Infinite volume

Two applications of the flucUiathig diffusion equation are made here to illustrate tlie additional infonnation the flucUiations provide over and beyond the detenninistic behaviour. Consider an infinite volume with an initial concentration, c, that is constant, Cq, everywhere. The solution to the averaged diffusion equation is then simply (c) = Cq for all t. However, the two-time correlation fiinction may be shown [26] to be... 

This covariance ftmetion vanishes as t-5 approaches because the initial density profile has a finite integral, that creates a vanishing density when it spreads out over the infinite volume. 

A substance is in the ideal gas state when the volume of its molecules is a zero fraction of the total volume taken up by the substance and when the individual molecules are far enough apart from each other so that there is no interaction between them. Although this only occurs at infinite volume and zero pressure, in practice, ideal gas properties can be used for gases up to a pressure of two atmospheres with little loss of accuracy. Thermal properties of ideal gas mixtures may be obtained by mole-fraction averaging the pure component values. 

Several boundary conditions have been used to prescribe the outer limit of an individual rhizosphere, (/ = / /,). For low root densities, it has been assumed that each rhizosphere extends over an infinite volume of. soil in the model //, is. set sufficiently large that the soil concentration at r, is never altered by the activity in the rhizosphere. The majority of models assume that the outer limit is approximated by a fixed value that is calculated as a function of the maximum root density found in the simulation, under the assumption that the roots are uniformly distributed in the soil volume. Each root can then extract nutrients only from this finite. soil cylinder. Hoffland (31) recognized that the outer limit would vary as more roots were formed within the simulated soil volume and periodically recalculated / /, from the current root density. This recalculation thus resulted in existing roots having a reduced //,. New roots were assumed to be formed in soil with an initial solute concentration equal to the average concentration present in the cylindrical shells stripped away from the existing roots. The effective boundary equation for all such assumptions is the same ... 

With these expansions, Eq. (259) gives, in the limit of an infinite volume,... 

To investigate spontaneous symmetry breaking, one ordinarily has to start at finite volume and insert a source which explicitly breaks the symmetry. The source is removed only after the infinite volume limit is taken. We stress that the source does not have to be a quark mass (it could be a higher dimension operator), so one can investigate symmetry breaking even when the quark mass is exactly zero throughout the calculation. (To be precise, a quark mass does not explicitly violate vector symmetries, so it cannot play the role of the source in the thermodynamic limit needed here.)... 

We have considered so far the energy transfer from a donor to a single acceptor. Extension to ensembles of donor and acceptor molecules distributed at random in an infinite volume will now be considered, paying special attention to the viscosity of the medium. Then, the effect of dimensionality and restricted geometry will be examined. Homotransfer among the donors or among the acceptors will be assumed to be negligible. 

For an ensemble of donor and acceptor molecules distributed at random in an infinite volume, it is easy to calculate the sum of the rate constants for transfer from donor to all acceptors because all donors of this ensemble are identical in the rapid diffusion limit ... 

Equation (9.29) for Forster kinetics is valid for randomly distributed acceptors in an infinite volume, i.e. in three dimensions. If the dimension is not 3, but 1, or 2, Eq. (9.29) must be rewritten in a more general form ... 

For assemblies of like chromophores in three dimensions in an infinite volume, many theories have provided various expressions of the survival probability Gs(t). Only one of them will be given here, owing to its simplicity and good accuracy. Huber (1981) obtained the following relationship ... 

A common feature of widely used apparatus like the paddle or basket method is their limited volume. Typical volumes used in these systems range from about 500 to 4000 mL, limiting their use for very poorly soluble substances. Theoretically at least, open systems may be operated with infinite volumes to complete the dissolution of even very poorly soluble com-... 

If, in an infinite plane flame, the flame is regarded as stationary and a particular flow tube of gas is considered, the area of the flame enclosed by the tube does not depend on how the term flame surface or wave surface in which the area is measured is defined. The areas of all parallel surfaces are the same, whatever property (particularly temperature) is chosen to define the surface and these areas are all equal to each other and to that of the inner surface of the luminous part of the flame. The definition is more difficult in any other geometric system. Consider, for example, an experiment in which gas is supplied at the center of a sphere and flows radially outward in a laminar manner to a stationary spherical flame. The inward movement of the flame is balanced by the outward flow of gas. The experiment takes place in an infinite volume at constant pressure. The area of the surface of the wave will depend on where the surface is located. The area of the sphere for which T = 500°C will be less than that of one for which T = 1500°C. So if the burning velocity is defined as the volume of unbumed gas consumed per second divided by the surface area of the flame, the result obtained will depend on the particular surface selected. The only quantity that does remain constant in this system is the product u,fj,An where ur is the velocity of flow at the radius r, where the surface area is An and the gas density is (>,. This product equals mr, the mass flowing through the layer at r per unit time, and must be constant for all values of r. Thus, u, varies with r the distance from the center in the manner shown in Fig. 4.14. 

When we make a plot of Z versus 1/V, we expect that the value of Z = 1 when 1/y = 0 or at infinite volume. The first virial coefficient B T) is the slope of the line at 1/y = 0. B T) is usually negative at very low temperatures due to the attractive forces. But when the temperature increases, B T) will turn positive, and the temperature at which B becomes zero is called the Boyle temperature, since that is the temperature where Boyle s law applies exactly (at infinite volume). 

These questions appear to be understandable in terms of both photon models. The wavepacket axisymmetric model has, however, an advantage of being more reconcilable with the dot-shaped marks finally formed by an individual photon impact on the screen of an interference experiment. If the photon would have been a plane wave just before the impact, it would then have to convert itself during the flight into a wavepacket of small radial dimensions, and this becomes a less understandable behavior from a simple physical point of view. Then it is also difficult to conceive how a single photon with angular momentum (spin) could be a plane wave, without spin and with the energy hv spread over an infinite volume. Moreover, with the plane-wave concept, each individual photon would be expected to create a continuous but weak interference pattern that is spread all over the screen, and not a pattern of dot-shaped impacts. 

Equation 13.22 holds for the quasi-steady rate at which B atoms diffuse to a spherical precipitate in a distribution of similar precipitates in a dilute supersaturated solution. Show that an equation of the same form holds for the quasi-steady rate of diffusion to a single spherical precipitate of constant radius R embedded in an infinite volume of a similar solution. [Note that a similar result is also obtained in Section 20.2.1 (Eq. 20.47) for the more realistic case where the particle is allowed to grow as the diffusion occurs.]... 

Rate laws for batch and flow conditions to describe FD and PD phenomena were developed by Boyd et al. (1947) and are given below. For PD with infinite volume (flow),... 

What will happen if a piece of Zn metal is dropped into an essentially infinite volume of distilled water Will all of the Zn dissolve If not, why Explain from a macro and a micro perspective. 

If the field is in an infinite volume, then the sum in Eq. (1.26) is replaced by an integral. It follows from Eq. (1.4) that the electric field (which in the absence of particles has only the transverse component) and the magnetic field are given as... 

When considering a coherent pulse of light, it is necessary to superimpose a collection of plane waves, as in Eqs. (1.27) and (1.28). In doing so it is reasonable to make the simplifying assumption that all the modes of the pulse propagate in the same direction (chosen as the z axis) and that all the pulse modes have the same polarization direction a. We can therefore eliminate the integration over the k directions and write Eq. (1-27) (in an infinite volume) as... 

The problem is inherently a finite-size problem. Results that otherwise would be considered as finite-size effects and should be neglected are in this case essential. At the limit of infinite volume there will be no release at all. Bunde et al. [84] found a power law also for the case of trapping in a model with a trap in the middle of the system, i.e., a classical trapping problem. In such a case, which is different from the model examined here, it is meaningful to talk about finite-size effects. In contrast, release from the surface of an infinite medium is impossible. 

The range of the voidage change extends from that for the single particle in an infinite volume of fluid, e = 1, to the lower limit for the fixed bed, e0. At this point, Eq. (2.1) would still be valid, although it possesses a different physical meaning Instead of particles suspended by the fluid, this relation now signifies a fixed bed with pressure drop less than that required to balance the particles in the suspended state ... 

The Flory equation of state does not reduce to the ideal gas equation of state at zero pressure and infinite volume. Flory and his coworkers derived the equation of state specifically for liquid polymer solutions and were not concerned with the performance of the equation in the vapor phase. Poor vapor phase performance of an equation of state causes considerable difficulty, however, when one tries to apply the equation to higher pressure, higher temperature situations. The Chen et al. equation of state was developed in order to remedy this deficiency of the Flory equation of state. 

Here p and are, respectively, the mean value and the dispersion (variance) with respect to a population. These characteristics establish all the integral properties of the normal random variable that is represented in our example by the value expected for the species concentration in identical samples. It is not feasible to calculate the exact values of p and because it is impossible to analyse the population of an infinite volume according to a single property. It is important to say that p and show physical dimensions, which are determined by the physical dimension of the random variable associated to the population. The dimension of a normal distribution is frequently transposed to a dimensionless state by using a new random variable. In this case, the current value is given by relation (5.21). Relations (5.22) and (5.23) represent the distribution and repartition of this dimensionless random variable. Relation (5.22) shows that this new variable takes the numerical value of x when the mean value and the dispersion are, respectively, p = 0 and = 1. 

This is the infinite-volume limit of the corresponding sum obtained from Eq. (5.43). 

Note that excess properties are zero in the limit of infinite volume V, so that... 

Thus, by measuring the value of rj may be found. The constant velocity u is often called the terminal velocity. The formula holds only if ur is small compared with rjy i.e. for very viscous liquids in the case of spheres of moderate size there are also corrections for the boundary conditions of the walls and base of the cylinder containing the liquid (the formula (1) being deduced for an infinite volume of liquid). If the liquid column is divided into three equal parts, and the centre one is used in timing the fall of the sphere, the correcting factor on the velocity for the wall effect is ... 

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