Differential distribution

If the differential distribution function is exponential in Q (Section XVII-14A), the resulting Q(P, T) is that known as the Freundlich isotherm... 

Chromatography is often used with advantage for the purification of small amounts of complex organic mixtures. Chromatography techniques all rely on the differential distribution of the various components in a mixture between the mobile phase and the stationary phase. The mobile phase can either be a gas or a liquid whereas the stationary phase can either be a solid or a liquid. 

The modal size is at the point of maximum frequeney on the differential distribution. The median size is at the 50 per eent point on the eumulative plot. Note that the modal size of a distribution is not neeessarily equal to the median. 

This result gives the flux for a small portion of the surface that has been exposed for exactly t seconds. Other portions of the surface will have been exposed for different times and thus will have different instantaneous fluxes. To And the average flux, we need the differential distribution of exposure times,/(f). Danckwerts assumed an exponential distribution ... 

The average size of the surviving particles is obtained by weighting R(t) by the differential distribution function and integrating over the range of possible times ... 

Impulse Response and the Differential Distribution. Suppose a small amount of tracer is instantaneously injected at time 1 = 0 into the inlet of a reactor. All the tracer molecules enter together but leave at varying times. The tracer concentration at the outlet is measured and integrated with respect to time. The integral will be finite and proportional to the total quantity of tracer that was injected. The concentration measurement at the reactor outlet is normalized by this integral to obtain the impulse response function. ... 

It is normally called the differential distribution function (of residence times). It is also known as the density function or frequency function. It is the analog for a continuous variable (e.g., residence time i) of the probabiUty distribution for a discrete variable (e.g., chain length /). The fraction that appears in Equations (15.2), (15.3), and (15.6) can be interpreted as a probability, but now it is the probability that t will fall within a specified range rather than the probability that t will have some specific value. Compare Equations (13.8) and (15.5). 

The differential distribution is related to the cumulative distribution and to the washout function by... 

A CSTR has an exponential distribution of residence times. The corresponding differential distribution can be found from Equation (15.7) ... 

The differential distribution is the response to a unit impulse. Setting A = gives the expected result. Equation (15.19). 

Example 15.4 The differential distribution can be defined as the outlet response of a system to a delta function input. 

Equation (8.11) gave the differential distribution function that corresponds to a parabolic velocity prohle in a tube. This specihc result is now derived in a more general way. 

Differentiating this result gives the differential distribution found in Equation (8.11). The washout function is... 

Thus, the fraction unreacted is the Laplace transform with respect to the transform parameter k of the differential distribution function. 

Given k fit) for nny reactor, you automatically have an expression for the fraction unreacted for a first-order reaction with rate constant k. Alternatively, given ttoutik), you also know the Laplace transform of the differential distribution of residence time (e.g., k[f(t)] = exp(—t/t) for a PER). This fact resolves what was long a mystery in chemical engineering science. What is f i) for an open system governed by the axial dispersion model Chapter 9 shows that the conversion in an open system is identical to that of a closed system. Thus, the residence time distributions must be the same. It cannot be directly measured in an open system because time spent outside the system boundaries does not count as residence but does affect the tracer measurements. 

Glycine is the simplest of all amino acids. It is involved in many metabolic pathways, is an essential component of proteins, and is found throughout the brain. A neurotransmitter role for glycine was first identified in the spinal cord, where it was found to be differentially distributed between dorsal and ventral regions and shown to cause hyperpolarisation of motoneurons (Werman et al. 1967). This inhibitory action of glycine is distinct from its... 

Probable errors in assigning the integral distribution curve, as indicated by scatter of the points in Fig. 57, are magnified in the process of taking the slope for the deduction of the differential distribution. Only the approximate location of the maximum and breadth of the latter are experimentally significant. 

Blue, M.E. Kosofsky, B.E. and Molliver, M.E. Regional differences in the serotonin innervation of rodent cerebral cortex Differential distribution of two morphologically distinct axon types. Abstr Soc Neurosci 14 209,... 

Krishna N.S., Getchell T.V. and Getchell M.L. (1992). Differential distribution of gamma-glutamyl cycle molecules in the vomeronasal organ of rats. Neurorep 3, 551-554. 

At a more molecular level, the influences of the composition of the membrane domains, which are characteristic of a polarized cell, on diffusion are not specifically defined. These compositional effects include the differential distribution of molecular charges in the membrane domains and between the leaflets of the membrane lipid bilayer (Fig. 3). The membrane domains often have physical differences in surface area, especially in the surface area that is accessible for participation in transport. For example, the surface area in some cells is increased by the presence of membrane folds such as microvilli (see Figs. 2 and 6). The membrane domains also have differences in metabolic selectivity and capacity as well as in active transport due to the asymmetrical distribution of receptors and transporters. 

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