Laplace coefficients

This book contains tables of the properties of water and steam from 0 to 800 and from 0 to 1000 bar which have been calculated using a set of equations accepted by the members of the Sixth International Conference on the Properties of Steam in 1967. Properties which are tabulated include the pressure, specific volume, density, specific enthalpy, specific heat of evaporation, specific entropy, specific isobaric heat capacity, dynamic viscosity, thermal conductivity, the Prandtl number, the ion-product of water, the dielectric constant, the isentropic exponent, the surface tension and Laplace coefficient. Also see items [43] and [70]. 

The implications of this decay behavior for the Coulomb-type products are illustrated in Figure 19 for the example of linear alkanes. For an alkane with four to five carbon atoms, the exact number of required transformed products (MP2, 6-31G, providing an accuracy of 0.1 mHartree for the first Laplace coefficient) scales already as low as approaching the asymptotic linear scaling. " However, the pseudo-Schwarz screening drastically overestimates the number of required products and no linear-scaling can be achieved using this criterion, because the distance dependence of the transformed products is not accounted for. 

Figure 19 Comparison of the number of significant Coulomb-type integral products (CnHin+i/b-dlG basis in units of 10 ) as estimated over shells by Schwarz-type screening (QQZZ 10 ) and MBIE (10 ) with the exact number of products selected via basis functions. For the latter, a threshold of has been selected to provide comparable accuracy in the absolute energies of 0.1 mH (only data for the first Laplace coefficient in computing the MP2 energy is listed).

In this expression, z is the distance from the surface into the sample, a(z) is the absorption coefficient, and S, the depth of penetration, is given by Eq. 2. A depth profile can be obtained for a given functional group by determining a(z), which is the inverse Laplace transform of A(S), for an absorption band characteristic of that functional group. 

With the Laplace operator V. The diffusion coefficient defined in Eq. (62) has the dimension [cm /s]. (For correct derivation of the Fokker-Planck equation see [89].) If atoms are initially placed at one side of the box, they spread as ( x ) t, which follows from (62) or from (63). 

Linear differential equations with constant coefficients can be solved by a mathematical technique called the Laplace transformation . Systems of zero-order or first-order reactions give rise to differential rate equations of this type, and the Laplaee transformation often provides a simple solution. 

The preceding two equations are examples of linear differential equations with constant coefficients and their solutions are often found most simply by the use of Laplace transforms [1]. 

Laplace had previously deduced from his theory that the temperature coefficient of surface tension should stand in a constant ratio to the coefficient of expansion this is in many cases verified, and shows that the effect of temperature is largely to be referred to the change of density (Cantor, 1892). 

Equation (15.34) is the system model. It is a linear PDE with constant coefficients and can be converted to an ODE by Laplace transformation. Define... 

In this section we reveal some properties of difference operators approximating the Laplace operator in a rectangle and derive several estimates for difference approximations to elliptic second-order operators with variable coefficients and mixed derivatives. 

In order to solve Laplace s equation we have to find also expressions for the metric coefficients. By definition, the elementary displacements along coordinate lines are... 

In this section, we will outline only those properties of the Laplace transform that are directly relevant to the solution of systems of linear differential equations with constant coefficients. A more extensive coverage can be found, for example, in the text book by Franklin [6]. 

The inverse transform Xp(t) in the time domain can be obtained by means of the method of indeterminate coefficients, which was presented above in Section 39.1.6. In this case the solution is the same as the one which was derived by conventional methods in Section 39.1.2 (eq. (39.16)). The solution of the two-compartment model in the Laplace domain (eq. (39.77)) can now be used in the analysis of more complex systems, as will be shown below. 

The inverse Laplace transform can be obtained again by means of the method of indeterminate coefficients. In this case the coefficients A, B and C must be solved by equating the corresponding terms in the numerators of the left- and right-hand parts of the expression ... 

The initial conditions are CD = CD(0) at t = 0 and CR = 0 at t = 0. Efforts to obtain analytical solutions are tedious and unnecessary. By applying the change in concentrations (or mass) in the donor and receiver solutions with time to the Laplace transforms of Eqs. (140) and (141), the inverse of the simultaneous transformed equations can be numerically calculated with appropriate software for best estimates of a, (3, and y. It is implicit here that P Pap, Pbh and Ke are functions of protein binding. Upon application of the transmonolayer flux model to the PNU-78,517 data in Figure 32, the effective permeability coefficients from the disappearance and appearance kinetics points of view are in good quantitative agreement with the permeability coefficients determined from independent studies involving uptake kinetics by MDCK cell monolayers cultured on a flat dish... 

An important technical development of the PFG and STD experiments was introduced at the beginning of the 1990s the Diffusion Ordered Spectroscopy, that is DOSY.69 70 It provides a convenient way of displaying the molecular self-diffusion information in a bi-dimensional array, with the NMR spectrum in one dimension and the self-diffusion coefficient in the other. While the chemical-shift information is obtained by Fast Fourier Transformation (FFT) of the time domain data, the diffusion information is obtained by an Inverse Laplace Transformation (ILT) of the signal decay data. The goal of DOSY experiment is to separate species spectroscopically (not physically) present in a mixture of compounds for this reason, DOSY is also known as "NMR chromatography."... 

Sometimes an equation out of this classification can be altered to fit by change of variable. The equations with separable variables are solved with a table of integrals or by numerical means. Higher order linear equations with constant coefficients are solvable with the aid of Laplace Transforms. Some complex equations may be solvable by series expansions or in terms of higher functions, for instance the Bessel equation encountered in problem P7.02.07, or the equations of problem P2.02.17. In most cases a numerical solution Is possible. 

The dynamic performance of a system can be deduced by merely observing the location of the roots of the system characteristic equation in the s plane. The time-domain specifications of time constants and damping coefficients for a closedloop system can be used directly in the Laplace domain. 

Equation (9.15) was written for macro-pore diffusion. Recognize that the diffusion of sorbates in the zeoHte crystals has a similar or even identical form. The substitution of an appropriate diffusion model can be made at either the macropore, the micro-pore or at both scales. The analytical solutions that can be derived can become so complex that they yield Httle understanding of the underlying phenomena. In a seminal work that sought to bridge the gap between tractabUity and clarity, the work of Haynes and Sarma [10] stands out They took the approach of formulating the equations of continuity for the column, the macro-pores of the sorbent and the specific sorption sites in the sorbent. Each formulation was a pde with its appropriate initial and boundary conditions. They used the method of moments to derive the contributions of the three distinct mass transfer mechanisms to the overall mass transfer coefficient. The method of moments employs the solutions to all relevant pde s by use of a Laplace transform. While the solutions in Laplace domain are actually easy to obtain, those same solutions cannot be readily inverted to obtain a complete description of the system. The moments of the solutions in the Laplace domain can however be derived with relative ease. 

P(Qol, t) is the conditional probability of the orientation being at time t, provided it was Qq a t time zero. The symbol — F is the rotational diffusion operator. In the simplest possible case, F then takes the form of the Laplace operator, acting on the Euler angles ( ml) specifying the orientation of the molecule-fixed frame with respect to the laboratory frame, multiplied with a rotational diffusion coefficient. Dr. Equation (44) then becomes identical to the isotropic rotational diffusion equation. The rotational diffusion coefficient is simply related to the rotational correlation time introduced earlier, by tr = 1I6Dr. 

Matrix D has I s along the diagonal, reflecting the use of a normalized discrete Laplace equation with a = —k /[2 h + 1 )], and B is a multiple of the identity matrix with j8 = —h /[2(h -I-1 )]. Matrix A displays a sparse block structure whose off-diagonal coefficients must be less than 1 to converge to a solution. 

Relaxation methods involve iteratively seeking a convergent solution to the Laplace equation. In the present case, for instance, if we rewrite the coefficient matrix A = I + E, where the latter matrix consists of elements that are all small compared to 1, the matrix Laplace equation takes the form = EU + b. One begins the calculation with values U = b [or, equivalently, U = 0] and iteratively computes successive values The calculation terminates when a specified limit of accuracy is achieved. One such measure involves calculating the proportional differences ... 

Using the rule developed in Appendix 2, the conversion we would expect from this system for a first-order reaction with a rate coefficient of 2.723 min is obtained by setting the Laplace variable, s, equal to this value in eqn. (49). Thus... 

In Example 4, we calculated the conversion from the RTD directly this was for a first-order reaction with rate coefficient 2.617 x 10 s". Converting this value to min and setting the Laplace variable equal to this value... 

The derivation of the electrostatic properties from the multipole coefficients given below follows the method of Su and Coppens (1992). It employs the Fourier convolution theorem used by Epstein and Swanton (1982) to evaluate the electric field gradient at the atomic nuclei. A direct-space method based on the Laplace expansion of 1/ RP — r has been described by Bentley (1981). 

The statement cA = c0/ (1 + K) in Eqs. (157a and b) above is tantamount to saying that cA + cB = Co, where c0 is the total concentration of both species of the dissolved solute. If the diffusivities SDA8 and DBs are assumed to be equal, then cB can be eliminated from Eqs. (155) and (156) and a fourth-order, linear partial-differential equation is obtained. The solution of this equation consistent with the conditions in Eq. (157) is obtainable by Laplace transform techniques (S9). Sherwood and Pigford discuss the results in terms of the behavior of the liquid-film mass transfer coefficient. 

Bamford and Tompa (93) considered the effects of branching on MWD in batch polymerizations, using Laplace Transforms to obtain analytical solutions in terms of modified Bessel functions of the first kind for a reaction scheme restricted to termination by disproportionation and mono-radicals. They also used another procedure which was to set up equations for the moments of the distribution that could be solved numerically the MWD was approximated as a sum of a number of Laguerre functions, the coefficients of which could be obtained from the moments. In some cases as many as 10 moments had to be computed in order to obtain a satisfactory representation of the MWD. The assumption that the distribution function decreases exponentially for large DP is built into this method this would not be true of the Beasley distribution (7.3), for instance. 

The analytical expression, Eq. 18-1, is not easy to evaluate for large values of n. Fortunately, the French mathematicians, DeMoivre and Laplace, found that with increasing n, the Bernoulli coefficients converge to the function ... 

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