as it would be for a pair of extended...

A spherical capacitor consists of two concentric conducting spherical shells of radii R1 (inner shell) and R2 (outer shell). The shells have  equal and opposite charges of +Q and −Q, respectively. For an isolated conducting spherical capacitor, the radius of the outer shell can be considered to be infinite.
Conventionally, considering the  symmetry, the electric field between the concentric shells of a spherical capacitor is directed radially outward. The magnitude of the field,...

Treating arylamines with nitrous acid gives aryldiazonium salts that are effective substrates in nucleophilic aromatic substitution reactions. The diazonio group in these salts can be easily displaced by different nucleophiles, yielding a wide variety of substituted benzenes. The leaving group departs as nitrogen gas, and this easy elimination is the driving force for the substitution reaction.
In the Sandmeyer reaction, for example, the diazonio group is replaced by a chloro, bromo,...