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Chemistry
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Chemistry is a branch of physical science that studies the composition, structure, properties and change of matter. Chemistry is sometimes called the science because it bridges other natural sciences, including physics. For the differences between chemistry and physics see comparison of chemistry and physics, the history of chemistry can be traced to alchemy, which had been practiced for several millennia in various parts of the world. The word chemistry comes from alchemy, which referred to a set of practices that encompassed elements of chemistry, metallurgy, philosophy, astrology, astronomy, mysticism. An alchemist was called a chemist in popular speech, and later the suffix -ry was added to this to describe the art of the chemist as chemistry, the modern word alchemy in turn is derived from the Arabic word al-kīmīā. In origin, the term is borrowed from the Greek χημία or χημεία and this may have Egyptian origins since al-kīmīā is derived from the Greek χημία, which is in turn derived from the word Chemi or Kimi, which is the ancient name of Egypt in Egyptian. Alternately, al-kīmīā may derive from χημεία, meaning cast together, in retrospect, the definition of chemistry has changed over time, as new discoveries and theories add to the functionality of the science. The term chymistry, in the view of noted scientist Robert Boyle in 1661, in 1837, Jean-Baptiste Dumas considered the word chemistry to refer to the science concerned with the laws and effects of molecular forces. More recently, in 1998, Professor Raymond Chang broadened the definition of chemistry to mean the study of matter, early civilizations, such as the Egyptians Babylonians, Indians amassed practical knowledge concerning the arts of metallurgy, pottery and dyes, but didnt develop a systematic theory. Greek atomism dates back to 440 BC, arising in works by such as Democritus and Epicurus. In 50 BC, the Roman philosopher Lucretius expanded upon the theory in his book De rerum natura, unlike modern concepts of science, Greek atomism was purely philosophical in nature, with little concern for empirical observations and no concern for chemical experiments. Work, particularly the development of distillation, continued in the early Byzantine period with the most famous practitioner being the 4th century Greek-Egyptian Zosimos of Panopolis. He formulated Boyles law, rejected the four elements and proposed a mechanistic alternative of atoms. Before his work, though, many important discoveries had been made, the Scottish chemist Joseph Black and the Dutchman J. B. English scientist John Dalton proposed the theory of atoms, that all substances are composed of indivisible atoms of matter. Davy discovered nine new elements including the alkali metals by extracting them from their oxides with electric current, british William Prout first proposed ordering all the elements by their atomic weight as all atoms had a weight that was an exact multiple of the atomic weight of hydrogen. The inert gases, later called the noble gases were discovered by William Ramsay in collaboration with Lord Rayleigh at the end of the century, thereby filling in the basic structure of the table. Organic chemistry was developed by Justus von Liebig and others, following Friedrich Wöhlers synthesis of urea which proved that organisms were, in theory
2.
Symmetry
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Symmetry in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, symmetry has a precise definition, that an object is invariant to any of various transformations. Although these two meanings of symmetry can sometimes be told apart, they are related, so they are discussed together. The opposite of symmetry is asymmetry, a geometric shape or object is symmetric if it can be divided into two or more identical pieces that are arranged in an organized fashion. This means that an object is symmetric if there is a transformation that moves individual pieces of the object, an object has rotational symmetry if the object can be rotated about a fixed point without changing the overall shape. An object has symmetry if it can be translated without changing its overall shape. An object has symmetry if it can be simultaneously translated and rotated in three-dimensional space along a line known as a screw axis. An object has symmetry if it does not change shape when it is expanded or contracted. Fractals also exhibit a form of symmetry, where small portions of the fractal are similar in shape to large portions. Other symmetries include glide reflection symmetry and rotoreflection symmetry, a dyadic relation R is symmetric if and only if, whenever its true that Rab, its true that Rba. Thus, is the age as is symmetrical, for if Paul is the same age as Mary. Symmetric binary logical connectives are and, or, biconditional, nand, xor, the set of operations that preserve a given property of the object form a group. In general, every kind of structure in mathematics will have its own kind of symmetry, examples include even and odd functions in calculus, the symmetric group in abstract algebra, symmetric matrices in linear algebra, and the Galois group in Galois theory. In statistics, it appears as symmetric probability distributions, and as skewness, symmetry in physics has been generalized to mean invariance—that is, lack of change—under any kind of transformation, for example arbitrary coordinate transformations. This concept has one of the most powerful tools of theoretical physics. See Noethers theorem, and also, Wigners classification, which says that the symmetries of the laws of physics determine the properties of the found in nature. Important symmetries in physics include continuous symmetries and discrete symmetries of spacetime, internal symmetries of particles, in biology, the notion of symmetry is mostly used explicitly to describe body shapes. Bilateral animals, including humans, are more or less symmetric with respect to the plane which divides the body into left
3.
Molecule
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A molecule is an electrically neutral group of two or more atoms held together by chemical bonds. Molecules are distinguished from ions by their lack of electrical charge, however, in quantum physics, organic chemistry, and biochemistry, the term molecule is often used less strictly, also being applied to polyatomic ions. In the kinetic theory of gases, the molecule is often used for any gaseous particle regardless of its composition. According to this definition, noble gas atoms are considered molecules as they are in fact monoatomic molecules. A molecule may be homonuclear, that is, it consists of atoms of one element, as with oxygen, or it may be heteronuclear. Atoms and complexes connected by non-covalent interactions, such as hydrogen bonds or ionic bonds, are not considered single molecules. Molecules as components of matter are common in organic substances and they also make up most of the oceans and atmosphere. Also, no typical molecule can be defined for ionic crystals and covalent crystals, the theme of repeated unit-cellular-structure also holds for most condensed phases with metallic bonding, which means that solid metals are also not made of molecules. In glasses, atoms may also be together by chemical bonds with no presence of any definable molecule. The science of molecules is called molecular chemistry or molecular physics, in practice, however, this distinction is vague. In molecular sciences, a molecule consists of a system composed of two or more atoms. Polyatomic ions may sometimes be thought of as electrically charged molecules. The term unstable molecule is used for very reactive species, i. e, according to Merriam-Webster and the Online Etymology Dictionary, the word molecule derives from the Latin moles or small unit of mass. Molecule – extremely minute particle, from French molécule, from New Latin molecula, diminutive of Latin moles mass, a vague meaning at first, the vogue for the word can be traced to the philosophy of Descartes. The definition of the molecule has evolved as knowledge of the structure of molecules has increased, earlier definitions were less precise, defining molecules as the smallest particles of pure chemical substances that still retain their composition and chemical properties. Molecules are held together by covalent bonding or ionic bonding. Several types of non-metal elements exist only as molecules in the environment, for example, hydrogen only exists as hydrogen molecule. A molecule of a compound is made out of two or more elements, a covalent bond is a chemical bond that involves the sharing of electron pairs between atoms
4.
Dipole
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In electromagnetism, there are two kinds of dipoles, An electric dipole is a separation of positive and negative charges. The simplest example of this is a pair of electric charges of equal magnitude but opposite sign, a permanent electric dipole is called an electret. A magnetic dipole is a circulation of electric current. A simple example of this is a loop of wire with some constant current through it. Dipoles can be characterized by their moment, a vector quantity. For the current loop, the dipole moment points through the loop. In addition to current loops, the electron, among other fundamental particles, has a dipole moment. That is because it generates a field that is identical to that generated by a very small current loop. However, the magnetic moment is not due to a current loop. It is also possible that the electron has a dipole moment although it has not yet been observed. A permanent magnet, such as a bar magnet, owes its magnetism to the magnetic dipole moment of the electron. The two ends of a bar magnet are referred to as poles, and may be labeled north and south, the dipole moment of the bar magnet points from its magnetic south to its magnetic north pole. The north pole of a bar magnet in a compass points north, however, that means that Earths geomagnetic north pole is the south pole of its dipole moment and vice versa. The only known mechanisms for the creation of magnetic dipoles are by current loops or quantum-mechanical spin since the existence of magnetic monopoles has never been experimentally demonstrated, the term comes from the Greek δίς, twice and πόλος, axis. A physical dipole consists of two equal and opposite point charges, in the sense, two poles. Its field at large distances depends almost entirely on the moment as defined above. A point dipole is the limit obtained by letting the separation tend to 0 while keeping the dipole moment fixed, the field of a point dipole has a particularly simple form, and the order-1 term in the multipole expansion is precisely the point dipole field. Although there are no magnetic monopoles in nature, there are magnetic dipoles in the form of the quantum-mechanical spin associated with particles such as electrons
5.
Spectroscopy
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Spectroscopy /spɛkˈtrɒskəpi/ is the study of the interaction between matter and electromagnetic radiation. Historically, spectroscopy originated through the study of visible light dispersed according to its wavelength, later the concept was expanded greatly to include any interaction with radiative energy as a function of its wavelength or frequency. Spectroscopic data is represented by an emission spectrum, a plot of the response of interest as a function of wavelength or frequency. Spectroscopy and spectrography are terms used to refer to the measurement of radiation intensity as a function of wavelength and are used to describe experimental spectroscopic methods. Spectral measurement devices are referred to as spectrometers, spectrophotometers, spectrographs or spectral analyzers, daily observations of color can be related to spectroscopy. Neon lighting is an application of atomic spectroscopy. Neon and other noble gases have characteristic emission frequencies, neon lamps use collision of electrons with the gas to excite these emissions. Inks, dyes and paints include chemical compounds selected for their characteristics in order to generate specific colors. A commonly encountered molecular spectrum is that of nitrogen dioxide, gaseous nitrogen dioxide has a characteristic red absorption feature, and this gives air polluted with nitrogen dioxide a reddish-brown color. Rayleigh scattering is a spectroscopic scattering phenomenon that accounts for the color of the sky, Spectroscopy is used in physical and analytical chemistry because atoms and molecules have unique spectra. As a result, these spectra can be used to detect, identify and quantify information about the atoms, Spectroscopy is also used in astronomy and remote sensing on earth. The measured spectra are used to determine the composition and physical properties of astronomical objects. One of the concepts in spectroscopy is a resonance and its corresponding resonant frequency. Resonances were first characterized in mechanical systems such as pendulums, mechanical systems that vibrate or oscillate will experience large amplitude oscillations when they are driven at their resonant frequency. A plot of amplitude vs. excitation frequency will have a peak centered at the resonance frequency and this plot is one type of spectrum, with the peak often referred to as a spectral line, and most spectral lines have a similar appearance. In quantum mechanical systems, the resonance is a coupling of two quantum mechanical stationary states of one system, such as an atom, via an oscillatory source of energy such as a photon. The coupling of the two states is strongest when the energy of the matches the energy difference between the two states. The energy of a photon is related to its frequency by E = h ν where h is Plancks constant, spectra of atoms and molecules often consist of a series of spectral lines, each one representing a resonance between two different quantum states
6.
Physical chemistry
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Some of the relationships that physical chemistry strives to resolve include the effects of, Intermolecular forces that act upon the physical properties of materials. Reaction kinetics on the rate of a reaction, the identity of ions and the electrical conductivity of materials. Surface chemistry and electrochemistry of cell membranes, interaction of one body with another in terms of quantities of heat and work called thermodynamics. Number of phases, number of components and degree of freedom can be correlated with one another with help of phase rule, the key concepts of physical chemistry are the ways in which pure physics is applied to chemical problems. Predicting the properties of compounds from a description of atoms. To describe the atoms and bonds precisely, it is necessary to know both where the nuclei of the atoms are, and how electrons are distributed around them, spectroscopy is the related sub-discipline of physical chemistry which is specifically concerned with the interaction of electromagnetic radiation with matter. Another set of important questions in chemistry concerns what kind of reactions can happen spontaneously and it can frequently be used to assess whether a reactor or engine design is feasible, or to check the validity of experimental data. To a limited extent, quasi-equilibrium and non-equilibrium thermodynamics can describe irreversible changes, however, classical thermodynamics is mostly concerned with systems in equilibrium and reversible changes and not what actually does happen, or how fast, away from equilibrium. Which reactions do occur and how fast is the subject of chemical kinetics, in general, the higher the barrier, the slower the reaction. A second is that most chemical reactions occur as a sequence of elementary reactions, the precise reasons for this are described in statistical mechanics, a specialty within physical chemistry which is also shared with physics. Statistical mechanics also provides ways to predict the properties we see in everyday life from molecular properties without relying on empirical correlations based on chemical similarities. The term physical chemistry was coined by Mikhail Lomonosov in 1752, modern physical chemistry originated in the 1860s to 1880s with work on chemical thermodynamics, electrolytes in solutions, chemical kinetics and other subjects. One milestone was the publication in 1876 by Josiah Willard Gibbs of his paper and this paper introduced several of the cornerstones of physical chemistry, such as Gibbs energy, chemical potentials, and Gibbs phase rule. Other milestones include the subsequent naming and accreditation of enthalpy to Heike Kamerlingh Onnes, together with Svante August Arrhenius, these were the leading figures in physical chemistry in the late 19th century and early 20th century. All three were awarded with the Nobel Prize in Chemistry between 1901–1909, developments in the following decades include the application of statistical mechanics to chemical systems and work on colloids and surface chemistry, where Irving Langmuir made many contributions. Another important step was the development of quantum mechanics into quantum chemistry from the 1930s, cathedrals of Science The Cambridge History of Science, The modern physical and mathematical sciences
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Inorganic chemistry
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Inorganic chemistry deals with the synthesis and behavior of inorganic and organometallic compounds. This field covers all chemical compounds except the myriad organic compounds, the distinction between the two disciplines is far from absolute, as there is much overlap in the subdiscipline of organometallic chemistry. It has applications in every aspect of the industry, including catalysis, materials science, pigments, surfactants, coatings, medications, fuels. Many inorganic compounds are compounds, consisting of cations and anions joined by ionic bonding. Examples of salts are magnesium chloride MgCl2, which consists of magnesium cations Mg2+ and chloride anions Cl−, or sodium oxide Na2O, in any salt, the proportions of the ions are such that the electric charges cancel out, so that the bulk compound is electrically neutral. The ions are described by their state and their ease of formation can be inferred from the ionization potential or from the electron affinity of the parent elements. Important classes of compounds are the oxides, the carbonates, the sulfates. Many inorganic compounds are characterized by high melting points, inorganic salts typically are poor conductors in the solid state. Other important features include their high meilting point and ease of crystallization, where some salts are very soluble in water, others are not. The simplest inorganic reaction is double displacement when in mixing of two salts the ions are swapped without a change in oxidation state, in redox reactions one reactant, the oxidant, lowers its oxidation state and another reactant, the reductant, has its oxidation state increased. The net result is an exchange of electrons, electron exchange can occur indirectly as well, e. g. in batteries, a key concept in electrochemistry. When one reactant contains hydrogen atoms, a reaction can take place by exchanging protons in acid-base chemistry, as a refinement of acid-base interactions, the HSAB theory takes into account polarizability and size of ions. Inorganic compounds are found in nature as minerals, soil may contain iron sulfide as pyrite or calcium sulfate as gypsum. Inorganic compounds are also found multitasking as biomolecules, as electrolytes, the first important man-made inorganic compound was ammonium nitrate for soil fertilization through the Haber process. Inorganic compounds are synthesized for use as such as vanadium oxide and titanium chloride. Subdivisions of inorganic chemistry are organometallic chemistry, cluster chemistry and bioinorganic chemistry and these fields are active areas of research in inorganic chemistry, aimed toward new catalysts, superconductors, and therapies. Inorganic chemistry is a highly practical area of science, traditionally, the scale of a nations economy could be evaluated by their productivity of sulfuric acid. The manufacturing of fertilizers is another application of industrial inorganic chemistry
8.
Group theory
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In mathematics and abstract algebra, group theory studies the algebraic structures known as groups. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra, linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right. Various physical systems, such as crystals and the hydrogen atom, thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is central to public key cryptography. The first class of groups to undergo a systematic study was permutation groups, given any set X and a collection G of bijections of X into itself that is closed under compositions and inverses, G is a group acting on X. If X consists of n elements and G consists of all permutations, G is the symmetric group Sn, in general, an early construction due to Cayley exhibited any group as a permutation group, acting on itself by means of the left regular representation. In many cases, the structure of a group can be studied using the properties of its action on the corresponding set. For example, in this way one proves that for n ≥5 and this fact plays a key role in the impossibility of solving a general algebraic equation of degree n ≥5 in radicals. The next important class of groups is given by matrix groups, here G is a set consisting of invertible matrices of given order n over a field K that is closed under the products and inverses. Such a group acts on the vector space Kn by linear transformations. In the case of groups, X is a set, for matrix groups. The concept of a group is closely related with the concept of a symmetry group. The theory of groups forms a bridge connecting group theory with differential geometry. A long line of research, originating with Lie and Klein, the groups themselves may be discrete or continuous. Most groups considered in the first stage of the development of group theory were concrete, having been realized through numbers, permutations, or matrices. It was not until the nineteenth century that the idea of an abstract group as a set with operations satisfying a certain system of axioms began to take hold. A typical way of specifying an abstract group is through a presentation by generators and relations, a significant source of abstract groups is given by the construction of a factor group, or quotient group, G/H, of a group G by a normal subgroup H. Class groups of algebraic number fields were among the earliest examples of factor groups, of much interest in number theory
9.
Molecular orbital
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In chemistry, a molecular orbital is a mathematical function describing the wave-like behavior of an electron in a molecule. This function can be used to calculate chemical and physical properties such as the probability of finding an electron in any specific region, the term orbital was introduced by Robert S. Mulliken in 1932 as an abbreviation for one-electron orbital wave function. At an elementary level, it is used to describe the region of space in which the function has a significant amplitude, Molecular orbitals are usually constructed by combining atomic orbitals or hybrid orbitals from each atom of the molecule, or other molecular orbitals from groups of atoms. They can be calculated using the Hartree–Fock or self-consistent field methods. A molecular orbital can be used to represent the regions in a molecule where an electron occupying that orbital is likely to be found, Molecular orbitals are obtained from the combination of atomic orbitals, which predict the location of an electron in an atom. A molecular orbital can specify the configuration of a molecule. Most commonly a MO is represented as a combination of atomic orbitals. They are invaluable in providing a model of bonding in molecules. Most present-day methods in computational chemistry begin by calculating the MOs of the system, a molecular orbital describes the behavior of one electron in the electric field generated by the nuclei and some average distribution of the other electrons. In the case of two electrons occupying the orbital, the Pauli principle demands that they have opposite spin. Necessarily this is an approximation, and highly accurate descriptions of the electronic wave function do not have orbitals. Molecular orbitals arise from allowed interactions between orbitals, which are allowed if the symmetries of the atomic orbitals are compatible with each other. Efficiency of atomic orbital interactions is determined from the overlap between two atomic orbitals, which is significant if the atomic orbitals are close in energy. Finally, the number of molecular orbitals that form must equal the number of orbitals in the atoms being combined to form the molecule. Here, the orbitals are expressed as linear combinations of atomic orbitals. Molecular orbitals were first introduced by Friedrich Hund and Robert S. Mulliken in 1927 and 1928, the linear combination of atomic orbitals or LCAO approximation for molecular orbitals was introduced in 1929 by Sir John Lennard-Jones. His ground-breaking paper showed how to derive the electronic structure of the fluorine and this qualitative approach to molecular orbital theory is part of the start of modern quantum chemistry. Linear combinations of atomic orbitals can be used to estimate the molecular orbitals that are formed upon bonding between the constituent atoms
10.
Ligand field theory
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Ligand field theory describes the bonding, orbital arrangement, and other characteristics of coordination complexes. It represents an application of molecular orbital theory to transition metal complexes, a transition metal ion has nine valence atomic orbitals - consisting of five nd, three p, and one s orbitals. These orbitals are of appropriate energy to form bonding interaction with ligands, the LFT analysis is highly dependent on the geometry of the complex, but most explanations begin by describing octahedral complexes, where six ligands coordinate to the metal. Other complexes can be described by reference to field theory. Griffith and Orgel championed ligand field theory as an accurate description of such complexes. In their paper, they propose that the cause of color differences in transition metal complexes in solution is the incomplete d orbital subshells. That is, the d orbitals of transition metals participate in bonding. In an octahedral complex, the orbitals created by coordination can be seen as resulting from the donation of two electrons by each of six σ-donor ligands to the d-orbitals on the metal. In octahedral complexes, ligands approach along the x-, y- and z-axes, so their σ-symmetry orbitals form bonding and anti-bonding combinations with the dz2, the dxy, dxz and dyz orbitals remain non-bonding orbitals. The irreducible representations that these span are a1g, t1u and eg, the six σ-bonding molecular orbitals result from the combinations of ligand SALCs with metal orbitals of the same symmetry. π bonding in octahedral complexes occurs in two ways, via any ligand p-orbitals that are not being used in σ bonding, and via any π or π* molecular orbitals present on the ligand. In the usual analysis, the p-orbitals of the metal are used for σ bonding, so the π interactions take place with the appropriate metal d-orbitals, i. e. dxy, dxz and these are the orbitals that are non-bonding when only σ bonding takes place. One important π bonding in coordination complexes is metal-to-ligand π bonding and it occurs when the LUMOs of the ligand are anti-bonding π* orbitals. These orbitals are close in energy to the dxy, dxz and dyz orbitals, the ligands end up with electrons in their π* molecular orbital, so the corresponding π bond within the ligand weakens. The other form of coordination π bonding is ligand-to-metal bonding and this situation arises when the π-symmetry p or π orbitals on the ligands are filled. They combine with the dxy, dxz and dyz orbitals on the metal and it is filled with electrons from the metal d-orbitals, however, becoming the HOMO of the complex. For that reason, ΔO decreases when ligand-to-metal bonding occurs, the greater stabilization that results from metal-to-ligand bonding is caused by the donation of negative charge away from the metal ion, towards the ligands. This allows the metal to accept the σ bonds more easily, the combination of ligand-to-metal σ-bonding and metal-to-ligand π-bonding is a synergic effect, as each enhances the other
11.
Crystal system
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In crystallography, the terms crystal system, crystal family and lattice system each refer to one of several classes of space groups, lattices, point groups or crystals. Informally, two crystals are in the crystal system if they have similar symmetries, though there are many exceptions to this. Space groups and crystals are divided into seven crystal systems according to their point groups, five of the crystal systems are essentially the same as five of the lattice systems, but the hexagonal and trigonal crystal systems differ from the hexagonal and rhombohedral lattice systems. The six crystal families are formed by combining the hexagonal and trigonal crystal systems into one hexagonal family, a lattice system is a class of lattices with the same set of lattice point groups, which are subgroups of the arithmetic crystal classes. The 14 Bravais lattices are grouped into seven lattice systems, triclinic, monoclinic, orthorhombic, tetragonal, rhombohedral, hexagonal, in a crystal system, a set of point groups and their corresponding space groups are assigned to a lattice system. Of the 32 point groups that exist in three dimensions, most are assigned to only one system, in which case both the crystal and lattice systems have the same name. However, five point groups are assigned to two systems, rhombohedral and hexagonal, because both exhibit threefold rotational symmetry. These point groups are assigned to the crystal system. In total there are seven crystal systems, triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, a crystal family is determined by lattices and point groups. It is formed by combining crystal systems which have space groups assigned to a lattice system. In three dimensions, the families and systems are identical, except the hexagonal and trigonal crystal systems. In total there are six families, triclinic, monoclinic, orthorhombic, tetragonal, hexagonal. Spaces with less than three dimensions have the number of crystal systems, crystal families and lattice systems. In one-dimensional space, there is one crystal system, in 2D space, there are four crystal systems, oblique, rectangular, square and hexagonal. The relation between three-dimensional crystal families, crystal systems and lattice systems is shown in the table, Note. To avoid confusion of terminology, the term trigonal lattice is not used, if the original structure and inverted structure are identical, then the structure is centrosymmetric. Still, even for non-centrosymmetric case, inverted structure in some cases can be rotated to align with the original structure and this is the case of non-centrosymmetric achiral structure. If the inverted structure cannot be rotated to align with the structure, then the structure is chiral
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Crystallography
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Crystallography is the experimental science of determining the arrangement of atoms in the crystalline solids. The word crystallography derives from the Greek words crystallon cold drop, frozen drop, with its meaning extending to all solids with some degree of transparency, and graphein to write. In July 2012, the United Nations recognised the importance of the science of crystallography by proclaiming that 2014 would be the International Year of Crystallography, X-ray crystallography is used to determine the structure of large biomolecules such as proteins. Before the development of X-ray diffraction crystallography, the study of crystals was based on measurements of their geometry. This involved measuring the angles of crystal faces relative to other and to theoretical reference axes. This physical measurement is carried out using a goniometer, the position in 3D space of each crystal face is plotted on a stereographic net such as a Wulff net or Lambert net. The pole to face is plotted on the net. Each point is labelled with its Miller index, the final plot allows the symmetry of the crystal to be established. Crystallographic methods now depend on analysis of the patterns of a sample targeted by a beam of some type. X-rays are most commonly used, other beams used include electrons or neutrons and this is facilitated by the wave properties of the particles. Crystallographers often explicitly state the type of beam used, as in the terms X-ray crystallography and these three types of radiation interact with the specimen in different ways. X-rays interact with the distribution of electrons in the sample. Electrons are charged particles and therefore interact with the charge distribution of both the atomic nuclei and the electrons of the sample. Neutrons are scattered by the atomic nuclei through the nuclear forces, but in addition. They are therefore also scattered by magnetic fields, when neutrons are scattered from hydrogen-containing materials, they produce diffraction patterns with high noise levels. However, the material can sometimes be treated to substitute deuterium for hydrogen, because of these different forms of interaction, the three types of radiation are suitable for different crystallographic studies. An image of an object is made using a lens to focus the beam. However, the wavelength of light is three orders of magnitude longer than the length of typical atomic bonds and atoms themselves
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X-ray crystallography
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By measuring the angles and intensities of these diffracted beams, a crystallographer can produce a three-dimensional picture of the density of electrons within the crystal. From this electron density, the positions of the atoms in the crystal can be determined, as well as their chemical bonds, their disorder. The method also revealed the structure and function of biological molecules, including vitamins, drugs, proteins. X-ray crystallography is still the method for characterizing the atomic structure of new materials. In a single-crystal X-ray diffraction measurement, a crystal is mounted on a goniometer, the goniometer is used to position the crystal at selected orientations. The crystal is illuminated with a finely focused monochromatic beam of X-rays, poor resolution or even errors may result if the crystals are too small, or not uniform enough in their internal makeup. X-ray crystallography is related to other methods for determining atomic structures. Similar diffraction patterns can be produced by scattering electrons or neutrons, for all above mentioned X-ray diffraction methods, the scattering is elastic, the scattered X-rays have the same wavelength as the incoming X-ray. By contrast, inelastic X-ray scattering methods are useful in studying excitations of the sample, Crystals, though long admired for their regularity and symmetry, were not investigated scientifically until the 17th century. Johannes Kepler hypothesized in his work Strena seu de Nive Sexangula that the symmetry of snowflake crystals was due to a regular packing of spherical water particles. The Danish scientist Nicolas Steno pioneered experimental investigations of crystal symmetry, hence, William Hallowes Miller in 1839 was able to give each face a unique label of three small integers, the Miller indices which remain in use today for identifying crystal faces. In the 19th century, a catalog of the possible symmetries of a crystal was worked out by Johan Hessel, Auguste Bravais, Evgraf Fedorov, Arthur Schönflies. Wilhelm Röntgen discovered X-rays in 1895, just as the studies of crystal symmetry were being concluded, physicists were initially uncertain of the nature of X-rays, but soon suspected that they were waves of electromagnetic radiation, in other words, another form of light. Single-slit experiments in the laboratory of Arnold Sommerfeld suggested that X-rays had a wavelength of about 1 angstrom, however, X-rays are composed of photons, and thus are not only waves of electromagnetic radiation but also exhibit particle-like properties. Albert Einstein introduced the concept in 1905, but it was not broadly accepted until 1922. Therefore, these properties of X-rays, such as their ionization of gases. Nevertheless, Braggs view was not broadly accepted and the observation of X-ray diffraction by Max von Laue in 1912 confirmed for most scientists that X-rays were a form of electromagnetic radiation, Crystals are regular arrays of atoms, and X-rays can be considered waves of electromagnetic radiation. Atoms scatter X-ray waves, primarily through the atoms electrons and this phenomenon is known as elastic scattering, and the electron is known as the scatterer
14.
Chirality (chemistry)
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Chirality /kaɪˈrælɪti/ is a geometric property of some molecules and ions. A chiral molecule/ion is non-superposable on its mirror image, the presence of an asymmetric carbon center is one of several structural features that induce chirality in organic and inorganic molecules. The term chirality is derived from the Greek word for hand, the mirror images of a chiral molecule/ion are called enantiomers or optical isomers. Individual enantiomers are often designated as either right- or left-handed, Chirality is an essential consideration when discussing the stereochemistry in organic and inorganic chemistry. The concept is of practical importance because most biomolecules and pharmaceuticals are chiral. Chirality is based on molecular symmetry elements, specifically, a chiral compound can contain no improper axis of rotation, which includes planes of symmetry and inversion center. Chiral molecules are always dissymmetric but not always asymmetric, in general, chiral molecules have point chirality at a single stereogenic atom, which has four different substituents. The two enantiomers of such compounds are said to have different absolute configurations at this center, the stereogenic atom is usually carbon, as in many biological molecules. However chirality can exist in any atom, including metals, phosphorus, Chiral nitrogen is equally possible, although the effects of nitrogen inversion can make many of these compounds impossible to isolate. While the presence of a stereogenic atom describes the great majority of cases, for instance it is not necessary for the chiral substance to have a stereogenic atom. Examples include 1-bromo-1-chloro-1-fluoroadamantane, methylethylphenyltetrahedrane, certain calixarenes and fullerenes, which have inherent chirality, the C2-symmetric species 1, 1-bi-2-naphthol,1, 3-dichloro-allene have axial chirality. -cyclooctene and many ferrocenes have planar chirality, when the optical rotation for an enantiomer is too low for practical measurement, the species is said to exhibit cryptochirality. Even isotopic differences must be considered when examining chirality, illustrative is the derivative of benzyl alcohol PhCHDOH is chiral. The S enantiomer has D = +0. 715°, many biologically active molecules are chiral, including the naturally occurring amino acids and sugars. In biological systems, most of these compounds are of the chirality, most amino acids are levorotatory. Typical naturally occurring proteins, made of L amino acids, are known as left-handed proteins, d-amino acids are very rare in nature and have only been found in small peptides attached to bacteria cell walls. The origin of this homochirality in biology is the subject of much debate, however, there is some suggestion that early amino acids could have formed in comet dust. Enzymes, which are chiral, often distinguish between the two enantiomers of a chiral substrate, one could imagine an enzyme as having a glove-like cavity that binds a substrate
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Rotation
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A rotation is a circular movement of an object around a center of rotation. A three-dimensional object always rotates around a line called a rotation axis. If the axis passes through the center of mass, the body is said to rotate upon itself. A rotation about a point, e. g. the Earth about the Sun, is called a revolution or orbital revolution. The axis is called a pole, mathematically, a rotation is a rigid body movement which, unlike a translation, keeps a point fixed. This definition applies to rotations within both two and three dimensions All rigid body movements are rotations, translations, or combinations of the two, a rotation is simply a progressive radial orientation to a common point. That common point lies within the axis of that motion, the axis is 90 degrees perpendicular to the plane of the motion. If the axis of the rotation lies external of the body in question then the body is said to orbit, there is no fundamental difference between a “rotation” and an “orbit” and or spin. The key distinction is simply where the axis of the rotation lies and this distinction can be demonstrated for both “rigid” and “non rigid” bodies. If a rotation around a point or axis is followed by a rotation around the same point/axis. The reverse of a rotation is also a rotation, thus, the rotations around a point/axis form a group. However, a rotation around a point or axis and a rotation around a different point/axis may result in something other than a rotation, Rotations around the x, y and z axes are called principal rotations. Rotation around any axis can be performed by taking a rotation around the x axis, followed by a rotation around the y axis and that is to say, any spatial rotation can be decomposed into a combination of principal rotations. In flight dynamics, the rotations are known as yaw, pitch. This terminology is used in computer graphics. In astronomy, rotation is an observed phenomenon. Stars, planets and similar bodies all spin around on their axes, the rotation rate of planets in the solar system was first measured by tracking visual features. Stellar rotation is measured through Doppler shift or by tracking active surface features and this rotation induces a centrifugal acceleration in the reference frame of the Earth which slightly counteracts the effect of gravity the closer one is to the equator
16.
Water
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Water is a transparent and nearly colorless chemical substance that is the main constituent of Earths streams, lakes, and oceans, and the fluids of most living organisms. Its chemical formula is H2O, meaning that its molecule contains one oxygen, Water strictly refers to the liquid state of that substance, that prevails at standard ambient temperature and pressure, but it often refers also to its solid state or its gaseous state. It also occurs in nature as snow, glaciers, ice packs and icebergs, clouds, fog, dew, aquifers, Water covers 71% of the Earths surface. It is vital for all forms of life. Only 2. 5% of this water is freshwater, and 98. 8% of that water is in ice and groundwater. Less than 0. 3% of all freshwater is in rivers, lakes, and the atmosphere, a greater quantity of water is found in the earths interior. Water on Earth moves continually through the cycle of evaporation and transpiration, condensation, precipitation. Evaporation and transpiration contribute to the precipitation over land, large amounts of water are also chemically combined or adsorbed in hydrated minerals. Safe drinking water is essential to humans and other even though it provides no calories or organic nutrients. There is a correlation between access to safe water and gross domestic product per capita. However, some observers have estimated that by 2025 more than half of the population will be facing water-based vulnerability. A report, issued in November 2009, suggests that by 2030, in developing regions of the world. Water plays an important role in the world economy, approximately 70% of the freshwater used by humans goes to agriculture. Fishing in salt and fresh water bodies is a source of food for many parts of the world. Much of long-distance trade of commodities and manufactured products is transported by boats through seas, rivers, lakes, large quantities of water, ice, and steam are used for cooling and heating, in industry and homes. Water is an excellent solvent for a variety of chemical substances, as such it is widely used in industrial processes. Water is also central to many sports and other forms of entertainment, such as swimming, pleasure boating, boat racing, surfing, sport fishing, Water is a liquid at the temperatures and pressures that are most adequate for life. Specifically, at atmospheric pressure of 1 bar, water is a liquid between the temperatures of 273.15 K and 373.15 K
17.
Cartesian coordinate system
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Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair. The coordinates can also be defined as the positions of the projections of the point onto the two axis, expressed as signed distances from the origin. One can use the principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes. In general, n Cartesian coordinates specify the point in an n-dimensional Euclidean space for any dimension n and these coordinates are equal, up to sign, to distances from the point to n mutually perpendicular hyperplanes. The invention of Cartesian coordinates in the 17th century by René Descartes revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes can be described by Cartesian equations, algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2, centered at the origin of the plane, a familiar example is the concept of the graph of a function. Cartesian coordinates are also tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering. They are the most common system used in computer graphics, computer-aided geometric design. Nicole Oresme, a French cleric and friend of the Dauphin of the 14th Century, used similar to Cartesian coordinates well before the time of Descartes. The adjective Cartesian refers to the French mathematician and philosopher René Descartes who published this idea in 1637 and it was independently discovered by Pierre de Fermat, who also worked in three dimensions, although Fermat did not publish the discovery. Both authors used a single axis in their treatments and have a length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes La Géométrie was translated into Latin in 1649 by Frans van Schooten and these commentators introduced several concepts while trying to clarify the ideas contained in Descartes work. Many other coordinate systems have developed since Descartes, such as the polar coordinates for the plane. The development of the Cartesian coordinate system would play a role in the development of the Calculus by Isaac Newton. The two-coordinate description of the plane was later generalized into the concept of vector spaces. Choosing a Cartesian coordinate system for a one-dimensional space – that is, for a straight line—involves choosing a point O of the line, a unit of length, and an orientation for the line. An orientation chooses which of the two half-lines determined by O is the positive, and which is negative, we say that the line is oriented from the negative half towards the positive half
18.
Sigma
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Sigma is the eighteenth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 200. When used at the end of a word, the form is used, e. g. Ὀδυσσεύς. The shape and alphabetic position of sigma is derived from Phoenician shin
19.
Perpendicular
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In elementary geometry, the property of being perpendicular is the relationship between two lines which meet at a right angle. The property extends to other related geometric objects, a line is said to be perpendicular to another line if the two lines intersect at a right angle. Perpendicularity can be shown to be symmetric, meaning if a first line is perpendicular to a second line, for this reason, we may speak of two lines as being perpendicular without specifying an order. Perpendicularity easily extends to segments and rays, in symbols, A B ¯ ⊥ C D ¯ means line segment AB is perpendicular to line segment CD. A line is said to be perpendicular to an if it is perpendicular to every line in the plane that it intersects. This definition depends on the definition of perpendicularity between lines, two planes in space are said to be perpendicular if the dihedral angle at which they meet is a right angle. Perpendicularity is one instance of the more general mathematical concept of orthogonality, perpendicularity is the orthogonality of classical geometric objects. Thus, in advanced mathematics, the perpendicular is sometimes used to describe much more complicated geometric orthogonality conditions. The word foot is used in connection with perpendiculars. This usage is exemplified in the top diagram, above, the diagram can be in any orientation. The foot is not necessarily at the bottom, step 2, construct circles centered at A and B having equal radius. Let Q and R be the points of intersection of two circles. Step 3, connect Q and R to construct the desired perpendicular PQ, to prove that the PQ is perpendicular to AB, use the SSS congruence theorem for and QPB to conclude that angles OPA and OPB are equal. Then use the SAS congruence theorem for triangles OPA and OPB to conclude that angles POA, to make the perpendicular to the line g at or through the point P using Thales theorem, see the animation at right. The Pythagorean Theorem can be used as the basis of methods of constructing right angles, for example, by counting links, three pieces of chain can be made with lengths in the ratio 3,4,5. These can be out to form a triangle, which will have a right angle opposite its longest side. This method is useful for laying out gardens and fields, where the dimensions are large, the chains can be used repeatedly whenever required. If two lines are perpendicular to a third line, all of the angles formed along the third line are right angles
20.
Parallel (geometry)
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In geometry, parallel lines are lines in a plane which do not meet, that is, two lines in a plane that do not intersect or touch each other at any point are said to be parallel. By extension, a line and a plane, or two planes, in three-dimensional Euclidean space that do not share a point are said to be parallel. However, two lines in space which do not meet must be in a common plane to be considered parallel. Parallel planes are planes in the same space that never meet. Parallel lines are the subject of Euclids parallel postulate, parallelism is primarily a property of affine geometries and Euclidean space is a special instance of this type of geometry. Some other spaces, such as space, have analogous properties that are sometimes referred to as parallelism. For example, A B ∥ C D indicates that line AB is parallel to line CD, in the Unicode character set, the parallel and not parallel signs have codepoints U+2225 and U+2226, respectively. In addition, U+22D5 represents the relation equal and parallel to, given parallel straight lines l and m in Euclidean space, the following properties are equivalent, Every point on line m is located at exactly the same distance from line l. Line m is in the plane as line l but does not intersect l. When lines m and l are both intersected by a straight line in the same plane, the corresponding angles of intersection with the transversal are congruent. Thus, the property is the one usually chosen as the defining property of parallel lines in Euclidean geometry. The other properties are consequences of Euclids Parallel Postulate. Another property that also involves measurement is that parallel to each other have the same gradient. The definition of parallel lines as a pair of lines in a plane which do not meet appears as Definition 23 in Book I of Euclids Elements. Alternative definitions were discussed by other Greeks, often as part of an attempt to prove the parallel postulate, proclus attributes a definition of parallel lines as equidistant lines to Posidonius and quotes Geminus in a similar vein. Simplicius also mentions Posidonius definition as well as its modification by the philosopher Aganis, at the end of the nineteenth century, in England, Euclids Elements was still the standard textbook in secondary schools. A major difference between these texts, both between themselves and between them and Euclid, is the treatment of parallel lines. These reform texts were not without their critics and one of them, Charles Dodgson, wrote a play, Euclid and His Modern Rivals, one of the early reform textbooks was James Maurice Wilsons Elementary Geometry of 1868
21.
Dihedral angle
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A dihedral angle is the angle between two intersecting planes. In chemistry it is the angle between planes through two sets of three atoms, having two atoms in common, in solid geometry it is defined as the union of a line and two half-planes that have this line as a common edge. In higher dimension, a dihedral angle represents the angle between two hyperplanes, a dihedral angle is an angle between two intersecting planes on a third plane perpendicular to the line of intersection. A torsion angle is an example of a dihedral angle. In stereochemistry every set of three atoms of a molecule defines a plane, when two such planes intersect, the angle between them is a dihedral angle. Dihedral angles are used to specify the molecular conformation, stereochemical arrangements corresponding to angles between 0° and ±90° are called syn, those corresponding to angles between ±90° and 180° anti. Similarly, arrangements corresponding to angles between 30° and 150° or between −30° and −150° are called clinal and those between 0° and ±30° or ±150° and 180° are called periplanar. The synperiplanar conformation is also known as the syn- or cis-conformation, antiperiplanar as anti or trans, for example, with n-butane two planes can be specified in terms of the two central carbon atoms and either of the methyl carbon atoms. The syn-conformation shown above, with an angle of 60° is less stable than the anti-configuration with a dihedral angle of 180°. For macromolecular usage the symbols T, C, G+, G−, A+, a Ramachandran plot, originally developed in 1963 by G. N. Ramachandran, C. Ramakrishnan, and V. Sasisekharan, is a way to visualize energetically allowed regions for backbone dihedral angles ψ against φ of amino acid residues in protein structure, the figure at right illustrates the definition of the φ and ψ backbone dihedral angles. In a protein chain three dihedral angles are defined as φ, ψ and ω, as shown in the diagram, the planarity of the peptide bond usually restricts ω to be 180° or 0°. The distance between the Cα atoms in the trans and cis isomers is approximately 3.8 and 2.9 Å, the cis isomer is mainly observed in Xaa–Pro peptide bonds. The sidechain dihedral angles tend to cluster near 180°, 60°, and −60°, which are called the trans, gauche+, the stability of certain sidechain dihedral angles is affected by the values φ and ψ. For instance, there are steric interactions between the Cγ of the side chain in the gauche+ rotamer and the backbone nitrogen of the next residue when ψ is near -60°. An alternative method is to calculate the angle between the vectors, nA and nB, which are normal to the planes. Cos φ = − n A ⋅ n B | n A | | n B | where nA · nB is the dot product of the vectors and |nA| |nB| is the product of their lengths. Any plane can also be described by two non-collinear vectors lying in that plane, taking their cross product yields a vector to the plane
22.
Xenon tetrafluoride
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Xenon tetrafluoride is a chemical compound with chemical formula XeF4. It was the first discovered binary compound of a noble gas. It is produced by the reaction of xenon with fluorine, F2, according to the chemical equation, Xe +2 F2 → XeF4 This reaction is exothermic. Xenon tetrafluoride is a crystalline substance under ordinary conditions. Its crystalline structure was determined by both NMR spectroscopy and X-ray crystallography in 1963, xenon tetrafluoride sublimes at a temperature of 115.7 °C. The formation of xenon tetrafluoride, like the other xenon fluorides, is exergonic and they are stable at normal temperatures and pressures. All of them react with water, releasing pure xenon gas, hydrogen fluoride. This reaction occurs in slightly moist air, hence, all xenon fluorides must be kept in anhydrous atmospheres, xenon tetrafluoride is produced by heating a mixture of xenon and fluorine in a 1,5 ratio in a nickel container to 400 °C. Some xenon hexafluoride, XeF6, is produced. Xenon tetrafluoride is hydrolyzed by water at low temperatures to form elemental xenon, oxygen, hydrofluoric acid, reaction with tetramethylammonium fluoride forms tetramethylammonium pentafluoroxenate, which contains the pentagonal XeF−5 anion. XeF4 reacts with the structure that makes up the backbone of silicone rubber to form simple gaseous products
23.
Benzene
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Benzene is an important organic chemical compound with the chemical formula C6H6. The benzene molecule is composed of 6 carbon atoms joined in a ring with 1 hydrogen atom attached to each, because it contains only carbon and hydrogen atoms, benzene is classed as a hydrocarbon. Benzene is a constituent of crude oil and is one of the elementary petrochemicals. Because of the cyclic continuous pi bond between the atoms, benzene is classed as an aromatic hydrocarbon, the second -annulene. Benzene is a colorless and highly flammable liquid with a sweet smell and it is used primarily as a precursor to the manufacture of chemicals with more complex structure, such as ethylbenzene and cumene, of which billions of kilograms are produced. Because benzene has a high number, it is an important component of gasoline. Because benzene is a carcinogen, most non-industrial applications have been limited. The word benzene derives historically from gum benzoin, a resin known to European pharmacists. An acidic material was derived from benzoin by sublimation, and named flowers of benzoin, the hydrocarbon derived from benzoic acid thus acquired the name benzin, benzol, or benzene. Michael Faraday first isolated and identified benzene in 1825 from the oily residue derived from the production of illuminating gas, in 1833, Eilhard Mitscherlich produced it by distilling benzoic acid and lime. He gave the compound the name benzin, in 1845, Charles Mansfield, working under August Wilhelm von Hofmann, isolated benzene from coal tar. Four years later, Mansfield began the first industrial-scale production of benzene, gradually, the sense developed among chemists that a number of substances were chemically related to benzene, comprising a diverse chemical family. In 1855, Hofmann used the word aromatic to designate this family relationship, in 1997, benzene was detected in deep space. The empirical formula for benzene was known, but its highly polyunsaturated structure. In 1865, the German chemist Friedrich August Kekulé published a paper in French suggesting that the structure contained a ring of six carbon atoms with alternating single and double bonds, the next year he published a much longer paper in German on the same subject. Kekulés symmetrical ring could explain these facts, as well as benzenes 1,1 carbon-hydrogen ratio. Here Kekulé spoke of the creation of the theory and he said that he had discovered the ring shape of the benzene molecule after having a reverie or day-dream of a snake seizing its own tail. This vision, he said, came to him years of studying the nature of carbon-carbon bonds
24.
Silicon tetrafluoride
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Silicon tetrafluoride or Tetrafluorosilane is the chemical compound with the formula SiF4. This tetrahedral molecule is notable for having a remarkably narrow liquid range and it was first synthesized by John Davy in 1812. SiF4 is a by-product of the production of fertilizers, resulting from the attack of HF on silicates. In the laboratory, the compound is prepared by heating BaSiF6 above 300 °C, whereupon the solid releases volatile SiF4, the required BaSiF6 is prepared by treating aqueous hexafluorosilicic acid with barium chloride. The corresponding GeF4 is prepared analogously, except that the thermal cracking requires 700 °C, volcanic plumes contain significant amounts of silicon tetrafluoride. Production can reach several tonnes per day, the silicon tetrafluoride is partly hydrolysed and forms hexafluorosilicic acid
25.
Staggered conformation
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In organic chemistry, a staggered conformation is a chemical conformation of an ethane-like moiety abcX–Ydef in which the substituents a, b, and c are at the maximum distance from d, e, and f. This requires the torsion angles to be 60°. Eliel, Ernest L. Wilen, such a conformation exists in any open chain single chemical bond connecting two sp3-hybridised atoms, and is normally a conformational energy minimum. For some molecules such as those of n-butane, there can be special versions of staggered conformations called gauche and anti, see first Newman projection diagram in Conformational isomerism
26.
Group (mathematics)
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In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element. The operation satisfies four conditions called the group axioms, namely closure and it allows entities with highly diverse mathematical origins in abstract algebra and beyond to be handled in a flexible way while retaining their essential structural aspects. The ubiquity of groups in areas within and outside mathematics makes them a central organizing principle of contemporary mathematics. Groups share a kinship with the notion of symmetry. The concept of a group arose from the study of polynomial equations, after contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Modern group theory—an active mathematical discipline—studies groups in their own right, to explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups. A theory has developed for finite groups, which culminated with the classification of finite simple groups. Since the mid-1980s, geometric group theory, which studies finitely generated groups as objects, has become a particularly active area in group theory. One of the most familiar groups is the set of integers Z which consists of the numbers, −4, −3, −2, −1,0,1,2,3,4. The following properties of integer addition serve as a model for the group axioms given in the definition below. For any two integers a and b, the sum a + b is also an integer and that is, addition of integers always yields an integer. This property is known as closure under addition, for all integers a, b and c, + c = a +. Expressed in words, adding a to b first, and then adding the result to c gives the final result as adding a to the sum of b and c. If a is any integer, then 0 + a = a +0 = a, zero is called the identity element of addition because adding it to any integer returns the same integer. For every integer a, there is a b such that a + b = b + a =0. The integer b is called the element of the integer a and is denoted −a. The integers, together with the operation +, form a mathematical object belonging to a class sharing similar structural aspects. To appropriately understand these structures as a collective, the abstract definition is developed
27.
Symmetry operation
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In the context of molecular symmetry, a symmetry operation is a permutation of atoms such that the molecule or crystal is transformed into a state indistinguishable from the starting state. Two basic facts follow from this definition, which emphasize its usefulness, physical properties must be invariant with respect to symmetry operations. Symmetry operations can be collected together in groups which are isomorphous to permutation groups, wavefunctions need not be invariant, because the operation can multiply them by a phase or mix states within a degenerate representation, without affecting any physical property. These are denoted by Cnm and are rotations of 360°/n, performed m times, the superscript m is omitted if it is equal to one. C1, rotation by 360°, is called the Identity operation and is denoted by E or I, cnn, n rotations 360°/n is also an Identity operation. These are denoted by Snm and are rotations of 360°/n followed by reflection in a perpendicular to the rotation axis. S1 is usually denoted as σ, an operation about a mirror plane. S2 is usually denoted as i, an operation about an inversion centre. When n is an even number Snn = E, but when n is odd Sn2n = E. Rotation axes, mirror planes and inversion centres are symmetry elements, the rotation axis of highest order is known as the principal rotation axis. It is conventional to set the Cartesian z axis of the molecule to contain the principal rotation axis, there is a C2 rotation axis which passes through the carbon atom and the midpoints between the two hydrogen atoms and the two chlorine atoms. Define the z axis as co-linear with the C2 axis, the xz plane as containing CH2, a C2 rotation operation permutes the two hydrogen atoms and the two chlorine atoms. Reflection in the yz plane permutes the hydrogen atoms while reflection in the xz plane permutes the chlorine atoms, the four symmetry operations E, C2, σand σ form the point group C2v. Note that if any two operations are carried out in succession the result is the same as if an operation of the group had been performed. In addition to the rotations of order 2 and 3 there are three mutually perpendicular S4 axes which pass half-way between the C-H bonds and six mirror planes. In crystals screw rotations and/or glide reflections are additionally possible and these are rotations or reflections together with partial translation. The Bravais lattices may be considered as representing translational symmetry operations, combinations of operations of the crystallographic point groups with the addition symmetry operations produce the 230 crystallographic space groups. Cotton Chemical applications of theory, Wiley,1962,1971
28.
Square planar molecular geometry
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The square planar molecular geometry in chemistry describes the stereochemistry that is adopted by certain chemical compounds. As the name suggests, molecules of this geometry have their atoms positioned at the corners of a square on the plane about a central atom. The addition of two ligands to linear compounds, ML2, can afford square planar complexes, for example, XeF2 adds fluorine to give square planar XeF4. In principle, square planar geometry can be achieved by flattening a tetrahedron, as such, the interconversion of tetrahedral and square planar geometries provides an intramolecular pathway for the isomerization of tetrahedral compounds. This pathway does not operate readily for hydrocarbons, but tetrahedral nickel complexes, e. g. NiBr22, square planar geometry can also be achieved by the removal of a pair of ligands from the z-axis of an octahedron, leaving four ligands in the xy plane. For transition metal compounds, the crystal field splitting diagram for square planar geometry can thus be derived from the octahedral diagram, the removal of the two ligands stabilizes the dz2 level, leaving the dx2−y2 level as the most destabilized. Consequently, the dx2−y2 remains unoccupied in complexes of metals with the d8 configuration and these compounds typically have 16 valence electrons. Numerous compounds adopt this geometry, examples being especially numerous for transition metal complexes, the noble gas compound XeF4 adopts this structure as predicted by VSEPR theory. The geometry is prevalent for transition metal complexes with d8 configuration, which includes Rh, Ir, Pd, Pt, notable examples include the anticancer drugs cisplatin and carboplatin. Many homogeneous catalysts are square planar in their state, such as Wilkinsons catalyst. Other examples include Vaskas complex and Zeises salt, when the two axial ligands are removed to generate a square planar geometry, the dz2 orbital is driven lower in energy as electron-electron repulsion with ligands on the z-axis is no longer present. However, for purely σ-donating ligands the dz2 orbital is still higher in energy than the dxy, dxz and it bears electron density on the x- and y-axes and therefore interacts with the filled ligand orbitals. The dxy, dxz and dyz orbitals are generally presented as degenerate and their relative ordering depends on the nature of the particular complex. Furthermore, the splitting of d-orbitals is perturbed by π-donating ligands in contrast to octahedral complexes
29.
Point group
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In geometry, a point group is a group of geometric symmetries that keep at least one point fixed. Point groups can exist in a Euclidean space with any dimension, point groups can be realized as sets of orthogonal matrices M that transform point x into point y, y = Mx where the origin is the fixed point. Point-group elements can either be rotations or else reflections, or improper rotations and these are the crystallographic point groups. Point groups can be classified into groups and achiral groups. The chiral groups are subgroups of the orthogonal group SO, they contain only orientation-preserving orthogonal transformations. The achiral groups contain also transformations of determinant −1, in an achiral group, the orientation-preserving transformations form a subgroup of index 2. Finite Coxeter groups or reflection groups are those point groups that are generated purely by a set of reflectional mirrors passing through the same point, a rank n Coxeter group has n mirrors and is represented by a Coxeter-Dynkin diagram. Coxeter notation offers a bracketed notation equivalent to the Coxeter diagram, with symbols for rotational. There are only two one-dimensional point groups, the identity group and the reflection group, point groups in two dimensions, sometimes called rosette groups. The subset of pure reflectional point groups, defined by 1 or 2 mirrors, can also be given by their Coxeter group and related polygons. The symmetry of the groups can be doubled by an isomorphism. Point groups in three dimensions, sometimes called point groups after their wide use in studying the symmetries of small molecules. They come in 7 infinite families of axial or prismatic groups, the reflection point groups, defined by 1 to 3 mirror planes, can also be given by their Coxeter group and related polyhedra. The group can be doubled, written as, mapping the first and last mirrors onto each other, doubling the symmetry to 48, the four-dimensional point groups are listed in Conway and Smith, Section 4, Tables 4. 1-4.3. The following list gives the four-dimensional reflection groups, each group is specified as a Coxeter group, and like the polyhedral groups of 3D, it can be named by its related convex regular 4-polytope. Front-back symmetric groups like and can be doubled, shown as double brackets in Coxeters notation, the following table gives the five-dimensional reflection groups, by listing them as Coxeter groups. The following table gives the six-dimensional reflection groups, by listing them as Coxeter groups, the following table gives the seven-dimensional reflection groups, by listing them as Coxeter groups. The following table gives the eight-dimensional reflection groups, by listing them as Coxeter groups, S. M. Coxeter, Kaleidoscopes, Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C
30.
Space group
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In mathematics, physics and chemistry, a space group is the symmetry group of a configuration in space, usually in three dimensions. In three dimensions, there are 219 distinct types, or 230 if chiral copies are considered distinct, Space groups are also studied in dimensions other than 3 where they are sometimes called Bieberbach groups, and are discrete cocompact groups of isometries of an oriented Euclidean space. In crystallography, space groups are called the crystallographic or Fedorov groups. A definitive source regarding 3-dimensional space groups is the International Tables for Crystallography, in 1879 Leonhard Sohncke listed the 65 space groups whose elements preserve the orientation. More accurately, he listed 66 groups, but Fedorov and Schönflies both noticed that two of them were really the same, the space groups in 3 dimensions were first enumerated by Fedorov, and shortly afterwards were independently enumerated by Schönflies. The correct list of 230 space groups was found by 1892 during correspondence between Fedorov and Schönflies, burckhardt describes the history of the discovery of the space groups in detail. The space groups in three dimensions are made from combinations of the 32 crystallographic point groups with the 14 Bravais lattices, the combination of all these symmetry operations results in a total of 230 different space groups describing all possible crystal symmetries. The elements of the space group fixing a point of space are rotations, reflections, the identity element, the translations form a normal abelian subgroup of rank 3, called the Bravais lattice. There are 14 possible types of Bravais lattice, the quotient of the space group by the Bravais lattice is a finite group which is one of the 32 possible point groups. Translation is defined as the moves from one point to another point. A glide plane is a reflection in a plane, followed by a parallel with that plane. This is noted by a, b or c, depending on which axis the glide is along. There is also the n glide, which is a glide along the half of a diagonal of a face, and the d glide, the latter is called the diamond glide plane as it features in the diamond structure. In 17 space groups, due to the centering of the cell, the glides occur in two directions simultaneously, i. e. the same glide plane can be called b or c, a or b. For example, group Abm2 could be also called Acm2, group Ccca could be called Cccb, in 1992, it was suggested to use symbol e for such planes. The symbols for five groups have been modified, A screw axis is a rotation about an axis. These are noted by a number, n, to describe the degree of rotation, the degree of translation is then added as a subscript showing how far along the axis the translation is, as a portion of the parallel lattice vector. So,21 is a rotation followed by a translation of 1/2 of the lattice vector
31.
Translation (geometry)
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In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure or a space by the same amount in a given direction. In Euclidean geometry a transformation is a correspondence between two sets of points or a mapping from one plane to another. )A translation can be described as a rigid motion. A translation can also be interpreted as the addition of a constant vector to every point, a translation operator is an operator T δ such that T δ f = f. If v is a vector, then the translation Tv will work as Tv. If T is a translation, then the image of a subset A under the function T is the translate of A by T, the translate of A by Tv is often written A + v. In a Euclidean space, any translation is an isometry, the set of all translations forms the translation group T, which is isomorphic to the space itself, and a normal subgroup of Euclidean group E. The quotient group of E by T is isomorphic to the orthogonal group O, E / T ≅ O, a translation is an affine transformation with no fixed points. Matrix multiplications always have the origin as a fixed point, similarly, the product of translation matrices is given by adding the vectors, T u T v = T u + v. Because addition of vectors is commutative, multiplication of matrices is therefore also commutative. In physics, translation is movement that changes the position of an object, for example, according to Whittaker, A translation is the operation changing the positions of all points of an object according to the formula → where is the same vector for each point of the object. When considering spacetime, a change of time coordinate is considered to be a translation, for example, the Galilean group and the Poincaré group include translations with respect to time
32.
VSEPR theory
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Valence shell electron pair repulsion theory is a model used in chemistry to predict the geometry of individual molecules from the number of electron pairs surrounding their central atoms. It is also named the Gillespie-Nyholm theory after its two main developers, the acronym VSEPR is pronounced either ves-pur or vuh-seh-per. Gillespie has emphasized that the repulsion due to the Pauli exclusion principle is more important in determining molecular geometry than the electrostatic repulsion. VSEPR theory is based on electron density rather than mathematical wave functions and hence unrelated to orbital hybridisation. While it is qualitative, VSEPR has a quantitative basis in quantum chemical topology methods such as the electron localization function. In 1957, Ronald Gillespie and Ronald Sydney Nyholm of University College London refined this concept into a detailed theory. In VSEPR theory, a bond or triple bond are treated as a single bonding group. The sum of the number of atoms bonded to a central atom, the number of electron pairs, therefore, determines the overall geometry that they will adopt. For example, when there are two electron pairs surrounding the atom, their mutual repulsion is minimal when they lie at opposite poles of the sphere. Therefore, the atom is predicted to adopt a linear geometry. If there are 3 electron pairs surrounding the atom, their repulsion is minimized by placing them at the vertices of an equilateral triangle centered on the atom. Therefore, the geometry is trigonal. Likewise, for 4 electron pairs, the arrangement is tetrahedral. The overall geometry is further refined by distinguishing between bonding and nonbonding electron pairs, the bonding electron pair shared in a sigma bond with an adjacent atom lies further from the central atom than a nonbonding pair of that atom, which is held close to its positively charged nucleus. VSEPR theory therefore views repulsion by the pair to be greater than the repulsion by a bonding pair. As such, when a molecule has 2 interactions with different degrees of repulsion, for instance, when 5 valence electron pairs surround a central atom, they adopt a trigonal bipyramidal molecular geometry with two collinear axial positions and three equatorial positions. The difference between pairs and bonding pairs may also be used to rationalize deviations from idealized geometries. For example, the H2O molecule has four pairs in its valence shell
33.
Enantiomer
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A single chiral atom or similar structural feature in a compound causes that compound to have two possible structures which are non-superposable, each a mirror image of the other. Each member of the pair is termed an enantiomorph, the property is termed enantiomerism. The presence of multiple features in a given compound increases the number of geometric forms possible. Enantiopure compounds refer to samples having, within the limits of detection and they are sometimes called optical isomers for this reason. Enantiomer members often have different chemical reactions with other enantiomer substances, since many biological molecules are enantiomers, there is sometimes a marked difference in the effects of two enantiomers on biological organisms. Owing to this discovery, drugs composed of one enantiomer can be developed to enhance the pharmacological efficacy. An example is eszopiclone, which is enantiopure and therefore administered in doses that are exactly 1/2 of the older, in the case of eszopiclone, the S enantiomer is responsible for all the desired effects, while the other enantiomer seems to be inactive. A dose of 2 mg of zopiclone must be administered to produce the therapeutic effect as 1 mg of eszopiclone. In chemical synthesis of enantiomeric substances, non-enantiomeric precursors inevitably produce racemic mixtures, in the absence of an effective enantiomeric environment, separation of a racemic mixture into its enantiomeric components is impossible. The R/S system is an important nomenclature system for denoting distinct enantiomers, another system is based on prefix notation for optical activity, - and - or d- and l-. The Latin for left and right is laevus and dexter, respectively, left and right have always had moral connotations, and the Latin words for these are sinister and rectus. The English word right is a cognate of rectus and this is the origin of the D, L and S, R notations, and the employment of prefixes levo- and dextro- in common names. Most compounds that one or more asymmetric carbon atoms show enantiomerism. There are a few compounds that do have asymmetric carbon atoms. An example of such an enantiomer is the sedative thalidomide, which was sold in a number of countries across the world from 1957 until 1961 and it was withdrawn from the market when it was found to cause of birth defects. One enantiomer caused the desirable effects, while the other, unavoidably present in equal quantities. The herbicide mecoprop is a mixture, with the --enantiomer possessing the herbicidal activity. Another example is the antidepressant drugs escitalopram and citalopram, citalopram is a racemate, escitalopram is a pure enantiomer
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Lysergic acid
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Amides of lysergic acid, lysergamides, are widely used as pharmaceuticals and as psychedelic drugs. Lysergic acid received its name as it was a product of the lysis of various ergot alkaloids, lysergic acid is generally produced by hydrolysis of natural lysergamides, but can also be synthesized in the laboratory by a complex total synthesis for example by Woodwards team in 1956. An enantioselective total synthesis based on a palladium catalyzed domino cyclization reaction has been described in 2011 by Fujii, lysergic acid monohydrate crystallizes in very thin hexagonal leaflets when recrystallized from water. Lysergic acid monohydrate, when dried forms anhydrous lysergic acid, the biosynthetic route is based on the alkylation of the amino acid tryptophan with dimethylallyl diphosphate giving 4-dimethylallyl-L-tryptophan which is N-methylated with S-adenosyl-L-methionine. Oxidative ring closure followed by decarboxylation, reduction, cyclization, oxidation, lysergic acid is a chiral compound with two stereocenters. The isomer with inverted configuration at carbon atom 8 close to the group is called isolysergic acid. Inversion at carbon 5 close to the nitrogen atom leads to L-lysergic acid and L-isolysergic acid, lysergic acid is listed as a Table I precursor under the United Nations Convention Against Illicit Traffic in Narcotic Drugs and Psychotropic Substances. Lysergic acid diethylamide Ergine Ergoline Lysergamides
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Leucine
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Leucine is an α-amino acid used in the biosynthesis of proteins. It contains a group, an α-carboxylic acid group. It is essential in humans—meaning the body cannot synthesize it and thus must obtain from the diet, leucine is a major component of the subunits in ferritin, astacin, and other buffer proteins. Leucine is used in the liver, adipose tissue, and muscle tissue, adipose and muscle tissue use leucine in the formation of sterols. Combined leucine use in these two tissues is seven times greater than in the liver, leucine is an essential amino acid in the diet of animals because they lack the complete enzyme pathway to synthesize it de novo from potential precursor compounds. Consequently, they must ingest it, usually as a component of proteins, in healthy individuals, approximately 60% of dietary L-leucine is metabolized after several hours, with roughly 5% of dietary L-leucine being converted to β-hydroxy β-methylbutyric acid. Around 40% of dietary L-leucine is converted to acetyl-CoA, which is used in the synthesis of other compounds. The vast majority of L-leucine metabolism is initially catalyzed by the amino acid aminotransferase enzyme. α-Ketoisocaproate is mostly metabolized by the mitochondrial enzyme branched-chain α-ketoacid dehydrogenase, isovaleryl-CoA is subsequently metabolized by isovaleryl-CoA dehydrogenase and converted to β-methylcrotonoyl-CoA, which is used in the synthesis of acetyl-CoA and other compounds. A relatively small amount of α-KIC is metabolized in the liver by the cytosolic enzyme 4-hydroxyphenylpyruvate dioxygenase, in healthy individuals, this minor pathway – which involves the conversion of L-leucine to α-KIC and then HMB – is the predominant route of HMB synthesis. HMB could be produced via certain metabolites that are generated along this pathway, the metabolism of HMB is initially catalyzed by an uncharacterized enzyme which converts it to HMB-CoA. HMB-CoA is metabolized by either enoyl-CoA hydratase or another uncharacterized enzyme, MC-CoA is then converted by the enzyme methylcrotonyl-CoA carboxylase to methylglutaconyl-CoA, which is subsequently converted to HMG-CoA by methylglutaconyl-CoA hydratase. HMG-CoA is then cleaved into to acetyl-CoA and acetoacetate by HMG-CoA lyase or used in the production of cholesterol via the mevalonate pathway and it is a dietary amino acid with the capacity to directly stimulate muscle protein synthesis. As a dietary supplement, leucine has been found to slow the degradation of tissue by increasing the synthesis of muscle proteins in aged rats. However, results of studies are conflicted. Long-term leucine supplementation does not increase muscle mass or strength in healthy elderly men, more studies are needed, preferably ones based on an objective, random sample of society. Until then, dietary supplemental leucine cannot be associated as the reason for muscular growth or optimal maintenance for the entire population. Leucine potently activates the mammalian target of rapamycin kinase that regulates cell growth, infusion of leucine into the rat brain has been shown to decrease food intake and body weight via activation of the mTOR pathway