Variation, infinitesimal

The infinitesimal variation of (jii ean be expressed in terms of its (small) eomponents along the other oeeupied (l)j and along the virtual (jim as follows ... 

Although intrinsic reaction coordinates like minima, maxima, and saddle points comprise geometrical or mathematical features of energy surfaces, considerable care must be exercised not to attribute chemical or physical significance to them. Real molecules have more than infinitesimal kinetic energy, and will not follow the intrinsic reaction path. Nevertheless, the intrinsic reaction coordinate provides a convenient description of the progress of a reaction, and also plays a central role in the calculation of reaction rates by variational state theory and reaction path Hamiltonians. 

The method gives directly the stability in the large instead of the infinitesimal stability yielded by the variational equations. 

The variation of specific heat with temperature was discovered by Dulong and Petit in 1819. It explains why so many different heat units exist (cf. 5), and requires the definition of specific heat to be so framed as to allow for this variation. For this purpose we replace the finite changes by infinitesimal ones. If SQ units of heat are absorbed when unit mass of a substance is raised in temperature from 6— SO) to 0- - SO) underspecified conditions, the true specific heat at the temperature 0 is ... 

The brute force method depends on a systematic variation of all involved coefficients over a reasonable parameter space. The combination yielding the lowest goodness-of-fit measure is picked as the center for a further round with a finer raster of coefficient variation. This sequence of events is repeated until further refinement will only infinitesimally improve the goodness-of-fit measure. This approach can be very time-consuming and produce reams of paper, but if carefully implemented, the global minimum will not be missed, cf. Figures 3.4 and 4.4. 

If the errors in a, b, c are not too large, a linearized error approximation (see later discussion) can then be made by expanding the squares and cross-products of the infinitesimally small derivatives to variances and covariances. Using to indicate the coefficient of variation, Cx/x, we obtain... 

For a problem for which we cannot obtain an analytical solution, you need to determine sensitivities numerically. You compute (1) the cost for the base case, that is, for a specified value of a parameter (2) change each parameter separately (one at a time) by some arbitrarily small value, such as plus 1 percent or 10 percent, and then calculate the new cost. You might repeat the procedure for minus 1 percent or 10 percent. The variation of the parameter, of course, can be made arbitrarily small to approximate a differential however, when the change approaches an infinitesimal value, the numerical error engendered may confound the calculations. 

The volumetric flow rate is determined by writing the equation for the volumetric flow rate across an infinitesimal element of the flow area then integrating the equation over the whole flow area, ie the cross-sectional area of the pipe. It is necessary to use an infinitesimal element of the flow area because the velocity varies over the cross section. Over the infinitesimal area, the velocity may be taken as uniform, and the variation with r is accommodated in the integration. 

The utility of the Fukui function for predicting chemical reactivity can also be described using the variational principle for the Fukui function [61,62], The Fukui function from the above discussion, /v (r), represents the best way to add an infinitesimal fraction of an electron to a system in the sense that the electron density pv/v(r) I has lower energy than any other N I -electron density... 

The second term of Eq. (40) gives the contribution from collisions. These are non-instantaneous processes since the variation of p 0> at the time t depends on the value of this function at the earlier instant t. The evolution is non-Markovian and the system remembers its earlier history. However, this memory extends only over a finite period, as one can see from the expression (44) for the kernel G, (t). This results from supposing that the poles z( are not infinitesimally close to the real axis and thus that the collision time tc is finite (see Eq. (39)). 

Consider a trajectory in the R" n-dimensional space of x t) and a nearby trajectory x t) + 6x t), where the symbol 6 means an infinitesimal variation, i.e. an arbitrary infinitesimal change not tangent to the initial trajectory. Eq.(55) can be linearized throughout the trajectory to obtain... 

We first derive a relation for total mass conservation. Consider an arbitrary volume V enclosed in a surface Q. The mass inside the volume is JpdV, where p is density (in kg/m ) and dt is an infinitesimal volume in the volume V. The time derivative of the mass in the volume (i.e., the rate of the variation of the mass with time) is... 

EXAMPLE 2.5 Unsteady State Variation of Concentration Profiles Due to Diffusion Gaussian Distribution. By consulting tables of the normal distribution function, draw curves that show the broadening of a band of material with time if the substance is initially at concentration c0 and in a plug of infinitesimal thickness at x = 0. Assume that the diffusion coefficient has the value 5 10 11 m2 s for this material. Use t = 106 and t = 3 106 s to see how the concentration profile changes with time. 

Relation to the variational method. As will be seen from our treatment of p.m. on the basis of the variational method, p.m. is in substance only a special case of the variational method. Moreover it is rather of a formal character because it regards the parameter k as infinitesimal, while in practical problems k is always finite if small. From this point of view, p.m. is only a convenient method for supplying ap-... 

Consider a material or system that is not at equilibrium. Its extensive state variables (total entropy number of moles of chemical component, i total magnetization volume etc.) will change consistent with the second law of thermodynamics (i.e., with an increase of entropy of all affected systems). At equilibrium, the values of the intensive variables are specified for instance, if a chemical component is free to move from one part of the material to another and there are no barriers to diffusion, the chemical potential, q., for each chemical component, i, must be uniform throughout the entire material.2 So one way that a material can be out of equilibrium is if there are spatial variations in the chemical potential fii(x,y,z). However, a chemical potential of a component is the amount of reversible work needed to add an infinitesimal amount of that component to a system at equilibrium. Can a chemical potential be defined when the system is not at equilibrium This cannot be done rigorously, but based on decades of development of kinetic models for processes, it is useful to extend the concept of the chemical potential to systems close to, but not at, equilibrium. 

Discontinuous transformations In this type of transformation, there is a free-energy barrier to infinitesimal variations and the system is initially metastable. However, a sufficiently large variation can cause the free energy to decrease. The transformation therefore can be initiated at a finite rate only by a variation that is large in degree but small in extent (i.e., nucleation is required). Examples include the formation of B-rich precipitates from a supersaturated A-B solution. 

Consider a two-phase system of fixed total volume, with constant T and p (an open system with respect to matter flow), as illustrated in Fig. C.5. Under these conditions, the function Cl = E — TS — piNi — P2N2 is the appropriate thermodynamic potential. For any small variation at equilibrium, such as an infinitesimal variation... 

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