What are the three known allotropes of carbon? How do their structures differ from one another, and how are they similar?

When we talk about the diamond structure, we're referring to how carbon atoms bond together in one of the strongest known substances. In a diamond, each carbon atom forms covalent bonds with four other carbon atoms, creating a tetrahedral lattice that extends throughout the entire crystal. This intricate three-dimensional geometric arrangement contributes to the diamond's famed hardness and high melting point.

The bonds within a diamond are equally spaced and incredibly strong, accounting for its remarkable properties. These properties include a high refractive index, which is why diamonds sparkle so magnificently, and excellent thermal conductivity. Scientists also utilize diamond's hardness in industrial applications, such as cutting and drilling.

Graphite's structure is drastically different from that of diamond. Here, carbon atoms are arranged in layers of hexagonal lattices. Each carbon atom bonds covalently with three neighbors within the same plane, forming planar sheets that can slide over one another due to weak van der Waals forces between the layers.

This arrangement provides graphite with its characteristic properties: it's a good electrical conductor, it's used as a lubricant thanks to its layers that can easily slide past each other, and it's soft and dark in appearance, which makes it ideal for pencil leads. Another notable feature of graphite is that it's the most stable form of carbon under standard conditions.

In contrast to the well-organized structure of diamond and graphite, amorphous carbon does not have a definitive geometric lattice. Instead, it consists of a chaotic network of carbon atoms linked in an irregular fashion. The lack of structure in amorphous carbon, which includes materials like charcoal and soot, leads to a wide range of properties.

Amorphous carbon can be considered a mixture of microscopic regions of graphite-like and diamond-like structures, resulting in varied mechanical and electrical properties. It's often used for its absorbent qualities, as in activated carbon filters, and for its hardness, as seen in some types of industrial coatings.

The strength and variety of carbon materials come from covalent bonding, where atoms share electrons to attain stability. Carbon's ability to form four covalent bonds makes it incredibly versatile. These bonds are what hold the carbon atoms together in the different allotropes.

Due to the nature of these bonds, carbon can create complex and stable structures like the diamond's rigid lattice. In graphite, covalent bonding results in layers that have delocalized electrons, allowing for electrical conductivity. The diversity of carbon's covalent bonding patterns is a cornerstone of both materials science and organic chemistry.

The properties of carbon allotropes are widely varied and depend on how the carbon atoms are bonded together. Diamonds are extremely hard and are excellent insulators, both thermally and electrically. Graphite, however, is a good conductor of electricity and is also soft and slippery. Amorphous carbon's properties, such as hardness and electrical conductivity, vary widely depending on the exact form and structure.

Allotropes of carbon also differ in other aspects, such as their opacity, where diamond is transparent and graphite is opaque. They all share the commonality of covalently bonded carbon, which lays the basis for their extensive and unique applications.