The Purnell equation is an equation used in analytical chemistry to calculate the resolution Rs between two peaks in a chromatogram. [1] [2]
where
The higher the resolution, the better the separation.
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions y(x) of Bessel's differential equation
In physics and thermodynamics, an equation of state is a thermodynamic equation relating state variables which describe the state of matter under a given set of physical conditions, such as pressure, volume, temperature (PVT), or internal energy. Equations of state are useful in describing the properties of fluids, mixtures of fluids, solids, and the interior of stars.
In physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, specifically in solids and some liquids. Often designated a quasiparticle, it is an excited state in the quantum mechanical quantization of the modes of vibrations for elastic structures of interacting particles. Phonons can be thought of as quantized sound waves, similar to photons as quantized light waves.
In mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for the purpose of modeling how a quantity such as heat diffuses through a given region.
Fractional calculus is a branch of mathematical analysis that studies the several different possibilities of defining real number powers or complex number powers of the differentiation operator D
In physics, specifically in quantum mechanics, a coherent state is the specific quantum state of the quantum harmonic oscillator, often described as a state which has dynamics most closely resembling the oscillatory behavior of a classical harmonic oscillator. It was the first example of quantum dynamics when Erwin Schrödinger derived it in 1926, while searching for solutions of the Schrödinger equation that satisfy the correspondence principle. The quantum harmonic oscillator and hence, the coherent states arise in the quantum theory of a wide range of physical systems. For instance, a coherent state describes the oscillating motion of a particle confined in a quadratic potential well. The coherent state describes a state in a system for which the ground-state wavepacket is displaced from the origin of the system. This state can be related to classical solutions by a particle oscillating with an amplitude equivalent to the displacement.
In mathematics and its applications, classical Sturm–Liouville theory is the theory of real second-order linear ordinary differential equations of the form:
In particle, atomic and condensed matter physics, a Yukawa potential is a potential of the form
In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are solutions of Laguerre's equation:
In signal processing, a comb filter is a filter implemented by adding a delayed version of a signal to itself, causing constructive and destructive interference. The frequency response of a comb filter consists of a series of regularly spaced notches, giving the appearance of a comb.
In condensed matter physics and crystallography, the static structure factor is a mathematical description of how a material scatters incident radiation. The structure factor is a critical tool in the interpretation of scattering patterns obtained in X-ray, electron and neutron diffraction experiments.
Paraboloidal coordinates are three-dimensional orthogonal coordinates that generalize two-dimensional parabolic coordinates. They possess elliptic paraboloids as one-coordinate surfaces. As such, they should be distinguished from parabolic cylindrical coordinates and parabolic rotational coordinates, both of which are also generalizations of two-dimensional parabolic coordinates. The coordinate surfaces of the former are parabolic cylinders, and the coordinate surfaces of the latter are circular paraboloids.
In classical mechanics, Appell's equation of motion is an alternative general formulation of classical mechanics described by Josiah Willard Gibbs in 1879 and Paul Émile Appell in 1900.
A hydrogen-like atom/ion (usually called a "hydrogenic atom") is any atomic nucleus bound to one electron and thus is isoelectronic with hydrogen. These atoms or ions can carry the positive charge , where is the atomic number of the atom. Examples of hydrogen-like atoms/ions are hydrogen itself, He+, Li2+, Be3+ and B4+. Because hydrogen-like atoms/ions are two-particle systems with an interaction depending only on the distance between the two particles, their (non-relativistic) Schrödinger equation can be solved in analytic form, as can the (relativistic) Dirac equation. The solutions are one-electron functions and are referred to as hydrogen-like atomic orbitals.
In electrochemistry, the Butler–Volmer equation, also known as Erdey-Grúz–Volmer equation, is one of the most fundamental relationships in electrochemical kinetics. It describes how the electrical current through an electrode depends on the voltage difference between the electrode and the bulk electrolyte for a simple, unimolecular redox reaction, considering that both a cathodic and an anodic reaction occur on the same electrode:
The fractional Schrödinger equation is a fundamental equation of fractional quantum mechanics. It was discovered by Nick Laskin (1999) as a result of extending the Feynman path integral, from the Brownian-like to Lévy-like quantum mechanical paths. The term fractional Schrödinger equation was coined by Nick Laskin.
An RLC circuit is an electrical circuit consisting of a resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel. The name of the circuit is derived from the letters that are used to denote the constituent components of this circuit, where the sequence of the components may vary from RLC.
The fundamental resolution equation is used in chromatography to help relate adjustable chromatographic parameters to resolution, and is as follows:
Ritz ballistic theory is a theory in physics, first published in 1908 by Swiss physicist Walther Ritz. In 1908, Ritz published Recherches critiques sur l'Électrodynamique générale, a lengthy criticism of Maxwell-Lorentz electromagnetic theory, in which he contended that the theory's connection with the luminiferous aether made it "essentially inappropriate to express the comprehensive laws for the propagation of electrodynamic actions."
In mathematics, the Romanovski polynomials are one of three finite subsets of real orthogonal polynomials discovered by Vsevolod Romanovsky within the context of probability distribution functions in statistics. They form an orthogonal subset of a more general family of little-known Routh polynomials introduced by Edward John Routh in 1884. The term Romanovski polynomials was put forward by Raposo, with reference to the so-called 'pseudo-Jacobi polynomials in Lesky's classification scheme. It seems more consistent to refer to them as Romanovski–Routh polynomials, by analogy with the terms Romanovski–Bessel and Romanovski–Jacobi used by Lesky for two other sets of orthogonal polynomials.
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