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In the mathematical study of heat conduction and diffusion, a heat kernel is the fundamental solution to the heat equation on a specified domain with appropriate boundary conditions. It is also one of the main tools in the study of the spectrum of the Laplace operator , and is thus of some auxiliary importance throughout mathematical physics .
For example, the shift of the phenol OH stretching frequency, Δχ, that occurs upon adduct formation has been analyzed using the following equation: Δχ = Ε A ∗ E B + C A ∗ C B + W ∗. where asterisks on the E A and C A for phenol indicate that the acceptor is held constant and the frequency shift is measured as the base is varied. The ...
In physical chemistry, the Arrhenius equation is a formula for the temperature dependence of reaction rates.The equation was proposed by Svante Arrhenius in 1889, based on the work of Dutch chemist Jacobus Henricus van 't Hoff who had noted in 1884 that the van 't Hoff equation for the temperature dependence of equilibrium constants suggests such a formula for the rates of both forward and ...
The delta potential is the potential = (), where δ(x) is the Dirac delta function. It is called a delta potential well if λ is negative, and a delta potential barrier if λ is positive. The delta has been defined to occur at the origin for simplicity; a shift in the delta function's argument does not change any of the following results.
In the equation, k B and h are the Boltzmann and Planck constants, respectively. Although the equations look similar, it is important to note that the Gibbs energy contains an entropic term in addition to the enthalpic one. In the Arrhenius equation, this entropic term is accounted for by the pre-exponential factor A.
The example below contains four reactions that can be related through their associated free energies. [ An example of the former is the dissolution of ammonium nitrate. This process is spontaneous even though it is endothermic. It occurs because the favored increase in disorder that accompanies dissolution outweighs the unfavored increase in ...
In physical chemistry, the Evans–Polanyi principle (also referred to as the Bell–Evans–Polanyi principle, Brønsted–Evans–Polanyi principle, or Evans–Polanyi–Semenov principle) observes that the difference in activation energy between two reactions of the same family is proportional to the difference of their enthalpy of reaction.
The definition of the Gibbs function is = + where H is the enthalpy defined by: = +. Taking differentials of each definition to find dH and dG, then using the fundamental thermodynamic relation (always true for reversible or irreversible processes): = where S is the entropy, V is volume, (minus sign due to reversibility, in which dU = 0: work other than pressure-volume may be done and is equal ...