Time-distance relationship

As we indicated earlier, diffusion in a solution is important for the movement of solutes across plant cells and tissues. How rapid are such processes For example, if we release a certain amount of material in one location, how long will it take before we can detect that substance at various distances To discuss such phenomena adequately, we must determine the dependence of the concentration on both time and distance. We can readily derive such a time-distance relationship if we first consider the conservation of mass, which is necessary if we are to transform Equation 1.1 into an expression that is convenient for describing the actual solute distributions caused by diffusion. In particular, we want to eliminate Jj from Equation 1.1 so that we can see how c depends on x and t. 

Because Photosystem II tends to occur in the grana and Photosystem I in the stromal lamellae, the intervening components of the electron transport chain need to diffuse in the lamellar membranes to link the two photosystems. We can examine such diffusion using the time-distance relationship derived in Chapter 1 (Eq. 1.6 x je = 4Djtife). In particular, the diffusion coefficient for plastocyanin in a membrane can be about 3 x 10 12 m2 s-1 and about the same in the lumen of the thylakoids, unless diffusion of plastocyanin is physically restricted in the lumen by the appres-sion of the membranes (Haehnel, 1984). For such a D , in 3 x 10-4 s (the time for electron transfer from the Cyt b(f complex to P ), plastocyanin could diffuse about [(4)(3 x 10-12 m2 s-1) (3 x 10-4 s)]1/2 or 60 nm, indicating that this complex in the lamellae probably occurs in relatively close proximity to its electron acceptor, Photosystem I. Plastoquinone is smaller and hence would diffuse more readily than plastocyanin, and a longer time (2 x 10-3 s) is apparently necessary to move electrons from Photosystem II to the Cyt b(f complex hence, these two components can be separated by greater distances than are the Cyt b f complex and Photosystem I. 

To analyze human behavior, blood samples are analyzed to determine alcohol or drug use or other medical conditions that would affect a driver s reaction time. Logbooks are analyzed to determine whether a commercial driver was sleep deprived. Impact injuries are examined to determine each occupant s position in the car, whether a seat belt was worn, and whether an air bag deployed. Evidence is examined to determine whether drivers accelerated, braked, or steered to avoid the accident. One fundamental outcome of this analysis is to calculate time-distance relationships and determine the point of perception for unimpaired and impaired drivers. 

As the acoustic pulse penetrates the sample, return echoes are produced from changes in material properties or changes In materials (i.e., interfaces) at specific depths within the sample. The echo return times are based on the thickness and acoustic velocity of the respective material(s). A digital oscilloscope displays these echoes, known as the A-Scan, along with their relative time/distance relationships, amplitude, polarity and frequency infoimation (Figure 3). These time/distance relationships within the A-scan provide the basis for investigating features at specific levels within a part. 

Probably the least flexible of all methods with respect to the time-step and distance relationship is the method of characteristics (MOC). It requires the pipe lengths in a network to be adjusted to satisfy the condition of a common time interval, but provides an accurate solution of the differential equations. MOC has been successfully implemented by Goacher (G4), Streeter and associates (S6), and Masliyah and Shook (M5). More recently,... 

If this relationship is not familiar or obvious to you, just keep in mind that energy has the dimensions of force times distance, so that differentiating with respect to distance gives you a force. 

It is desirable to express the calibration data in functional form with a curve fit for real-time processing. Polynomial fits of various orders by the least-square method may be used in various regions of the data to represent the intensity-to-distance relationship with the following form ... 

The term dqljdt represents the overall rate of mass transfer for component / (at time t and distance averaged over a particle. This is governed by a mass transfer rate expression which may be thought of as a general functional relationship of the form... 

Seam correlations, measurements of rank and geologic history, interpretation of petroleum (qv) formation with coal deposits, prediction of coke properties, and detection of coal oxidation can be deterrnined from petrographic analysis. Constituents of seams can be observed over considerable distances, permitting the correlation of seam profiles in coal basins. Measurements of vitrinite reflectance within a seam permit mapping of variations in thermal and tectonic histories. Figure 2 indicates the relationship of vitrinite reflectance to maximum temperatures and effective heating time in the seam (11,15). 

If samples of two metals widr polished faces are placed in contact then it is clear that atomic transport must occur in both directions until finally an alloy can be formed which has a composition showing die relative numbers of gram-atoms in each section. It is vety unlikely that the diffusion coefficients, of A in B and of B in A, will be equal. Therefore there will be formation of an increasingly substantial vacancy concentration in the metal in which diffusion occurs more rapidly. In fact, if chemically inert marker wires were placed at the original interface, they would be found to move progressively in the direction of slowest diffusion widr a parabolic relationship between the displacement distance and time. 

Equation 18 defmes a parabolic relationship between filtrate volume and time. The expression is valid for any type of cake (i.e., compressible and incompressible). From a plot of V + C versus (t+Tq), the filtration process may be represented by a parabola with its apex at the origin as illustrated in Figure 5. Moving the axes to distances C and Tq provides the characteristic filtration curve for the system in terms of volume versus time. Because the parabola s apex is not located at the origin of this new system, it is clear why the filtration rate at the beginning of the process will have a finite value, which corresponds to actual practice. 

Mendeleev went to some length to distance himself from the use of numerical relationships such as Prout s relationship and the notion of triads. However, it is quite clear that many of his predictions of the properties of new elements involve the notion of triads. The triads he considered were sometimes vertical, or horizontal, or at times the combination of both vertical and horizontal triads. 

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