Relationship to time

The time-dependent Hartree-Fock theory was first discussed by Dirac (1930b) and subsequently in perturbative form by Dalgamo and Victor (1966). Its relationship to time-dependent perturbation theory has been discussed by Langhoff, Epstein and Karplus (1972). 

Donnan GA, Davis SM, Chambers BR, Gates PC, Hankey GJ, McNeil JJ, Rosen D, Stewart-Wynne EG, Tuck RR. Streptokinase for acute ischemic stroke with relationship to time of administration Australian Streptokinase (ASK) trial study group. JAMA. 1996 276 961-966. 

In hairless albino mice treated topically with the compound bergapten at doses of 0 to 50 ppm, a dose-dependent relationship to time of tumor onset was observed and bergapten was reported to have phototumorigenic potential at the lowest tested dose of 5 ppm. The addition of UVA and UVB sunscreens to the treatment vehicle significantly reduced the tumorigenic effects (Young et al. 1990). 

Table 13.1 Outcomes in maternal PKU in relationship to timing of maternal blood phenylalanine control ...

P.A. Vernon, S. Nador, L. Kantor, Reaction times and speed of processing their relationship to timed and untimed measures of intelligence. Intelligence, 9 357-374 (1985). 

Plastic materials—when compared to other materials— have a unique relationship to time. What do 1 mean by that ... 

Because of their widespread use for simple process control, hydrometers are frequently calibrated, not in specific gravity, but in some units related to it, which bear (or bore at one time) some relationship to the concentration being so measured. 

Now that we have defined capacity factor, selectivity, and column efficiency we consider their relationship to chromatographic resolution. Since we are only interested in the resolution between solutes eluting with similar retention times, it is safe to assume that the peak widths for the two solutes are approximately the same. Equation 12.1, therefore, is written as... 

Signal conditioning. The process of altering the relationship of a transducer (ion or neutral detector) output with respect to time or other parameters (frequency, voltage, or current). 

In a shear experiment the first of these is given by Eq. (3.50). For the viscous component we do not have an expression for 7, only for the way 7 varies with time. Hence it is not possible to develop this relationship any further as an explicit equation, but only as a differential equation. Differentiating Eq. (3.53) with respect to time, we obtain... 

An ideal gas obeys Dalton s law that is, the total pressure is the sum of the partial pressures of the components. An ideal solution obeys Raoult s law that is, the partial pressure of the ith component in a solution is equal to the mole fraction of that component in the solution times the vapor pressure of pure component i. Use these relationships to relate the mole fraction of component 1 in the equilibrium vapor to its mole fraction in a two-component solution and relate the result to the ideal case of the copolymer composition equation. 

Traditionally, the average specific cake and medium resistances have been deterrnined from constant pressure experiments and the solution of the basic filtration equation for constant pressure which relates filtrate volume to time. This relationship is, in theory, paraboHc but deviations occur in practice. 

Pump vibration measurements and analysis of the ampHtude—frequency spectmm have led to the deterrnination of possible equipment ailments. Examples of frequency relationships to particular pump vibration problems are available (62). Eor example, a one-time (lx) frequency corresponding to a mnning speed usually indicates the presence of rotor unbalance, a 2 x peak may mean a bent shaft, and a - O.dx may be a symptom of an oil whip at the sleeve bearing. 

This relationship is exponential with respect to time t and with increasing time quickly approaches equation 7, where is known as the terminal settling velocity under gravity. 

From the time function F t) and the calculation of [IT], the values of G may be found. One way to calculate the G matrix is by a fast Fourier technique called the Cooley-Tukey method. It is based on an expression of the matrix as a product of q square matrices, where q is again related to N by = 2 . For large N, the number of matrix operations is greatly reduced by this procedure. In recent years, more advanced high-speed processors have been developed to carry out the fast Fourier transform. The calculation method is basically the same for both the discrete Fourier transform and the fast Fourier transform. The difference in the two methods lies in the use of certain relationships to minimize calculation time prior to performing a discrete Fourier transform. 

Adsorption for gas purification comes under the category of dynamic adsorption. Where a high separation efficiency is required, the adsorption would be stopped when the breakthrough point is reached. The relationship between adsorbate concentration in the gas stream and the solid may be determined experimentally and plotted in the form of isotherms. These are usually determined under static equilibrium conditions but dynamic adsorption conditions operating in gas purification bear little relationship to these results. Isotherms indicate the affinity of the adsorbent for the adsorbate but do not relate the contact time or the amount of adsorbent required to reduce the adsorbate from one concentration to another. Factors which influence the service time of an adsorbent bed include the grain size of the adsorbent depth of adsorbent bed gas velocity temperature of gas and adsorbent pressure of the gas stream concentration of the adsorbates concentration of other gas constituents which may be adsorbed at the same time moisture content of the gas and adsorbent concentration of substances which may polymerize or react with the adsorbent adsorptive capacity of the adsorbent for the adsorbate over the concentration range applicable over the filter or carbon bed efficiency of adsorbate removal required. 

Sinee the relationship between dT/dt and dX /dt is impbeit with respeet to time t, Equation 6-57 is expressed as... 

The quantities VC and VC., represent the mass of biomass and substrate respeetively in the reaetor. Dividing these masses by the volume V gives the eoneentrations and as a funetion of time, whieh are required in the kinetie relationships to determine r, and r,. 

Therefore, in seale-up, the use of the relationship to produee a larger system with the same t using various eonfigurations of D , D, H, Hj, and Np ean result in an enor. Henee, the use of equal mixing times... 

In systems such as the 2- and 6-hydroxypteridines, sudden addition of an alkaline solution to a neutral buffer, or of a neutral solution to an alkaline buffer, displaces the equilibrium between hydrated and anhydrous species (because the anions are less hydrated than the neutral molecules). By measuring the time-dependent change of optical density at a selected wavelength, a first-order rate constant, obs5 can be obtained. This constant is a composite one, and to see its relationship to other quantities some discussion is necessary. 

Variable-speed machinery Variable-speed machine-train narrowband windows should be converted to their relationship to the running speed (lx). For example, if the frequency of the ball-pass inner-race rolling-element bearing is calculated to be 5.9 times the primary shaft running speed, then the narrowband window should be set as 5.3x to 6.2x. This allows the microprocessor to track the actual bearing rotational frequency regardless of the variation in running speed. 

The oxidation of aluminium at room temperature is reported to conform to an inverse logarithmic equation for growth periods up to 5 years duration. At elevated temperatures, oxidation studies over shorter periods illustrate conformity to parabolic, linear and logarithmic relationships according to time and temperature. These kinetic variations are attributed to different mechanisms of film formation . ... 

If the PBR is less than unity, the oxide will be non-protective and oxidation will follow a linear rate law, governed by surface reaction kinetics. However, if the PBR is greater than unity, then a protective oxide scale may form and oxidation will follow a reaction rate law governed by the speed of transport of metal or environmental species through the scale. Then the degree of conversion of metal to oxide will be dependent upon the time for which the reaction is allowed to proceed. For a diffusion-controlled process, integration of Pick s First Law of Diffusion with respect to time yields the classic Tammann relationship commonly referred to as the Parabolic Rate Law ... 

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