Mean penetration distance

This discussion of geometric effects ignored the attenuation of radiation by material through which the radiation must travel to reach the receptor. The number of particles, dN, penetrating material, equals the number of particles incident N times a small penetration distance, dx, divided by the mean free path length of the type of particle in the type of material (equation 8.3-8). Integrating gives the transmission coefficient for the radiation (equation 8.3-9). 

Consider Fig. 4.32, a graph of flame velocity 5L as a function of distance, for a wave inside a tube. In this case, the flame has entered the tube. The distance from the burner wall is called the penetration distance dv (half the quenching diameter dT). If iij is the mean velocity of the gas flow in the tube and the line labeled (7, is the graph of the velocity profile near the tube wall, the local flame velocity is not greater than the local gas velocity at any point therefore, any flame that finds itself inside the tube will then blow out of the tube. At a lower velocity u2, which is just tangent to the SL curve, a stable point is reached. Then u2 is the minimum mean velocity before flashback occurs. The line for the mean velocity % indicates a region where the flame speed is greater than the velocity in the tube represented by in this case,... 

The physical meaning of Equation (17) is simply this if the specific rate constant goes up, say nine times, because of an increase of temperature, the concentration profile must be steeper in order to diffuse in the extra amount of reactant gas. Consequently, the penetration into the carbon decreases. Obviously, equilibrium is reached when the concentration gradient increa.scs threefold, the penetration distance decreases threefold, and the over-all reaction rate (proportional to fc times penetration distance) increases threefold, where 3, in this example, is the square root of the factor of specific-rate-constant increase. For these conditions, the over-all reaction rate has increased threefold, but so has the diffusion gradient that is, equilibrium has been reached. 

The problem of the charge state of ions penetrating matter is one of the most relevant questions for studies on the interaction of ions with solids. It is known that after some penetration distance the ions reach a state of charge equilibrium determined by the competition between capture and loss processes [2,7]. As a result of this equiUbrium the ions acquire a mean ionization charge as well as a stationary distribution of charge states around q. 

A high absorption coefficient for a particular wavelength means that the light penetrates the polymer to a limited distance. The intensity falls exponentially with penetration distance x, according to... 

To find experimentally the free volume distribution, a variant of gel chromatography was used that allowed estimation of the dimensions of molecules dissolved in eluent from the retention time tR in gel In [126,127], inverse gel-permeation chromatography was used and the dimensions of free space or vacancies in the network were determined from the retention time of test molecules in the gel. These molecvdes were of various sizes up to monomeric ones. In such a way, the part of space between the cross-links in the network was found, available for permeation of molecules (macromolecules) with known mean-square distance between the chain ends, . The larger the macromolecule dimension, the lower the total fraction of vacancies available for their penetration. Thus the distribution of vacancy size can be estimated. The main problem in the apphcation of this method consists in the transition from the distribution obtained for swollen gel (the necessary condition for gel chromatography) to distribution for pure network without solvent. 

Diffusion coefficients in liquids are about ten thousand times slower than those in dilute gases. To see what this means, we again calculate the penetration distance which was the distance we found central to unsteady diffusion. As an example, consider benzene diffusing into cyclohexane with a diffusion coefficient of about 2 10" cm /sec. At time zero, we bring the benzene and cyclohexane into contact. After 1 second, the diffusion has penetrated 0.004 cm, compared with 0.3 cm for gases after 1 minute, the penetration is 0.03 cm, compared with 4 cm after 1 hour, it is 0.3 cm, compared with 30 cm. 

Xps is a surface sensitive technique as opposed to a bulk technique because electrons caimot travel very far in soHds without undergoing energy loss. Thus, even though the incident x-rays penetrate the sample up to relatively large depths, the depth from which the electron information is obtained is limited by the "escape depth" of the photoemitted electrons. This surface sensitivity of xps is quantitatively defined by the inelastic mean free path parameter which is given the symbol X. This parameter is defined to be the distance an electron travels before engaging in an interaction in which it experiences an energy loss. 

As indicated above, the penetration depth is on the order of a micrometer. That means that in ATR, absorption of infrared radiation mostly occurs within a distance 8 of the surface and ATR is not as surface sensitive as some other surface analysis techniques. However, ATR, like all forms of infrared spectroscopy, is very sensitive to functional groups and is a powerful technique for characterizing the surface regions of polymers. 

The high degree of X-ray polarisation in the electron orbit plane provides means of controlling both the signal/noise ratio and the penetration of the X-rays into the specimen. Depending on whether the incidence plane is chosen vertically or horizontally, sigma or pi polarisation may be selected. The strain sensitivity and the extinction distance can thus be varied while the normal photoelectric absorption conditions remain identical. 

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