Mean square concentration

According to the thermodynamic theory of fluctuations, the mean-square concentration fluctuation is given - by... 

The mean square concentration is given by Eq. (2.10), which can be rewritten as... 

This method examines the degree of mixing of two fluids on the molecular scale by observing the rate of a second-order reaction in the mixture. The method gives the mean square concentration of component A in a nonhomo-geneous mixture of A and B and the actual distribution of deviations from the mean concentration in the mixture. 

Use the data in Table 4.3 to calculate the zero wavevector hmit of the scattering function 5 (0) and the mean-square concentration fluctuation S4>) ) at a 50 A scale (at q = 2nj50 = 0.126 A ) assuming the Ornstein-... 

When approaching the critical points in a binary system (generally, approaching the spinodal), dpijdx —> 0. From Equation 1.4-33, the mean square concentration fluctuation ((Axq)) —> 00. In terms of the order parameter, it means > 00 in accordance with ik e"" and (- ) ". ... 

FIGURE 25.2 A comparison of centerline mean-square concentrations measured in a heated plume in grid turbulence, 0, with values calculated, x, for k = 0 [see Eq. (25.18)]. (Source Moseley, 1991)... 

The mean-squared concentration fluctuation 8c can be calculated using the expression relating the fluctuation of c to the free energy ... 

Polymer chains at low concentrations in good solvents adopt more expanded confonnations tlian ideal Gaussian chains because of tire excluded-volume effects. A suitable description of expanded chains in a good solvent is provided by tire self-avoiding random walk model. Flory 1151 showed, using a mean field approximation, that tire root mean square of tire end-to-end distance of an expanded chain scales as... 

On average, the reflectivity decays in proportion to 0 or since both interfaces are sharp. However, if either surfaces is rough, then marked deviations are seen and the reflectivity is damped by a factor of exp -2k o, where a is the root-mean-square roughness. Thus, the reflectivity is very sensitive to surface roughness and to concentration gradients at interfaces. 

A similar treatment applies for the unstable regime of the phase diagram (v / < v /sp), where the mixture decays via spinodal decomposition.For the linearized theory of spinodal decomposition to hold, we must require that the mean square amplitude of the growing concentration waves is small in comparison with the distance from the spinodal curve. 

The average value of the asymmetrical fluctuation is defined as the root mean square (rms) value. Using the Rayleigh theorem,94 the average value of the surface concentration fluctuation, for example, can be written as... 

Atmospheric turbulence is a dynamic process and the wavefront aberrations are constantly changing. A characteristic timescale may be defined as the time over which the mean square change in wavefront error is less than 1 rad. If the turbulence were concentrated in a single layer with Fried parameter ro moving with a horizontal speed of v ms then the characteristic time, ro, is given by... 

Some alternative method had to be devised to quantify the TCDD measurements. The problem was solved with the observation, illustrated in Figure 9, that the response to TCDD is linear over a wide concentration range as long as the size and nature of the sample matrix remain the same. Thus, it is possible to divide a sample into two equal portions, run one, then add an appropriate known amount of TCDD to the other, run it, and by simply noting the increase in area caused by the added TCDD to calculate the amount of TCDD present in the first portion. Figure 9 illustrates the reproducibility of the system. Each point was obtained from four or five independent analyses with an error (root mean square) of 5-10%, as indicated by the error flags, which is acceptable for the present purposes. 

Gaussian shape the mean-squared displacement of spins or molecules follows the same Gaussian evolution as the concentration of a pointlike source. It has the shape... 

NMR Self-Diffusion of Desmopressin. The NMR-diffusion technique (3,10) offers a convenient way to measure the translational self-diffusion coefficient of molecules in solution and in isotropic liquid crystalline phases. The technique is nonperturbing, in that it does not require the addition of foreign probe molecules or the creation of a concentration-gradient in the sample it is direct in that it does not involve any model dependent assumptions. Obstruction by objects much smaller than the molecular root-mean-square displacement during A (approx 1 pm), lead to a reduced apparent diffusion coefficient in equation (1) (10). Thus, the NMR-diffusion technique offers a fruitful way to study molecular interactions in liquids (11) and the phase structure of liquid crystalline phases (11,12). 

Diffusion provides an effective basis for net migration of solute molecules over the short distances encountered at cellular and subcellular levels. Since the diffu-sional flux is linearly related to the solute concentration gradient across a transport barrier [Eq. (5)], a mean diffusion time constant (reciprocal first-order rate constant) can be obtained as the ratio of the mean squared migration distance (L) to the effective diffusivity in the transport region of interest. 

The ultrasound-assisted experiment of Figure 2(a) is again not typical in that the reagent concentrations have an inflection point mid-way through the reaction. We have performed a kinetic analyses of the stirred (blank) data in Figure 2(b) and found the following equations reproduce the data well with a root-mean-square error of 2.8% ... 

Here the square brackets indicate the concentration of the chemical species within the bracket. That is, [A] means the concentration of A, and so forth. [A]" means the concentration of A raised to the a power, where a is the value of the coefficient of A in the balanced equation for the chemical equilibrium. The value of the ratio of concentration terms is symbolized by the letter K, called the equilibrium constant. For example, for the reaction of nitrogen and hydrogen referred to in Sec. 19.3,... 

Here c is the polymer concentration by weight. < the density of the polymer, a an effective bond length or measure of the coil dimensions, and to the monomeric friction factor. The subscript zero indicates the pure polymer. Since 2 (H), the mean-square end-to-end chain separation, the viscosity will be directly proportional to the polymer concentration unless the plasticizer modifies the coil swelling. At high molecular weight the monomeric friction factor is increased by the factor (MIMf)" and M, is increased relative to the undiluted polymer [equation (55)]. Thus... 

Bueche (16,172) proposed that the viscosity is proportional to the fourth power of the polymer concentration and a complex function of the free volume of the mixture. Kraus and Gruver (170) find that the 3.4 power fits experimental data better than does the fourth power. They used equation (58) with (r2) replaced by the mean-square radius of gyration (s2). The term r2)/(rf) indicates that poor solvents should lower the viscosity more than a good solvent. As the temperature increases, the factor increases as a function of the ratio (T - 7 (tJJ)/(7 - 7 ). The glass transition temperatures of the polymer and diluent are andT o, respectively. 

It is of considerable practical importance to have some idea of how far an atom or ion will diffuse into a solid during a diffusion experiment. An approximate estimate of the depth to which diffusion is significant is given by the penetration depth, xp, which is the depth where an appreciable change in the concentration of the tracer can be said to have occurred after a diffusion time t. A reasonable estimate can be given with respect to the root mean square displacement of the diffusing... 

From a well-known result of calculus, the definite integral on the right-hand side is s/n so M is just equal to the quantity of diffusing substance. The present solution is therefore applicable to the case where M grams (or moles) per unit surface is deposited on the plane x=x at t=0. In terms of concentration, the initial distribution is an impulse function (point source) centered at x=x which evolves with time towards a gaussian distribution with standard deviation JlQit (Figure 8. 13). Since the standard deviation is the square-root of the second moment, it is often stated that the mean squared distance traveled by the diffusion species is 22t. 

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