query: what is the k in the rate law equation

pos: Rate equation positive integers, but they may also be zero, fractional, or negative. The constant "k" is the reaction rate constant or "rate coefficient" of the reaction and has units of 1/time. Its value may depend on conditions such as temperature, ionic strength, surface area of an adsorbent, or light irradiation. Elementary (single-step) reactions and reaction steps have reaction orders equal to the stoichiometric coefficients for each reactant. The overall reaction order, i.e. the sum of stoichiometric coefficients of reactants, is always equal to the molecularity of the elementary reaction. However complex (multi-step) reactions may or may not have reaction orders equal

pos: Molecularity reactions or steps. In a unimolecular reaction, a single molecule rearranges atoms forming different molecules. This is illustrated by the equation where P means Product(s). The reaction or reaction step is an isomerization if there is only one product molecule, or a dissociation if there is more than one product molecule. In either case the rate of the reaction or step is described by the first order rate law where [A] is the concentration of species A, t is time, and k is the reaction rate constant. As can be deduced from the rate law equation, the number of A

pos: Brønsted catalysis equation concluded from Brønsted analysis that phenalene acidity is very different from either indene acidity or phenylene acidity . Brønsted catalysis equation The Brønsted catalysis equation or law of correlation, after Johannes Nicolaus Brønsted, gives the relationship between acid strength and catalytic activity in general acid catalysis. formula_1 A plot of the common logarithm of the reaction rate constant k versus the logarithm of the ionization constant K for a series of acids (for example a group of substituted phenols or carboxylic acids) gives a straight line with slope α and intercept C. The Brønsted equation is a free-energy relationship. The

pos: Brønsted catalysis equation Brønsted catalysis equation The Brønsted catalysis equation or law of correlation, after Johannes Nicolaus Brønsted, gives the relationship between acid strength and catalytic activity in general acid catalysis. formula_1 A plot of the common logarithm of the reaction rate constant k versus the logarithm of the ionization constant K for a series of acids (for example a group of substituted phenols or carboxylic acids) gives a straight line with slope α and intercept C. The Brønsted equation is a free-energy relationship. The relationship implies that the Gibbs free energy for proton dissociation is proportional to the activation energy for the

pos: Bioreactor d(VS)/dt Now, apply general mass balance equation i.e., "Rate of Mass In – Rate of Mass Out + Rate of Generation = Accumulation" d(VS)/dt = FS – FS – Vr (6) rearranging above equation, we get dS/dt = D(S –S ) – r (7) where r is the rate of substrate consumption. For the chemical reaction, A ----> P We can write Where, ( –r) = rate of disappearance of A (r) = rate of formation of A k = reaction rate constant C = Concentration of reactant A n = order of reaction with respect to component A For

pos: Nucleophile that, in a nucleophilic displacement on benzyl chloride, the azide anion reacts 3000 times faster than water. The Ritchie equation, derived in 1972, is another free-energy relationship: where "N" is the nucleophile dependent parameter and "k" the reaction rate constant for water. In this equation, a substrate-dependent parameter like "s" in the Swain–Scott equation is absent. The equation states that two nucleophiles react with the same relative reactivity regardless of the nature of the electrophile, which is in violation of the reactivity–selectivity principle. For this reason, this equation is also called the "constant selectivity relationship". In the original publication the

pos: Rate equation several elementary steps can, of course, be of any (including non-integer) order. Where M stands for concentration in molarity (mol · L), "t" for time, and "k" for the reaction rate constant. The half-life of a first order reaction is often expressed as "t" = 0.693/"k" (as ln2 = 0.693). In fractional order reactions, the order is a non-integer, which often indicates a chemical chain reaction or other complex reaction mechanism. For example, the pyrolysis of acetaldehyde (CHCHO) into methane and carbon monoxide proceeds with an order of 1.5 with respect to acetaldehyde: "r" = "k"[CHCHO]. The decomposition of phosgene

neg: Rate equation occasionally pseudo–second-order) rate equation. For a typical second-order reaction with rate equation r = k[A][B], if the concentration of reactant B is constant then r = k[A][B] = k'[A], where the pseudo–first-order rate constant k' = k[B]. The second-order rate equation has been reduced to a pseudo–first-order rate equation, which makes the treatment to obtain an integrated rate equation much easier. One way to obtain a pseudo-first order reaction is to use a large excess of one reactant (say, [B]≫[A]) so that, as the reaction progresses, only a small fraction of the reactant in excess (B) is consumed, and its

neg: Quantum superposition up to 1: So that the total probability will be preserved, K is what is called a stochastic matrix. When no time passes, nothing changes: for 0 elapsed time formula_43, the K matrix is zero except from a state to itself. So in the case that the time is short, it is better to talk about the rate of change of the probability instead of the absolute change in the probability. where formula_45 is the time derivative of the K matrix: The equation for the probabilities is a differential equation that is sometimes called the master equation: The R matrix

neg: Rate equation the pyridine and the ruthenium center. A "first order reaction" depends on the concentration of only one reactant (a "unimolecular reaction"). Other reactants can be present, but each will be zero order. The rate law for such a reaction is The half-life is independent of the starting concentration and is given by formula_20. Examples of such reactions are: In organic chemistry, the class of S1 (nucleophilic substitution unimolecular) reactions consists of first-order reactions. For example, in the reaction of aryldiazonium ions with nucleophiles in aqueous solution ArN + X → ArX + N, the rate equation is "r" = "k"[ArN],

neg: Enzyme kinetics unimolecular rate constant "k". If the reaction path proceeds over one or several intermediates, "k" will be a function of several elementary rate constants, whereas in the simplest case of a single elementary reaction (e.g. no intermediates) it will be identical to the elementary unimolecular rate constant "k". The apparent unimolecular rate constant "k" is also called "turnover number" and denotes the maximum number of enzymatic reactions catalysed per second. The Michaelis–Menten equation describes how the (initial) reaction rate "v" depends on the position of the substrate-binding equilibrium and the rate constant "k". with the constants This Michaelis–Menten equation is

neg: Reaction rate reaction rate. For this example, a possible mechanism is: Reactions 1 and 3 are very rapid compared to the second, so the slow reaction 2 is the rate determining step. This is a bimolecular elementary reaction whose rate is given by the second order equation: where "k" is the rate constant for the second step. However NO is an unstable intermediate whose concentration is determined by the fact that the first step is in equilibrium, so that , where "K" is the equilibrium constant of the first step. Substitution of this equation in the previous equation leads to a rate

neg: Rate equation of the concentration is given by The time dependence for a rate proportional to two unequal concentrations is if the concentrations are equal, they satisfy the previous equation. The second type includes nucleophillic addition-elimination reactions, such as the alkaline hydrolysis of ethyl acetate: This reaction is first-order in each reactant and second-order overall: "r" = "k"[CHCOOCH][OH] If the same hydrolysis reaction is catalyzed by imidazole, the rate equation becomes "r" = "k"[imidazole][CHCOOCH]. The rate is first-order in one reactant (ethyl acetate), and also first-order in imidazole which as a catalyst does not appear in the overall chemical equation. Another well-known

neg: Reaction rate equation expressed in terms of the original reactants This agrees with the form of the observed rate equation if it is assumed that . In practice the rate equation is used to suggest possible mechanisms which predict a rate equation in agreement with experiment. The second molecule of H does not appear in the rate equation because it reacts in the third step, which is a rapid step "after" the rate-determining step, so that it does not affect the overall reaction rate. Each reaction rate coefficient k has a temperature dependency, which is usually given by the Arrhenius equation: "E"

neg: Golden Rule savings rate what is needed to (1) equip any additional workers and (2) replace any worn out capital. In a steady state, therefore: formula_2, where "n" is the constant exogenous population growth rate, and "d" is the constant exogenous rate of depreciation of capital. Since "n" and "d" are constant and formula_3 satisfies the Inada conditions, this expression may be read as an equation connecting "s" and "k" in steady state: any choice of "s" implies a unique value for "k" (thus also for "y") in steady state. Since consumption is proportional to output ( formula_4 ), then a choice of value

neg: Thermal conduction varies with spatial location. For many simple applications, Fourier's law is used in its one-dimensional form. In the "x"-direction, By integrating the differential form over the material's total surface formula_13, we arrive at the integral form of Fourier's law: where (including the SI units): The above differential equation, when integrated for a homogeneous material of 1-D geometry between two endpoints at constant temperature, gives the heat flow rate as: where This law forms the basis for the derivation of the heat equation. Writing where "U" is the conductance, in W/(m K). Fourier's law can also be stated as: The reciprocal

neg: Reaction rate + "b" B → "p" P + "q" Q, the rate equation or rate law is a mathematical expression used in chemical kinetics to link the rate of a reaction to the concentration of each reactant. It is often of the type: For gas phase reaction the rate is often alternatively expressed by partial pressures. In these equations "k"("T") is the "reaction rate coefficient" or "rate constant", although it is not really a constant, because it includes all the parameters that affect reaction rate, except for concentration, which is explicitly taken into account. Of all the parameters influencing reaction rates,

neg: Logistic function for the growth of bacteria in broth and experimentally tested using a technique for nonlinear parameter estimation. The equation is also sometimes called the "Verhulst-Pearl equation" following its rediscovery in 1920 by Raymond Pearl (1879–1940) and Lowell Reed (1888–1966) of the Johns Hopkins University. Another scientist, Alfred J. Lotka derived the equation again in 1925, calling it the "law of population growth". Letting "P" represent population size ("N" is often used in ecology instead) and "t" represent time, this model is formalized by the differential equation: where the constant "r" defines the growth rate and "K" is the carrying capacity.

neg: E-folding to describe this first order kinetic process: What this ordinary differential equation states is that a change (in this case the disappearance) of the concentration of A, "d"[A]/"dt", is equal to the rate constant "k" multiplied by the concentration of A. Consider what the units of "k" would be. On the left hand side, we have a concentration divided by a unit of time. The units for "k" would need to allow for these to be replicated on the right hand side. For this reason, the units of "k", here, would be 1/time. Because this is a linear, homogeneous and

neg: Schwarzschild's equation for radiative transfer when working with a laboratory spectrophotometer, where the sample is near 300 °K and the light source is a filament at several thousand °K. If the medium is homogeneous, "nσ" doesn't vary with location. Integration over a path of length s affords the form of Beer's Law used most often in the laboratory experiments: Schwarzschild’s equation provides a simple explanation for the existence of the greenhouse effect and demonstrates that it requires a non-zero lapse rate. Rising air in the atmosphere expands and cools as the pressure on it falls, producing a negative temperature gradient in the Earth's troposphere. When

neg: Hammett equation equation k is the reference reaction rate of the unsubstituted reactant, and k that of a substituted reactant. A plot of log(K/K) for a given equilibrium versus log(k/k) for a given reaction rate with many differently substituted reactants will give a straight line. The starting point for the collection of the substituent constants is a chemical equilibrium for which both the substituent constant and the reaction constant are arbitrarily set to 1: the ionization of benzoic acid (R and R' both H) in water at 25 °C. Having obtained a value for K, a series of equilibrium constants (K) are

neg: Rate equation Rate equation The rate law or rate equation for a chemical reaction is an equation that links the reaction rate with the concentrations or pressures of the reactants and constant parameters (normally rate coefficients and partial reaction orders). For many reactions the rate is given by a power law such as where [A] and [B] express the concentration of the species A and B (usually in moles per liter (molarity, M)). The exponents "x" and "y" are the partial "orders of reaction" for A and B and the "overall" reaction order is the sum of the exponents. These are often

neg: Associative substitution 200 pm. The result of the rate law is that at high concentrations of Y, the rate approximates k[M] while at low concentrations the result is kK[M][Y]. The Eigen-Fuoss equation shows that higher values of K (and thus a faster pre-equilibrium) are obtained for large, oppositely-charged ions in solution. Associative substitution Associative substitution describes a pathway by which compounds interchange ligands. The terminology is typically applied to coordination and organometallic complexes, but resembles the Sn2 mechanism in organic chemistry. The opposite pathway is dissociative substitution, being analogous to the Sn1 pathway. Intermediate pathways exist between the pure associative and pure

neg: Luigi Pasinetti only refer to that part of profits that accrue to the capitalists. Specifically, the modified Cambridge equation would take this peculiar form: Note that the preceding equation now shows not the profit rate of the economy, "P/K", but a ratio, "P /K", which has no economic significance if taken as such. To fix this anomaly, Pasinetti added "P /K", to both sides of the equality: To complete the formulation, Pasinetti simply assumes that, in the long-term, the variable "i", representing the interest rate earned by workers when they lend their savings to capitalists, is equal to the profit rate, "P/K".

neg: Radical polymerization mechanisms. "Effects of chain transfer:" The most obvious effect of chain transfer is a decrease in the polymer chain length. If the rate of transfer is much larger than the rate of propagation, then very small polymers are formed with chain lengths of 2-5 repeating units (telomerization). The Mayo equation estimates the influence of chain transfer on chain length ("x"): formula_1. Where "k" is the rate constant for chain transfer and "k" is the rate constant for propagation. The Mayo equation assumes that transfer to solvent is the major termination pathway. There are four industrial methods of radical polymerization: Other

neg: Receptor–ligand kinetics can be determined analytically. Choosing [R] as the independent concentration and representing the concentrations by italic variables for brevity (e.g., formula_10), the kinetic rate equation can be written Dividing both sides by "k" and introducing the constant "2E = R - L - K", the rate equation becomes where the two equilibrium concentrations formula_13 are given by the quadratic formula and "D" is defined However, only the formula_15 equilibrium has a positive concentration, corresponding to the equilibrium observed experimentally. Separation of variables and a partial-fraction expansion yield the integrable ordinary differential equation whose solution is or, equivalently, formula_19 for association,

neg: Catalysis overall reaction equation: As a catalyst is regenerated in a reaction, often only small amounts are needed to increase the rate of the reaction. In practice, however, catalysts are sometimes consumed in secondary processes. The catalyst does usually appear in the rate equation. For example, if the rate-determining step in the above reaction scheme is the first step X + C → XC, the catalyzed reaction will be second order with rate equation v = k[X][C], which is proportional to the catalyst concentration [C]. However [C] remains constant during the reaction so that the catalyzed reaction is pseudo-first order: v

neg: Plateau principle used for any substance or quantity that is being produced at a measurable rate and degraded with first-order kinetics. Because the equation applies in many instances of mass balance, it has very broad applicability in addition to pharmacokinetics. The most important inference derived from the steady state equation and the equation for fractional change over time is that the elimination rate constant ("k") or the sum of rate constants that apply in a model determine the time course for change in mass when a system is perturbed (either by changing the rate of inflow or production, or by changing the

neg: Power-law fluid Hagen–Poiseuille equation), a power-law fluid will result in a power-law velocity profile, where "u"("r") is the (radially) local axial velocity, "" is the pressure gradient along the pipe, and "R" is the pipe radius. Power-law fluid A power-law fluid, or the Ostwald–de Waele relationship, is a type of generalized Newtonian fluid (time independent Non-Newtonian fluid) for which the shear stress, "τ", is given by where: The quantity represents an "apparent" or "effective viscosity" as a function of the shear rate (SI unit Pa s). The value of K and n can be obtained from graph off log(µ ) and formula_3.

neg: Rate equation concentration can be considered to stay constant. For example, the hydrolysis of esters by dilute mineral acids follows pseudo-first order kinetics where the concentration of water is present in large excess: The hydrolysis of sucrose in acid solution is often cited as a first-order reaction with rate r = k[sucrose]. The true rate equation is third-order, r = k[sucrose][H][HO]; however, the concentrations of both the catalyst H and the solvent HO are normally constant, so that the reaction is pseudo–first-order. Elementary reaction steps with order 3 (called "ternary reactions") are rare and unlikely to occur. However, overall reactions composed of

neg: Charles's law does not permit the assignment of any law stating the linear relation. Both Dalton's and Gay-Lussac's main conclusions can be expressed mathematically as: where "V" is the volume occupied by a given sample of gas at 100 °C; "V" is the volume occupied by the same sample of gas at 0 °C; and "k" is a constant which is the same for all gases at constant pressure. This equation does not contain the temperature and so has nothing to do with what became known as Charles' Law. Gay-Lussac's value for "k" (), was identical to Dalton's earlier value for vapours

neg: Rate equation deduced from the chemical equation and must be determined by experiment. A common form for the rate equation is a power law: The constant is called the "rate constant". The exponents, which can be fractional, are called "partial orders of reaction" and their sum is the overall order of reaction. In a dilute solution, an elementary reaction (one having a single step with a single transition state) is empirically found to obey the law of mass action. This predicts that the rate depends only on the concentrations of the reactants, raised to the powers of their stoichiometric coefficients. The natural

neg: Dilution (equation) replaced with "clean" air or gas. For example, the equation can be used to calculate the time required at a certain ventilation rate to reduce a high carbon monoxide concentration in a room. Sometimes the equation is also written as: The basic room purge equation can be used only for purge scenarios. In a scenario where a liquid continuously evaporates from a container in a ventilated room, a differential equation has to be used: where the ventilation rate has been adjusted by a mixing factor "K": The dilution in welding terms is defined as the weight of the base metal

neg: Tafel equation reaction rate is negligible compared to the forward reaction rate. The Tafel equation is applicable to the region where the values of polarization are high. At low values of polarization, the dependence of current on polarization is usually linear (not logarithmic): This linear region is called "polarization resistance" due to its formal similarity to Ohm's law. Tafel equation The Tafel equation is an equation in electrochemical kinetics relating the rate of an electrochemical reaction to the overpotential. The Tafel equation was first deduced experimentally and was later shown to have a theoretical justification. The equation is named after Swiss chemist

neg: Rate-determining step is second-order in and zero-order in CO, with rate equation "r" = "k"[]. This suggests that the rate is determined by a step in which two molecules react, with the CO molecule entering at another, faster, step. A possible mechanism in two elementary steps that explains the rate equation is: In this mechanism the reactive intermediate species is formed in the first step with rate "r" and reacts with CO in the second step with rate "r". However can also react with NO if the first step occurs in the "reverse direction" (NO + → 2 ) with rate "r",

neg: Equation of exchange often omitted for simplicity.) The Cambridge equation for demand for cash balances is thus: which, given the classical dichotomy and that real income must equal expenditures formula_22, is equivalent to Assuming that the economy is at equilibrium (formula_36), that real income is exogenous, and that "k" is fixed in the short run, the Cambridge equation is equivalent to the equation of exchange with velocity equal to the inverse of "k": The money demand function is often conceptualized in terms of a "liquidity function", formula_38, where formula_40 is real income and formula_41 is the real rate of interest. If formula_21 is

neg: Dilatant parameters, all shear thickening fluids are stabilized suspensions and have a volume fraction of solid that is relatively high. Viscosity of a solution as a function of shear rate is given via the Power Law equation, where η is the viscosity, K is a material-based constant, and γ̇ is the applied shear rate. Dilatant behavior occurs when n is greater than 1. Below is a table of viscosity values for some common materials. A suspension is composed of a fine, particulate phase dispersed throughout a differing, heterogeneous phase. Shear-thickening behavior is observed in systems with a solid, particulate phase dispersed

neg: Non-linear effects heterochiral complexes at K=4. Obeying the same kinetic rate law, each of the three catalytic complexes catalyze the desired reaction to form product. As enantiomers of each other, the homochiral complexes catalyze the reaction at the same rate, although opposite absolute configuration of the product is induced (i.e. r=r). The heterochiral complex, however, forms a racemic product at a different rate constant (i.e. r). In order to describe the ML model in quantitative parameters, Kagan and coworkers described the following formula: In the correction factor, Kagan and co-workers introduced two new parameters absent in Equation 1, β and g. In

neg: Scattering rate finding formula_13 is found by evaluating formula_14. In case of constant perturbation,formula_15 is calculated by Using the equation which is The transition rate of an electron from the initial state formula_19 to final state formula_20 is given by where formula_22 and formula_23 are the energies of the initial and final states including the perturbation state and ensures the formula_24-function indicate energy conservation. The scattering rate w(k) is determined by summing all the possible finite states k' of electron scattering from an initial state k to a final state k', and is defined by The integral form is Scattering rate A

neg: Grain growth holds that the rate of growth is proportional to the driving force and that the driving force is proportional to the total amount of grain boundary energy, then it can be shown that the time "t" required to reach a given grain size is approximated by the equation formula_8 where "d" is the initial grain size, "d" is the final grain size and k is a temperature dependent constant given by an exponential law: formula_9 where "k" is a constant, "T" is the absolute temperature and "Q" is the activation energy for boundary mobility. Theoretically, the activation energy for boundary

neg: Master equation As always, the first subscript represents the row, the second subscript the column. That is, the source is given by the second subscript, and the destination by the first subscript. This is the opposite of what one might expect, but it is technically convenient. For each state "k", the increase in occupation probability depends on the contribution from all other states to "k", and is given by: where formula_12 is the probability for the system to be in the state formula_13, while the matrix formula_3 is filled with a grid of transition-rate constants. Similarly, formula_15 contributes to the occupation of

neg: Bicarbonate buffer system easier to obtain from measurement than carbonic acid, the Henry’s Law solubility constant – which relates the partial pressure of a gas to its solubility – for CO in plasma is used in lieu of the carbonic acid concentration. After rearranging the equation and applying Henry’s Law, the equation becomes: Where K’ is the dissociation constant from the pK of carbonic acid, 6.1, which is equal to 800nmol/L (since K’ = 10 = 10 ≈ 8.00X10mol/L = 800nmol/L). By multiplying K’ (expressed as nmol/L) and 0.03 (800 X 0.03 = 24) and rearranging with respect to HCO, the equation is

neg: R/K selection theory is the population, "r" is the maximum growth rate, "K" is the carrying capacity of the local environment, and "dN/dt", the derivative of "N" with respect to time "t", is the rate of change in population with time. Thus, the equation relates the growth rate of the population "N" to the current population size, incorporating the effect of the two constant parameters "r" and "K". (Note that decrease is negative growth.) The choice of the letter "K" came from the German "Kapazitätsgrenze" (capacity limit), while "r" came from "rate". "r"-selected species are those that emphasize high growth rates, typically exploit

neg: Graph equation solutions of the graph equation "L"("K") = "T"("K") and "G" = "K", and "H" = "K" are the solutions of the graph equation "L"("K") = "T"("K"). Note that "T"("K") is a 4-regular graph on 6 vertices. Graph equation In graph theory, Graph equations are equations in which the unknowns are graphs. One of the central questions of graph theory concerns the notion of isomorphism. We ask: When are two graphs the same (i.e., graph isomorphism)? The graphs in question may be expressed differently in terms of graph equations. What are the graphs (solutions) "G" and "H" such that the line

neg: Hypercycle (chemistry) consequence, the system loses its internal stability and cannot live on. The reason for inability to survive is the lack of mutual control of constituent abundances. The dynamics of the elementary hypercycle can be modelled using the following differential equation: where In the equation above, "x" is the concentration of template "I"; "x" is the total concentration of all templates; "k" is the excess production rate of template "I", which is a difference between formation "f" by self-replication of the template and its degradation "d", usually by hydrolysis; "k" is the production rate of template "I" catalysed by "I"; and

neg: Nicholson–Bailey model explained based on probability. formula_2 is the probability that the host will survive formula_3 predators; whereas formula_4 is that they will not, bearing in mind the parasitoid eventually will hatch into larva and escape. Density dependence can be added to the model, by assuming that the growth rate of the host decreases at high abundances. The equation for the parasitoid is unchanged, and the equation for the host is modified: formula_5 The host rate of increase "k" is replaced by "r", which decreases when the host population density reaches "K". Simulating this system usually results in oscillations moving away from

neg: Rate equation the concentration of one reactant is measured with all other reactants in large excess so that their concentration remains essentially constant. For a reaction a·A + b·B → c·C with rate law: formula_16, the partial order x with respect to A is determined using a large excess of B. In this case formula_17 with formula_18, and x may be determined by the integral method. The order y with respect to B under the same conditions (with B in excess) is determined by a series of similar experiments with a range of initial concentration [B] so that the variation of k'

neg: Edwards equation attacking atoms (e.g. oxygen, carbon, nitrogen) and each of these atoms has different nucleophilic characteristics. The Edwards equation attempts to account for this additional parameter by introducing a polarizability term. The first generation of the Edwards equation was where k and k are the rate constants for a nucleophile and a standard (HO). H is a measure of the basicity of the nucleophile relative to protons, as defined by the equation: where the pK is that of the conjugate acid of the nucleophile and the constant 1.74 is the correction for the pK of HO. E is the term Edwards

neg: Elimination rate constant Elimination rate constant The elimination rate constant "K" is a value used in pharmacokinetics to describe the rate at which a drug is removed from the system. It is often abbreviated "K" or "K". It is equivalent to the fraction of a substance that is removed per unit time measured at any particular instant and has units of T. This can be expressed mathematically with the differential equation where formula_2 is the blood plasma concentration of drug in the system at a given point in time formula_3, formula_4 is an infinitely small change in time, and formula_5 is the concentration

neg: Rate equation expression for the above reactions (assuming each one is elementary) can be expressed as: where: k is the rate coefficient for the reaction that consumes A and B; k is the rate coefficient for the backwards reaction, which consumes X and Y and produces A and B. The constants k and k are related to the equilibrium coefficient for the reaction (K) by the following relationship (set "r"=0 in balance): In a simple equilibrium between two species: Where the reactions starts with an initial concentration of A, <chem>[A]0</chem>, with an initial concentration of 0 for B at time "t"=0. Then

neg: Transition state theory In the development of TST, three approaches were taken as summarized below In 1884, Jacobus van't Hoff proposed the Van 't Hoff equation describing the temperature dependence of the equilibrium constant for a reversible reaction: where Δ"U" is the change in internal energy, "K" is the equilibrium constant of the reaction, "R" is the universal gas constant, and "T" is thermodynamic temperature. Based on experimental work, in 1889, Svante Arrhenius proposed a similar expression for the rate constant of a reaction, given as follows: Integration of this expression leads to the Arrhenius equation where "k" is the rate constant. "A"

neg: Turbulence modeling is the specific rate of dissipation (of the turbulence kinetic energy k into internal thermal energy). SST (Menter's shear stress transport) turbulence model is a widely used and robust two-equation eddy-viscosity turbulence model used in computational fluid dynamics. The model combines the k-omega turbulence model and K-epsilon turbulence model such that the k-omega is used in the inner region of the boundary layer and switches to the k-epsilon in the free shear flow. The Reynolds stress equation model (RSM), also referred to as second moment closure model , is the most complete classical turbulence modelling approach. Popular eddy-viscosity based models

neg: Chemical kinetics derived for others. Elementary reactions follow the law of mass action, but the rate law of stepwise reactions has to be derived by combining the rate laws of the various elementary steps, and can become rather complex. In consecutive reactions, the rate-determining step often determines the kinetics. In consecutive first order reactions, a steady state approximation can simplify the rate law. The activation energy for a reaction is experimentally determined through the Arrhenius equation and the Eyring equation. The main factors that influence the reaction rate include: the physical state of the reactants, the concentrations of the reactants, the temperature

neg: Fundamental resolution equation Fundamental resolution equation The fundamental resolution equation is used in chromatography to help relate adjustable chromatographic parameters to resolution, and is as follows: R = [N/4][(α-1)/α][k'/(1+k')], where N = Number of theoretical plates α = Selectivity Term = k'/k' The [N/4] term is the column factor, the [(α-1)/α] term is the thermodynamic factor, and the [k'/(1+k')] term is the retention factor. The 3 factors are not completely independent, but they are very close, and can be treated as such. So what does this mean? It means that to increase resolution of two peaks on a chromatogram, one of the three

neg: Rate equation are consumed) are known as stoichiometric coefficients. One molecule of A combines with two of B to form 3 of C, so if we use the symbol [X] for the number of moles of chemical X, If the reaction takes place in a closed system at constant temperature and volume, without a build-up of reaction intermediates, the "reaction rate" is defined as where is the stoichiometric coefficient for chemical X. The reaction rate has some functional dependence on the concentrations of the reactants, and this dependence is known as the "rate equation" or "rate law". This law generally cannot be

neg: Rate equation concentrations measured over a longer time (several half-lives) with the integrated form of the rate equation. For example, the integrated rate law for a first-order reaction is where [A] is the concentration at time t and [A] is the initial concentration at zero time. The first-order rate law is confirmed if formula_14 is in fact a linear function of time. In this case the rate constant formula_15 is equal to the slope with sign reversed. The partial order with respect to a given reactant can be evaluated by the method of flooding (or of isolation) of Ostwald. In this method,

neg: Taft equation proposed that polar effects would only influence the reaction rate of the base catalyzed reaction since a new charge was formed. He defined the polar substituent constant σ* as: where log(k/k) is the ratio of the rate of the base catalyzed reaction compared to the reference reaction, log(k/k) is ratio of a rate of the acid catalyzed reaction compared to the reference reaction, and ρ* is a reaction constant that describes the sensitivity of the reaction series. For the definition reaction series, ρ* was set to 1 and R = methyl was defined as the reference reaction (σ* = zero).

neg: Eadie–Hofstee diagram Eadie–Hofstee diagram In biochemistry, an Eadie–Hofstee diagram (also Woolf–Eadie–Augustinsson–Hofstee or Eadie–Augustinsson plot) is a graphical representation of enzyme kinetics in which reaction rate is plotted as a function of the ratio between rate and substrate concentration: where "v" represents reaction rate, "K" is the Michaelis–Menten constant, ["S"] is the substrate concentration, and "V" is the maximum reaction rate. It can be derived from the Michaelis–Menten equation as follows: invert and multiply with formula_3: Rearrange: Isolate v: A plot of v against "v"/[S] will hence yield "V" as the y-intercept, "V"/K as the x-intercept, and "K" as the negative slope. Like

neg: Gas laws its container is directly proportional to its absolute temperature. As a mathematical equation, Gay-Lussac's law is written as either: K=P divided by T Avogadro's law states that the volume occupied by an ideal gas is directly proportional to the number of molecules of the gas present in the container. This gives rise to the molar volume of a gas, which at STP (273.15 K, 1 atm) is about 22.4 L. The relation is given by The Combined gas law or General Gas Equation is obtained by combining Boyle's Law, Charles's law, and Gay-Lussac's Law. It shows the relationship between the

neg: Haken-Kelso-Bunz model the inverse of the movement rate or oscillation frequency in the experiment. The equation: ϕ˙=−sinϕ−2ksin2ϕ The equation predicts that for k > 0.25 relative phase values of 0 ±π are both stable, a condition coined as bistability. An increase in movement rate, starting in parallel-phase, leads to a switch to anti-parallel phase at a critical frequency. Starting with a large k and decreasing k leads to a destabilization of the fixed point at π which becomes unstable at the value kc=0.25. The HKB model has had a profound effect on many conceptual, methodological, and practical models since its inception. HKB

neg: Differential equation a quantum system (usually atoms, molecules, and subatomic particles whether free, bound, or localized). It is not a simple algebraic equation, but in general a linear partial differential equation, describing the time-evolution of the system's wave function (also called a "state function"). The Lotka–Volterra equations, also known as the predator–prey equations, are a pair of first-order, non-linear, differential equations frequently used to describe the population dynamics of two species that interact, one as a predator and the other as prey. The "rate law" or rate equation for a chemical reaction is a differential equation that links the reaction rate with

neg: Uncompetitive inhibitor the inhibitor and "K" is an inhibition constant characteristic of the inhibitor. The Michaelis-Menten equation is altered to: where As described by the equation above, at high concentrations of substrate, "V" approaches "V"/α'. Thus, an uncompetitive inhibitor lowers the measured "V". Apparent "K" also decreases, because [S] required to reach one-half "V" decreases by the factor α'. It is important to note that V and K decrease at the same rate as a result of the inhibitor. This is apparent when viewing a Lineweaver-Burk plot of uncompetitive enzyme inhibition: the ratio between V and K remains the same with or

neg: Solow residual which can be deduced in a simple regression. In a regression analysis, the equation one would estimate is: where: "y" is (log) output, ln(Y) "k" is capital, ln(K) "ℓ" is labour, ln(L) "C" can be interpreted as the co-efficient on log("A") – the rate of technological change – (1 − "α"). Given the form of the regression equation, we can interpret the coefficients as elasticities. For calculation of the actual quantity/ level of technology formula_11 we simply refer back to our equation in levels. Knowing quantities of output formula_13, Capital formula_14, Labor formula_15 and estimates for formula_16,formula_17 and formula_18 we

neg: Edwards equation 1953: This equation relates the rate constant "k", of a reaction, normalized to that of a standard reaction with water as the nucleophile ("k"), to a nucleophilic constant "n" for a given nucleophile and a substrate constant "s" that depends on the sensitivity of a substrate to nucleophilic attack (defined as 1 for methyl bromide). This equation was modeled after the Hammett equation. However, both the Swain–Scott equation and the Brønsted relationship make the rather inaccurate assumption that all nucleophiles have the same reactivity with respect to a specific reaction site. There are several different categories of nucleophiles with different

neg: Weighted average cost of capital component is stated as k" and since there is a tax benefit from interest payments then the after tax WACC component is k(1-T); where T is the tax rate. Advantages: Disadvantages: 3 ways of calculating KK: "Cost of new equity should be the adjusted cost for any underwriting fees terme flotation costs (F)" K = D/P(1-F) + g; where F = flotation costs, D is dividends, P is price of the stock, and g is the growth rate. Weighted average cost of capital equation: WACC= (W)[(K)(1-t)]+ (W)(K)+ (W)(K) Marginal cost of capital (MCC) schedule or an investment opportunity curve is

neg: Rate equation straight line with slope "k" + "k". By measurement of A and B the values of "K" and the two reaction rate constants will be known. If the concentration at the time "t" = 0 is different from above, the simplifications above are invalid, and a system of differential equations must be solved. However, this system can also be solved exactly to yield the following generalized expressions: When the equilibrium constant is close to unity and the reaction rates very fast for instance in conformational analysis of molecules, other methods are required for the determination of rate constants for instance

neg: Arrhenius equation vibrational) energy will all determine the chance that the collision will produce a product molecule AB. Macroscopic measurements of E and "k" are the result of many individual collisions with differing collision parameters. To probe reaction rates at molecular level, experiments are conducted under near-collisional conditions and this subject is often called molecular reaction dynamics. There are deviations from the Arrhenius law during the glass transition in all classes of glass-forming matter. The Arrhenius law predicts that the motion of the structural units (atoms, molecules, ions, etc.) should slow down at a slower rate through the glass transition than is

neg: Bioreactor first order reaction, n = 1 and accordingly, –r = k C The reaction kinetics involved in biochemical operations is comparatively difficult to obtain than the chemical reaction kinetics. In biochemical operations, the cell kinetics is used for the unstructured models where balanced growth condition is assumed. The following equation is used to represent the net rate of cell mass growth: r = "μx (10)" "where μ" is the specific growth rate or specific growth rate coefficient(s). Here, "μ" is analogous to first order rate constant k but however, "μ" is not a constant. In biochemical engineering, yield is defined

neg: Activation energy absolute temperature (usually in kelvins), and "k" is the reaction rate coefficient. Even without knowing "A", "E" can be evaluated from the variation in reaction rate coefficients as a function of temperature (within the validity of the Arrhenius equation). At a more advanced level, the net Arrhenius activation energy term from the Arrhenius equation is best regarded as an experimentally determined parameter that indicates the sensitivity of the reaction rate to temperature. There are two objections to associating this activation energy with the threshold barrier for an elementary reaction. First, it is often unclear as to whether or not reaction

neg: Autocatalysis letters are stoichiometric coefficients and the capital Latin letters represent chemical species. The chemical reaction proceeds in both the forward and reverse direction. This equation is easily generalized to any number of reactants, products, and reactions. In chemical equilibrium the forward and reverse reaction rates are such that each chemical species is being created at the same rate it is being destroyed. In other words, the rate of the forward reaction is equal to the rate of the reverse reaction. Here, the brackets indicate the concentration of the chemical species, in moles per liter, and k and k are rate

neg: Rate equation logarithm of the power-law rate equation is This can be used to estimate the order of reaction of each reactant. For example, the initial rate can be measured in a series of experiments at different initial concentrations of reactant A with all other concentrations [B], [C], ... kept constant, so that The slope of a graph of formula_11 as a function of formula_12 then corresponds to the order x with respect to reactant A. However, this method is not always reliable because The tentative rate equation determined by the method of initial rates is therefore normally verified by comparing the

neg: Enzyme kinetics thus formula_5. A complete derivation can be found here. The Michaelis constant "K" is experimentally defined as the concentration at which the rate of the enzyme reaction is half "V", which can be verified by substituting [S] = "K" into the Michaelis–Menten equation and can also be seen graphically. If the rate-determining enzymatic step is slow compared to substrate dissociation (formula_6), the Michaelis constant "K" is roughly the dissociation constant "K" of the ES complex. If <chem>[S]</chem> is small compared to formula_7 then the term formula_8 and also very little ES complex is formed, thus <chem>[E]_{\rm tot} \approx [E]</chem>. Therefore,

neg: Surface and bulk erosion whether a certain material will undergo surface or bulk erosion by looking at the ratio between the two rates. The rate of diffusion of water is modeled by the equation formula_1 Where <x> is the mean length of the material and D is the diffusion coefficient of water inside the polymer. The rate of degradation is modeled by the following equation formula_2 Where M= molecular weight of polymer, N=Avogadro's number, N=degree of polymerization, p=density of polymer, k=degradation rate The ratio between diffusion time and degradation time gives us a dimensionless parameter ε called the erosion number. formula_3 If ε≫1, surface

neg: Dissociative substitution not affect the rate of reaction, leading to the simple rate equation: However, in some cases, the back reaction (k) becomes important, and [L'] can exert an effect on the overall rate of reaction. The backward reaction k therefore competes with the second forward reaction (k), thus the fraction of intermediate (denoted as "Int") that can react with L' to form the product is given by the expression <chem> \frac{\mathit k_2[L'][Int]} = \left(\frac{k_2[\ce L'][Int]}</ref> Rates vary by a factor of 10, [Ir(HO)] being the slowest and [Na(HO)] being one of the fastest for octahedral complexes. Charge has a significant influence

neg: Dynamic equilibrium in which B is converted into A. If both reactions are elementary reactions, then the rate of reaction is given by where "k" is the rate constant for the forward reaction and "k" is the rate constant for the backward reaction and the square brackets, [..] denote concentration . If only A is present at the beginning, time "t"=0, with a concentration [A], the sum of the two concentrations, [A] and [B], at time "t", will be equal to [A]. The solution to this differential equation is and is illustrated at the right. As time tends towards infinity, the concentrations

neg: Black's equation to the predominance of other failure modes. Black's equation Black's Equation is a mathematical model for the mean time to failure (MTTF) of a semiconductor circuit due to electromigration: a phenomenon of molecular rearrangement (movement) in the solid phase caused by an electromagnetic field. The equation is: formula_1 formula_2 is a constant formula_3 is the current density formula_4 is a model parameter formula_5 is the activation energy formula_6 is Boltzmann's constant formula_7 is the absolute temperature in K The model is abstract, not based on a specific physical model, but flexibly describes the failure rate dependence on the temperature, the

neg: Black's equation Black's equation Black's Equation is a mathematical model for the mean time to failure (MTTF) of a semiconductor circuit due to electromigration: a phenomenon of molecular rearrangement (movement) in the solid phase caused by an electromagnetic field. The equation is: formula_1 formula_2 is a constant formula_3 is the current density formula_4 is a model parameter formula_5 is the activation energy formula_6 is Boltzmann's constant formula_7 is the absolute temperature in K The model is abstract, not based on a specific physical model, but flexibly describes the failure rate dependence on the temperature, the electrical stress, and the specific technology and

neg: Diffusion the concentration, "m" > 0 (usually "m" > 1, the case "m" = 1 corresponds to Fick's law). For diffusion of gases in porous media this equation is the formalisation of Darcy's law: the velocity of a gas in the porous media is where "k" is the permeability of the medium, "μ" is the viscosity and "p" is the pressure. The flux "J" = "nv" and for formula_82 Darcy's law gives the equation of diffusion in porous media with "m" = "γ" + 1. For underground water infiltration, the Boussinesq approximation gives the same equation with "m" = 2. For

neg: Transition state theory is still a flow from left to right. Hence, to be technically correct, the reactants are in equilibrium only with [AB], the activated complexes that were reactants in the immediate past. The activated complexes do not follow a Boltzmann distribution of energies, but an "equilibrium constant" can still be derived from the distribution they do follow. The equilibrium constant "K" for the quasi-equilibrium can be written as So, the concentration of the transition state AB is Therefore, the rate equation for the production of product is where the rate constant "k" is given by Here, "k" is directly proportional to

neg: Radiative transfer equation and diffusion theory for photon transport in biological tissue infinite homogeneous medium is presented in this section. The source term in the diffusion equation becomes formula_50, where formula_2 is the position at which fluence rate is measured and formula_52 is the position of the source. The pulse peaks at time formula_53. The diffusion equation is solved for fluence rate to yield The term formula_55 represents the exponential decay in fluence rate due to absorption in accordance with Beer's law. The other terms represent broadening due to scattering. Given the above solution, an arbitrary source can be characterized as a superposition of short-pulsed point sources. Taking time variation out of

neg: Smoluchowski coagulation equation of the coagulated particles are continuous variables, the equation involves an integral: If "dy" is interpreted as a discrete measure, i.e. when particles join in discrete sizes, then the discrete form of the equation is a summation: There exists a unique solution for a chosen kernel function. The operator, "K", is known as the coagulation kernel and describes the rate at which particles of size formula_3 coagulate with particles of size formula_4. Analytic solutions to the equation exist when the kernel takes one of three simple forms: known as the constant, additive, and multiplicative kernels respectively. However, in most practical

neg: Edwards equation have greater nucleophilic character than suggested by their basicity because their electron density can be shifted with relative ease to concentrate in one area. Prior to Edwards developing his equation, other scientists were also working to define nucleophilicity quantitatively. Brønsted and Pederson first discovered the relationship between basicity, with respect to protons, and nucleophilicity in 1924: where k is the rate constant for nitramide decomposition by a base (B) and β is a parameter of the equation. Swain and Scott later tried to define a more specific and quantitative relationship by correlating nucleophilic data with a single-parameter equation derived in

neg: Metabolic theory of ecology or J/K: While "B" in the previous equation is mass-independent, it is not explicitly independent of temperature. To explain the relationship between body mass and temperature, building on earlier work showing that the effects of both body mass and temperature could be combined multiplicatively in a single equation, the two equations above can be combined to produce the primary equation of the MTE, where "b" is a normalization constant that is independent of body size or temperature: According to this relationship, metabolic rate is a function of an organism’s body mass and body temperature. By this equation, large organisms have

neg: International Fisher effect 〖∆E〗_(1/2)/E_(1/2) = i_1-i_2 〖=π〗_1-π_2. Combining the international Fisher effect with uncovered interest rate parity yields the following equation: where Combining the international Fisher effect with covered interest rate parity yields the equation for unbiasedness hypothesis, where the forward exchange rate is an unbiased predictor of the future spot exchange rate.: where Suppose the current spot exchange rate between the United States and the United Kingdom is 1.4339 GBP/USD. Also suppose the current interest rates are 5 percent in the U.S. and 7 percent in the U.K. What is the expected spot exchange rate 12 months from now according to the

neg: Arrhenius equation dependence of the pre-exponential factor is negligible compared to the temperature dependence of the formula_4 factor; except in the case of "barrierless" diffusion-limited reactions, in which case the pre-exponential factor is dominant and is directly observable. Taking the natural logarithm of Arrhenius' equation yields: Rearranging yields: This has the same form as an equation for a straight line: where x is the reciprocal of T. So, when a reaction has a rate constant that obeys Arrhenius' equation, a plot of ln("k") versus "T" gives a straight line, whose gradient and intercept can be used to determine "E" and "A" .

neg: Fick's laws of diffusion Fick's first law (one-dimensional case) can be written as: where the index denotes the th species, is the concentration (mol/m), is the universal gas constant (J/K/mol), is the absolute temperature (K), and is the chemical potential (J/mol). If the primary variable is mass fraction (, given, for example, in kg/kg), then the equation changes to: where is the fluid density (for example, in kg/m). Note that the density is outside the gradient operator. Fick's second law predicts how diffusion causes the concentration to change with time. It is a partial differential equation which in one dimension reads: where In two

neg: Rate equation the constant K at equilibrium is expressed as: Where formula_33 and formula_34 are the concentrations of A and B at equilibrium, respectively. The concentration of A at time t, formula_35, is related to the concentration of B at time t, formula_36, by the equilibrium reaction equation: Note that the term <chem>[B]0</chem> is not present because, in this simple example, the initial concentration of B is 0. This applies even when time t is at infinity; i.e., equilibrium has been reached: then it follows, by the definition of K, that and, therefore, These equations allow us to uncouple the system of

neg: Monod equation are empirical coefficients to the Monod equation. They will differ between species and based on the ambient environmental conditions. The rate of substrate utilization is related to the specific growth rate as follows: where: r is negative by convention. In some applications, multiple terms of the form [S/(K + S)] are multiplied together where more than one nutrient or growth factor has the potential to be limiting (e.g. organic matter and oxygen are both necessary to Heterotrophic bacteria). When the yield coefficient, being the ratio of mass of microorganisms to mass of substrate utilized, becomes very large this signifies that

neg: Monte Carlo methods for electron transport time dt. Because of the complexity of the integral at the exponent, it is impractical to generate stochastic free flights with the distribution of the equation above. In order to overcome this difficulty, people use a fictitious “self-scattering” scheme. By doing this, the total scattering rate, including this self-scattering, is constant and equal to, say, formula_15. By random selection, if self-scattering is selected, k′ after the collision is the same as k and the carrier continues its flight without perturbation. Introducing a constant formula_16, the above equation reduces to Random numbers "r" can be used very simply to generate stochastic

neg: K–omega turbulence model K–omega turbulence model In computational fluid dynamics, the k–omega ("k"–ω) turbulence model is a common two-equation turbulence model, that is used as a closure for the Reynolds-averaged Navier–Stokes equations (RANS equations). The model attempts to predict turbulence by two partial differential equations for two variables, "k" and ω, with the first variable being the turbulence kinetic energy ("k") while the second (ω) is the specific rate of dissipation (of the turbulence kinetic energy "k" into internal thermal energy). The eddy viscosity ν, as needed in the RANS equations, is given by: , while the evolution of "k" and ω is

neg: Microbial enhanced oil recovery been followed. For instance, coupled nonlinear parabolic differential equations: adding equation for the rate of diffusion of microbes and their capture by porous medium; differential balance equations for nutrient transport, including the effect of adsorption; and the assumption of bacterial growth kinetic based on Monod equation. Monod equation is commonly used in modelling software but it has a limited behaviour for being inconsistent with the law of mass action that form the basis of kinetic characterization of microbial growth. Application of law of mass action to microbial populations results in the linear logistic equation. If the law of mass action

neg: Thermal resistance of Fourier's law, the physical significance of treating formula_30 as a variable becomes evident when the rate at which energy is conducted across a cylindrical surface, this is represented as Where formula_33 is the area that is normal to the direction of where the heat transfer occurs. Equation 1 implies that the quantity formula_34 is not dependent of the radius formula_35, it follows from equation 5 that the heat transfer rate, formula_36 is a constant in the radial direction. In order to determine the temperature distribution in the cylinder, equation 4 can be solved applying the appropriate boundary conditions. With

neg: Chemical specificity equation (E = enzyme, S = substrate, P = product) k1 k2 E + S <--> ES <--> E + P k-1 kd would be equivalent to k-1/k1, where k1 and k-1 are the rates of the forward and backward reaction, respectively in the conversion of individual E and S to the enzyme substrate complex. The chemical specificity of an enzyme for a particular substrate can be found using two variables that are derived from the Michaelis-Menten equation. km approximates the dissociation constant of enzyme-substrate complexes. kcat represents the turnover rate, or the number of reactions catalyzed by an enzyme

neg: Enthalpy–entropy compensation free-energy equation, In both forms, it is apparent that the difference in Gibbs free-energies of activations (δΔ"G") will be zero when the temperature is at the isokinetic (or isoequilibrium) temperature and hence identical for all members of the reaction set at that temperature. Beginning with the Arrhenius equation and assuming kinetic compensation (obeying ln"A" = ln"A" + αΔ"E"), the isokinetic temperature may also be given by "k"β = 1/α. The reactions will have approximately the same value of their rate constant "k" at an isokinetic temperature. In a 1925 paper, F.H. Constable described the linear relationship observed for the reaction

neg: Solder fatigue predict solder fatigue, such as testing and simple acceleration transforms, it is now generally acknowledged that Engelmaier and other models that are based on strain range do not provide a sufficient degree of accuracy. Darveaux proposed a model relating the quantity of volume weighted average inelastic work density, the number of cycles to crack initiation, and the crack propagation rate to the characteristic cycles to failure. formula_5 formula_6 formula_7 In the first equation "N" represents the number of cycles to crack initiation, ∆W represents inelastic work density, "K" and "K" are material model constants. In the second equation, da/dN represents

neg: Reaction mechanism for the reaction). Information about the mechanism of a reaction is often provided by the use of chemical kinetics to determine the rate equation and the reaction order in each reactant. Consider the following reaction for example: In this case, experiments have determined that this reaction takes place according to the rate law formula_1. This form suggests that the rate-determining step is a reaction between two molecules of NO. A possible mechanism for the overall reaction that explains the rate law is: Each step is called an elementary step, and each has its own rate law and molecularity. The elementary

neg: Kleiber's law considered, appearing instead as the relation of the cell's metabolic rate to the organism's ME, where the ME of the organism is the same as for its cells. Major proponents of the equation, in the form of 'quarter power scaling', always limit themselves to mass specific metabolic rates, where the mass is one gram. The equation shows that at one gram mass, metabolic rate is the same for all MEs. This is why one gram is favored. It eliminates the role of ME in the equation, and makes the exponent ¾ or ⅔ at least plausible to the initiate concerned

neg: Transition state theory surface of the initial ground state. One of the most important features introduced by Eyring, Polanyi and Evans was the notion that activated complexes are in quasi-equilibrium with the reactants. The rate is then directly proportional to the concentration of these complexes multiplied by the frequency ("k""T"/"h") with which they are converted into products. Below, a non-rigorous plausibility argument is given for the functional form of the Eyring equation. However, the key statistical mechanical factor "k""T"/"h" will not be justified, and the argument presented below does not constitute a true "derivation" of the Eyring equation. It should be noted that

neg: Ideal gas law the number of moles of gas and "R" = "N""k" is the gas constant. Ideal gas law The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. It is a good approximation of the behavior of many gases under many conditions, although it has several limitations. It was first stated by Émile Clapeyron in 1834 as a combination of the empirical Boyle's law, Charles's law, Avogadro's law, and Gay-Lussac's law. The ideal gas law is often written as where formula_2, formula_3 and formula_4 are the pressure, volume and absolute temperature;

neg: Transition state theory is proportional to the "critical increment". His ideas were further developed by Richard Chace Tolman. In 1919, Austrian physicist Karl Ferdinand Herzfeld applied statistical mechanics to the equilibrium constant and kinetic theory to the rate constant of the reverse reaction, "k", for the reversible dissociation of a diatomic molecule. He obtained the following equation for the rate constant of the forward reaction where formula_8 is the dissociation energy at absolute zero, "k" is the Boltzmann constant, "h" is the Planck constant, "T" is thermodynamic temperature, "υ" is vibrational frequency of the bond. This expression is very important since it is

neg: Ritz ballistic theory of Ampere's law. The second derivative of Ritz's equation is Consider the diagram on the right, and note that formula_31, Plugging these expressions into Ritz's equation, we obtain the following Comparing to the original expression for Ampere's force law we obtain the coefficients in Ritz's equation From this we obtain the full expression of Ritz's electrodynamic equation with one unknown In a footnote at the end of Ritz's section on "Gravitation" (English translation) the editor says, "Ritz used "k" = 6.4 to reconcile his formula (to calculate the angle of advancement of perihelion of planets per century) with the observed