Exponential relationships time constant

If C is approximately constant during the initial drydown period, as it is in many closed-system apphcations, then the water concentration decays exponentially with time. The rate equation can be integrated to give the relationship between water concentration and time ... 

An attempt is often made to relate T] and T2 to the molecular dynamics of a system. For this purpose a relationship is sought between T1 or T2 and the correlation time tc of the nuclei under investigation. The correlation time is the time constant for exponential decay of the fluctuations in the medium that are responsible for relaxation of the magnetism of the nuclei. In general, l/xc can be thought of as a rate constant made up of the sum of all the rate constants for various independent processes that lead to relaxation. One of the most important of these (1 /t2) is for molecular tumbling. 

Here, w = m, n, and S. V represents the membrane potential, n is the opening probability of the potassium channels, and S accounts for the presence of a slow dynamics in the system. Ic and Ik are the calcium and potassium currents, gca = 3.6 and gx = 10.0 are the associated conductances, and Vca = 25 mV and Vk = -75 mV are the respective Nernst (or reversal) potentials. The ratio r/r s defines the relation between the fast (V and n) and the slow (S) time scales. The time constant for the membrane potential is determined by the capacitance and typical conductance of the cell membrane. With r = 0.02 s and ts = 35 s, the ratio ks = r/r s is quite small, and the cell model is numerically stiff. The calcium current Ica is assumed to adjust immediately to variations in V. For fixed values of the membrane potential, the gating variables n and S relax exponentially towards the voltage-dependent steady-state values noo (V) and S00 (V). Together with the ratio ks of the fast to the slow time constant, Vs is used as the main bifurcation parameter. This parameter determines the membrane potential at which the steady-state value for the gating variable S attains one-half of its maximum value. The other parameters are assumed to take the following values gs = 4.0, Vm = -20 mV, Vn = -16 mV, 9m = 12 mV, 9n = 5.6 mV, 9s = 10 mV, and a = 0.85. These values are all adjusted to fit experimentally observed relationships. In accordance with the formulation used by Sherman et al. [53], there is no capacitance in Eq. (6), and all the conductances are dimensionless. To eliminate any dependence on the cell size, all conductances are scaled with the typical conductance. Hence, we may consider the model to represent a cluster of closely coupled / -cells that share the combined capacity and conductance of the entire membrane area. 

Normally, for semiconductors, Csc < CH so CT Csc. Roat may be varied systematically and the decay of j can often be approximated by a single exponential form, i.e. kr 1/RtCi. or kT potentials well positive of V, the long-time transient time-constant t (Rm + Rout)Csc, and a plot of x vs. R]oad (sflin + Rout) is linear, as shown in Fig. 105. Confirmation of this is obtained from the fact that 1/t2 obeys the Mott-Schottky relationship. At potentials close to V, kec becomes much larger and the decay law more complex. 

Finally, Delancey and Chiang5 3,54 reported a general mathematical evaluation of multicomponent non isothermal mass transfer in the presence as well as absence of a chemical reaction. These studies followed the matrix approach to the problem. The chemical reaction considered was a simple first-order irreversible reaction. The problem was solved assuming time dependence of the rate constant and an exponential relationship between the temperature and the distance. 

Constant force field provides for highest resolution of particles in the sample with resulting highest precision. However, characterization of samples with wide size distributions is difficult and time consuming. Force field programming [89,90] removes these limitations to ensure that the entire distribution can be analyzed in a convenient time. In time delayed exponential decay the initial force field is held constant for a time equal to r and after this the force field is decayed exponentially with a time constant r. In this mode, a log-linear relationship is obtained of particle mass against retention time. This simple relationship permits a convenient calculation of the quantitative information needed for the sample. Retention time is given by ... 

The probability builds up exponentially in time to t = (ro -I- r)/vo, after which it decays exponentially. The decay-time constant is t = h/3. For the Lorentzian wave-packet shape (4.158) the uncertainty principle is an exact relationship if the energy uncertainty is the full width at half maximum 3 and the time uncertainty is the decay time t. 

The field-induced velocity of the analyte in the separation channel is constant and comparable with its diffusive motion (t/ = constant, Uxt V2Dt, where t is time). The resulting concentration profile of the analyte is given by the exponential relationship [9]... 

Total circulating blood volume was calculated by summing plasma volume and total circulating red blood cell volume. Plasma volume changes in the model are represented by an empirically derived relationship. Upon application of a stimulus, plasma volume changes from the control value to an experimentally determined steady-state value in an exponential manner with a time constant of 1.5 days (2). 

The present model also allows calculation of the relationship between flow velocity at constant relative conversion and operating time in a fixed bed. In the past linear as well as exponential functions have been proposed for this property. In figure 8 the expected relationships according to the present model, linear decay and exponential decay are shown. The results of one representative experiment have also been shown. The present theory results in a decay curve which is between the linear and exponential relationships. The assumption of linear decay involves an underestimation of the activity half life the assumption of exponential decay, however, overestimates the half life of the initial flow velocity. 

Many biological happenings are nonlinear. They may oscillate, but not with any set frequency. They may form exponential-like responses, but cannot be characterized by one time constant. Input-output relationships may not follow idealized forms. In these cases, the biological engineer must either resort to nonlinear equations or to numerical solutions to describe these phenomena. 

Fig. 1.16 (A) Fluorescence (so/id curve) from a molecule that is excited with sinusoidally modulated light (dotted curve). If the fluorescence decays exponentially with single time constant T, the phase shift (4>) and the relative modulation of the fluorescence amplitude (w) are related to t and the angular frequency of the modulation m) by < = arctan((OT) and w = (1 + o> ) (Appendix A4). The curves shown here are calculated for t = 8 ns, cd = 1.257 x 10 rad/s (20 MHz) and 100% modulation of the excitation light (< = 0.788 rad, i = 0.705). (B) Phase shift (4>, solid curve) and relative modulation (m, dotted curve) of the fluorescence of a molecule that decays with a single exponential time constant, plotted as a functirai of the product on. The relationships among , m, r and m become more complicated if the fluOTescence decays with multiexponential kinetics (Appendix A4)...

As it was demonstrated [18], the rate of photodegradation follows an exponential relationship, where the certain intensity of applied light acts partially in the direct dependence of experimental conditional including exposure time (Eq. 1), where k represents the oxidation rate, K means the rate constant, I denotes the exposure intensity, a is a material constant that describes the physical and chemical peculiarities and t is exposure time. 

From the Arrhenius form of Eq. (70) it is intuitively expected that the rate constant for chain scission kc should increase exponentially with the temperature as with any thermal activation process. It is practically impossible to change the experimental temperature without affecting at the same time the medium viscosity. The measured scission rate is necessarily the result of these two combined effects to single out the role of temperature, kc must be corrected for the variation in solvent viscosity according to some known relationship, established either empirically or theoretically. 

The longest relaxation time. t,. corresponds to p = 1. The important characteristics of the polymer are its steady-state viscosity > at zero rate of shear, molecular weight A/, and its density p at temperature 7" R is the gas constant, and N is the number of statistical segments in the polymer chain. For vinyl polymers N contains about 10 to 20 monomer units. This equation holds only for the longer relaxation times (i.e., in the terminal zone). In this region the stress-relaxation curve is now given by a sum of exponential terms just as in equation (10), but the number of terms in the sum and the relationship between the T S of each term is specified completely. Thus... 

Recently, Orosz et al. [136] reviewed and critically reevaluated some of the known mechanistic studies. Detailed mathematical expressions for rate constants were presented, and these are used to derive relationships, which can then be used as guidelines in the optimization procedure of the POCL response. A model based on the time-window concept, which assumes that only a fraction of the exponential light emission curve is captured and integrated by the detector, was presented. Existing data were used to simulate the detector response for different reagent concentrations and flow rates. 

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