Zero time

The mixture was prepared and allowed to achieve equilibrium to it was added an excess of urea which caused the immediate precipitation as urea nitrate of the free nitric acid present. As a result of the sudden removal of the nitric acid from the mixture, the system underwent change to re-establish the equilibrium however, the use of an excess of urea removed the nitric acid as it was produced from acetyl nitrate and acetic acid, and the consumption of acetyl nitrate proceeded to completion. Thus, by following the production of urea nitrate with the time from the addition of urea, the rate of the back reaction could be determined, and by extrapolating the results to zero time the equilibrium... 

The nitric acid used in this work contained 10% of water, which introduced a considerable proportion of acetic acid into the medium. Further dilution of the solvent wnth acetic acid up to a concentration of 50 moles % had no effect on the rate, but the addition of yet more acetic acid decreased the rate, and in the absence of acetic anhydride there was no observed reaction. It was supposed from these results that the adventitious acetic acid would have no effect. The rate coefficients of the nitration diminished rapidly with time in one experiment the value of k was reduced by a factor of 2 in i h. Corrected values were obtained by extrapolation to zero time. The author ascribed the decrease to the conversion of acetyl nitrate into tetranitromethane, but this conversion cannot be the explanation because independent studies agree in concluding that it is too slow ( 5.3.1). 

For stand-alone or hybrid TOF mass spectrometry, the ions examined must all start from some point at the same instant. From this zero time, the ions are accelerated through a short region by applying a short pulse of electric potential of several kilovolts. The acceleration gives the ions velocities that vary in proportion to the square root of their m/z values. 

In time-of-flight (TOF) mass spectrometers, a pulse of ions is accelerated electrically at zero time. Having attained a maximum velocity, the ions drift along the flight tube of the analyzer. The times of arrival of ions at a detector are noted. 

A rotational viscometer connected to a recorder is used. After the sample is loaded and allowed to come to mechanical and thermal equiUbtium, the viscometer is turned on and the rotational speed is increased in steps, starting from the lowest speed. The resultant shear stress is recorded with time. On each speed change the shear stress reaches a maximum value and then decreases exponentially toward an equiUbrium level. The peak shear stress, which is obtained by extrapolating the curve to zero time, and the equiUbrium shear stress are indicative of the viscosity—shear behavior of unsheared and sheared material, respectively. The stress-decay curves are indicative of the time-dependent behavior. A rate constant for the relaxation process can be deterrnined at each shear rate. In addition, zero-time and equiUbrium shear stress values can be used to constmct a hysteresis loop that is similar to that shown in Figure 5, but unlike that plot, is independent of acceleration and time of shear. 

Fig. 7. Decay of shear stress during steady shear at various shear rates. Determination of zero-time shear stresses or yield stresses and equiUbrium shear...

The well-known difficulty with batch reactors is the uncertainty of the initial reaction conditions. The problem is to bring together reactants, catalyst and operating conditions of temperature and pressure so that at zero time everything is as desired. The initial reaction rate is usually the fastest and most error-laden. To overcome this, the traditional method was to calculate the rate for decreasingly smaller conversions and extrapolate it back to zero conversion. The significance of estimating initial rate was that without any products present, rate could be expressed as the function of reactants and temperature only. This then simplified the mathematical analysis of the rate fianction. 

The small volumes of the feed line from the switching valve to the reactor and the discharge line to the analyzer can be corrected for, since they will have plug flow. This will only displace the zero time on the recorder and it can be easily corrected. 

Extrapolate the compression curve to the critical point or zero time. 

Locate the time when the upper interface (between the supernatant liquid and slurry) is at height Z g, halfway between the initial height, Zg, and the extrapolated zero-time compression zone height, Z g. This time represents the period in which all the solids were at the critical dilution and went into compression. The retention time is computed as t -1,., where t is the time when the solids reach the specified... 

To be strictly accurate, the right hand side should have the form (Aq -I- At") to allow for the instantaneous strain at zero time. However, for long creep times, sufficient accuracy can be obtained by ignoring Aq. 

The predicted strain variation is shown in Fig. 2.43(b). The constant strain rates predicted in this diagram are a result of the Maxwell model used in this example to illustrate the use of the superposition principle. Of course superposition is not restricted to this simple model. It can be applied to any type of model or directly to the creep curves. The method also lends itself to a graphical solution as follows. If a stress is applied at zero time, then the creep curve will be the time dependent strain response predicted by equation (2.54). When a second stress, 0 2 is added then the new creep curve will be obtained by adding the creep due to 02 to the anticipated creep if stress a had remained... 

With Eq. (2-42) the first-order rate constant can be calculated from concentrations at any two times. Of course, usually concentrations are measured at many times during the course of a reaction, and then one has choices in the way the estimates will be calculated. One possibility is to let r, be zero time for all calculations in this case the same value c° is employed in each calculation, so error in this quantity is transmitted to each rate constant estimate. Another possibility is to apply Eq. (2-42) to successive time intervals. If, as often happens, the time intervals are all... 

The value of the integration constant is determined by the magnitude of the displacement from the equilibrium position at zero time. King also gives a solution for Scheme IV, and Pladziewicz et al. show how these equations can be used with a measured instrumental signal to estimate the rate constants by means of nonlinear regression. 

As the initial conditions choose [AJo = 100, [B]o = 0, [C]o = 0, the brackets representing mole percent, which is numerically equal to the number of molecules in a grid, because each grid contains 100 spaces. At zero time we load 100 molecules of A in grid A, by writing an A in each of the 100 cells of the grid. 

One of the problems that arises is specification of zero time, the time of initiation... 

The results for the intermittent titration are shown in Figure 10. The progressive and linear increase in the apparent moisture with time, observed for most materials, was ascribed to some side reaction. The true moisture was obtained by extrapolation of the straight-line portion to zero time. 

Fig. 7.1.3 Influence of the buffer and the type of peroxide on the luminescence reaction of Chaetopterus photoprotein. The reaction was initiated at zero time by the addition of a peroxide (old dioxane or H2O2) and FeSC>4 in each case, with successive additions of FeSC>4 or H2O2 at the time indicated with an arrow. In the experiments of the two upper curves, 10 pi of old dioxane and 1 pi of lOmM FeSC>4 were added at zero time, followed by 1 pi of 10 mM FeSC>4 at each arrow. In the experiments of the two lower curves, 50 pi of 10 mM H2O2 and 20 pi of 10 mM FeSC>4 were added at zero time, followed by 50 pi of 10 mM H2O2 or 20 pi of 10 mM FeSC>4 at each arrow. All in 5 ml of 10 mM phosphate or Tris buffer, pH 7.2. From Shimomura and Johnson, 1966.

Fig. 7.2.5 Spectral change of Odontosyllis luciferin in 50 mM NaOH in air A, at zero time B, after 15 min C, after 30 min and D, after 80 min (Xmax 385 nm). From Shimomura et ai, 1963d, with permission from John Wiley Sons Ltd.

Note incidentally that in any particular Lorentz frame we could choose as the representative of an equivalence class that vector that has zero time component. For example, for the vectors equivalent to we could choose as the representative of that equivalence class the vector... 

For experimental purposes it is customary to define a second-order rate coefficient, k, usually determined in the limit of zero time, as follows,... 

Experimentally k is normally determined in the limit of zero time, at which time the special case defined by (25) might be expected to hold, since [R2NH2 ] 0 and [BH+] 0. Moreover, k9 > k 9 and k10 > k 10, since usually B and R2NH are probably stronger bases than IV. 

Since a first-order rate constant does not depend on [A]o, one need not know either the initial concentration or the exact instant at which the reaction began. This characteristic should not be used to rationalize experimentation on impure materials. These features do allow, however, a procedure in which measurements of slower reactions are not taken until the sample has reached temperature equilibrium with the thermostating bath. The first sample is simply designated as t = 0. Likewise, for rapidly decaying reaction transients, knowing the true zero time is immaterial. 

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