Combinatorial principlesIn proving results in combinatorics several useful combinatorial rules or combinatorial principles are commonly recognized and used. The rule of sum, rule of product, and inclusion–exclusion principle are often used for enumerative purposes. Bijective proofs are utilized to demonstrate that two sets have the same number of elements. The pigeonhole principle often ascertains the existence of something or is used to determine the minimum or maximum number of something in a discrete context.
Allotropes of carbonCarbon is capable of forming many allotropes (structurally different forms of the same element) due to its valency. Well-known forms of carbon include diamond and graphite. In recent decades, many more allotropes have been discovered and researched, including ball shapes such as buckminsterfullerene and sheets such as graphene. Larger-scale structures of carbon include nanotubes, nanobuds and nanoribbons. Other unusual forms of carbon exist at very high temperatures or extreme pressures.
Carbon–hydrogen bondIn chemistry, the carbon-hydrogen bond ( bond) is a chemical bond between carbon and hydrogen atoms that can be found in many organic compounds. This bond is a covalent, single bond, meaning that carbon shares its outer valence electrons with up to four hydrogens. This completes both of their outer shells, making them stable. Carbon–hydrogen bonds have a bond length of about 1.09 Å (1.09 × 10−10 m) and a bond energy of about 413 kJ/mol (see table below). Using Pauling's scale—C (2.55) and H (2.
OrganocatalysisIn organic chemistry, organocatalysis is a form of catalysis in which the rate of a chemical reaction is increased by an organic catalyst. This "organocatalyst" consists of carbon, hydrogen, sulfur and other nonmetal elements found in organic compounds. Because of their similarity in composition and description, they are often mistaken as a misnomer for enzymes due to their comparable effects on reaction rates and forms of catalysis involved.
Knoevenagel condensationIn organic chemistry, the Knoevenagel condensation (ˈknøːvənaːɡl̩) reaction is a type of chemical reaction named after German chemist Emil Knoevenagel. It is a modification of the aldol condensation. A Knoevenagel condensation is a nucleophilic addition of an active hydrogen compound to a carbonyl group followed by a dehydration reaction in which a molecule of water is eliminated (hence condensation). The product is often an α,β-unsaturated ketone (a conjugated enone). In this reaction the carbonyl group is an aldehyde or a ketone.
Reactive oxygen speciesIn chemistry, reactive oxygen species (ROS) are highly reactive chemicals formed from diatomic oxygen (). Examples of ROS include peroxides, superoxide, hydroxyl radical, singlet oxygen, and alpha-oxygen. The reduction of molecular oxygen () produces superoxide (), which is the precursor to most other reactive oxygen species: O2{} + e^- -> \ ^\bullet O2- Dismutation of superoxide produces hydrogen peroxide (): 2 H+{} + 2 \ ^\bullet O2^-{} -> H2O2{} + O2 Hydrogen peroxide in turn may be partially reduced, thus forming hydroxide ions and hydroxyl radicals (), or fully reduced to water: H2O2{} + e^- -> HO^-{} + \ ^\bullet OH 2 H+ + 2 e- + H2O2 -> 2 H2O In a biological context, ROS are byproducts of the normal metabolism of oxygen.
Amorphous carbonAmorphous carbon is free, reactive carbon that has no crystalline structure. Amorphous carbon materials may be stabilized by terminating dangling-π bonds with hydrogen. As with other amorphous solids, some short-range order can be observed. Amorphous carbon is often abbreviated to aC for general amorphous carbon, aC:H or HAC for hydrogenated amorphous carbon, or to ta-C for tetrahedral amorphous carbon (also called diamond-like carbon).
ImineIn organic chemistry, an imine (ᵻˈmiːn or ˈɪmɪn) is a functional group or organic compound containing a carbon–nitrogen double bond (). The nitrogen atom can be attached to a hydrogen or an organic group (R). The carbon atom has two additional single bonds. Imines are common in synthetic and naturally occurring compounds and they participate in many reactions. For ketimines and aldimines, respectively, the five core atoms (C2C=NX and C(H)C=NX, X = H or C) are coplanar.
Combinatorial proofIn mathematics, the term combinatorial proof is often used to mean either of two types of mathematical proof: A proof by double counting. A combinatorial identity is proven by counting the number of elements of some carefully chosen set in two different ways to obtain the different expressions in the identity. Since those expressions count the same objects, they must be equal to each other and thus the identity is established. A bijective proof. Two sets are shown to have the same number of members by exhibiting a bijection, i.
Orbital hybridisationIn chemistry, orbital hybridisation (or hybridization) is the concept of mixing atomic orbitals to form new hybrid orbitals (with different energies, shapes, etc., than the component atomic orbitals) suitable for the pairing of electrons to form chemical bonds in valence bond theory. For example, in a carbon atom which forms four single bonds the valence-shell s orbital combines with three valence-shell p orbitals to form four equivalent sp3 mixtures in a tetrahedral arrangement around the carbon to bond to four different atoms.
