Exponential decay first-order chemical

The maximum fluorescence quantum yield is 1.0 (100 %) every photon absorbed results in a photon emitted. Compounds with quantum yields of 0.10 are still considered quite fluorescent. The fluorescence lifetime is an instance of exponential decay. Thus, it is similar to a first-order chemical reaction in which the first-order rate constant is the sum of all of the rates (a parallel kinetic model). Thus, the lifetime is related to the facility of the relaxation pathway. If the rate of spontaneous emission or any of the other rates are fast, the lifetime is short (for commonly used fluorescent compounds, typical excited state decay times for fluorescent compounds that emit photons with energies from the UV to near infrared are within the range of 0.5-20 ns). The fluorescence lifetime is an important parameter for practical applications of fluorescence such as fluorescence resonance energy transfer. There are several rules that deal with fluorescence. 

A rate equation was derived for the dispersion of carbon black (as a function of time), which fits the kinetic data well. It is analogous to a first-order chemical-reaction rate equation and describes the disappearance of undispersed carbon black as an exponential decay. The rate equation is valid for both low- and high-structure carbon black, over a wide range of mixer speeds. 

The Gaussian plume foimulations, however, use closed-form solutions of the turbulent version of Equation 5-1 subject to simplifying assumptions. Although these are not treated further here, their description is included for comparative purposes. The assumptions are reflection of species off the ground (that is, zero flux at the ground), constant value of vertical diffusion coefficient, and large distance from the source compared with lateral dimensions. This Gaussian solution to Equation 5-1 is obtained under the assumption that chemical transformation source and sink terms are all zero. In some cases, an exponential decay factor is applied for reactions that obey first-order kinetics. A typical solution (with the time-decay factor) is ... 

Such a chemical reaction, in which molecules are not colliding with other atoms or molecules, is called a first-order reaction because the rate at which chemical concentration changes at any instant in time is proportional to the concentration raised to the first power. Certain chemical processes, such as radioactive decay, are described by first-order kinetics. In the absence of any other sources of the chemical, first-order kinetics may lead to exponential decay or first-order decay of the chemical concentration (i.e., the concentration of the parent compound decreases exponentially with time) ... 

In the expressions for the gas exchange coefficient employed previously, it is evident that the air-water gas exchange flux density is proportional to the difference between a chemical concentration in the water (Cw) and the corresponding equilibrium concentration (Cw H) in air. Consequently, the difference between actual and equilibrium concentration in the water tends to decay exponentially, as expected for any first-order process. In many situations, exponential decay may provide a useful model of a volatile chemical concentration in a surface water. A classic example is degassing of a dissolved gas from a stream if the gas is present at concentration C0 upstream, atmospheric concentration of the gas is negligible, and flow is steady and uniform along the stream, then the gas concentration in the stream is given by... 

Based on their abihty to convert reactants to products via first-order irreversible chemical kinetics, in rectangular channels with various aspect ratios at large Damkohler numbers (i.e., p = 1000) in the diffusion-limited regime. Reactant molar density vs. channel length follows a single exponential decay for those deposition profiles that are not underlined. 

Bc2 produced by flash photolysis of seawater decays by parallel first- and second-order reactions. The environmentally important exponential decay is a pseudo first-order reaction of Br2 with the carbonate/ bicarbonate system in seawater. A chemical speciation model for the free ions and ion-pairs in seawater and in solutions at seawater ionic strength allowed us to measure the dependence of the pseudo first-order rate term, a, on individual C02-containing species. A predictive equation based on reaction of Br2 with free C03 and the hgC03°, NaC03 and CaC03 ion pairs accounts for the mean seawater a at pH 8.1 within experimental uncertainty. The reaction productfs) are unknown. 

We can also turn the question around. In chemical kinetics, we need a model to fit the data. This model can be simple, as in first-order reactions where the decay is exponential, or more complicated depending on a complex mechanism. If we do not have a model, our data are just that, data. We could try to fit to a variety of functions, but as there is an infinite number of different functions, that is a pointless exercise. As we have seen in the classical part of this chapter, even for a simple reaction a variety of models are possible, based on dissipative classical dynamics, and we can use these models to try to understand our data. This often involves varying the external parameters, temperature, pH, viscosity, and polarizabihty, but our model should tell us what to expect for such variations for instance, how the rate constant for a reaction depends on those parameters. If our models are quantum mechanical in nature, it is mandatory that we also provide a mechanism for decay, and show how the decay constant or constants depend on external parameters. 

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